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has t for */#c-value while K ...Kr' ...&\ where r', ...,r(r) are respectively p' or Np'', ...,p(v) or Np{v) but the set / , ...,rW is different from the set qf, ...,#(j;), h a s / for *JKC-value, on the same replacement. Now form the disjunction of all K ...Kq' ...q(v) for just those sets q', ...,q(v) for which PROP.
} there may have been several cancellations of (Eg) $'{£}. K . lid'
" be the conjunction of the following three /^-statements: {A%",..., gW) (A^') (EC, C) W, £2] = * if and only if £ is the place number of the ordered pair <£1? £2) m o u r ordering of all ordered pairs of natural numbers, then p',p", ...,p^K+2) clearly satisfy <j>". We now show that if
...,&Sd)
(e) if = then C^tp'... ftse)i/r is a theorem. Use of this metatheorem frequently enables us to shorten proofs, the full proof can be
2.5 Deductions
41
found from the meta-theorem. The condition will hold in a propositional calculus if the following condition: (f) if
' "*' = t h e n = ^ " '
for some primitive or defined symbol C
6' 6 of type ooo. For suppose (/) t h e n : if Y'"'*Y
t h e n Cfi(C...
(CftS0ty)...)
is a theorem. We can then define C^ by for C^'( Thus we obtain (e). In a propositional calculus in which (iv) below holds a necessary and sufficient condition that (/) hold is that the conditions (i), (ii), (iii) below should hold, (i) Ctfxfi be a theorem. (ii) Ccfrx be a theorem whenever x is an axiom. (iii) = . (iv) If —^—^
is a rule then
=
0
For suppose (i), (ii), (iii), (iv) and —J In the tree obtained by writing out
— in full replace each
premiss and conclusion x by C^se)x- The tops of the branches become C^^sd) o r CfjFVfi, ...tC^fi*, or Cft^x, where X is an axiom. In the first and last cases we add the proofs of the theorems given in (i) and (ii), in the other cases we add the deductions given in (iii). In the rest of the tree we replace use of a rule by the full proof of the derived rule 6',...,6<® 6 given in (iv). This gives us . Conversely suppose (/): we have = / whence Ccjxj) by (/), that is Ccjxj) is a theorem hence (i). Again we have = where x is a n axiom hence Ctfix by (/) and so (iii). Hence the result.
42
Ch. 2 Propositional calculi
2.6 The classical propositional calculus A pure propositional calculus is called classical if it is equivalent to the following propositional calculus, £PC. Symbols: Noo Dooo p0 ' ( )
negation symbol, disjunction symbol, variable, generating sign, left parenthesis, right parenthesis.
The parentheses can be omitted because we only use application, so that if a formula is well-formed the parentheses can be omitted because there is a unique method of replacing them so that the result is well-formed. This was shown in Ch. 1, lemma (iii). Axiom scheme. DnNn, where TT is a variable. Rules: Remodelling la DDDa)(f)}Jr(o' permutation. Building
n.
Jr.. dilution
m
M*??*" composition
ne
"*»
double negation
o), o)' are called subsidiary formulae and may be omitted, X is a secondary formula and must be present. 0, i/r are called the main formulae. If we omit a subsidiary formula then one occurrence of D is struck out. The premiss la written in full without omission of parentheses is ((D((D((DOJ) $)) ft)) 0)'), which is the only way of replacing the parentheses so that the result is well-formed. Notice that any symbol introduced into a ^ a -proof remains in that 0*c-proof from that place onwards. Hence any ^-theorem must contain the variables which occur in the axioms used in that proof. A formal system S£ is called direct if any symbol introduced into an o5f-proof remains in the J^-proof from that place onwards. Thus the system 2PC is direct.
2.7 Some properties of the remodelling and building schemes
43
2.7 Some properties of the remodelling and building schemes 1 Disjunction is communtative and associative. Commutativity of disjunction:
PROP.
jyTi
I& with both subsidiary formulae absent.
Associativity of disjunction:
Again
D<j)D\jrx
la with left subsidiary formula absent, la with right subsidiary formula absent, la with both subsidiary formulae absent.
—-—— la with both subsidiary formulae absent, ^-^- la with right subsidiary formula absent, LJLA \a with left subsidiary formula absent. Hence disjunction is associative. 2. The schemes la, 116, c are reversible. That is to say, if we have a ^ c -proof of the lower formula of one of these schemes then from that ^ c -proof we can obtain a ^ c -proof of the upper formula (upper formulae) by carrying out the procedure that we are about to describe. Thus in our notation PROP.
DNNcjxi)
The scheme l a is reversible. This is clear. The scheme 116 is reversible. Suppose we have a ^ c -proof of then we have a tree whose base is DNDfii/ro). Follow corresponding occurrences of ND(j)^jr up the tree. If ND
44
Ch. 2 Propositional calculi
there fails to be a corresponding occurrence of ND
3. We have the derived rule 16 77 , . * cancellation.
Suppose we have a ^ c -proof of Dtfxf). Using la repeatedly we obtain D... DO®, where O is a formula of the form i/r'n... ni/ASd\ parentheses are omitted as in la, each ^(A), 1 ^ A ^ S6 is either a variable or the negation of a variable or is NNx^x) where ^(A) is a ^-statement
2.7 Some properties of the remodelling and building schemes {X)
{X
(A)
45
A)
or is NDx to \ where ^ and o/ are ^-statements. By repeatedly using I a and the reversibility of II c, then I a repeatedly we may replace NNx(X) by #(A). (la is used to bring NNx{X) to the left.) By repeatedly using the reversibility of 116 then la repeatedly we may replace NDx^co^ by Nx(A) or NOJ^\ Continuing these three operations as long as possible (this amounts to moving occurrences of N as far to the right as possible) we obtain ^ o -proofs of formulae DD... DD... DHWW, 1 < /i ^ v, (1) where m^\ 1 < /u, < v is a formula formed from a sequence of variables and negated variables. The process must cease because at each of the last two steps we decrease the length of the formula dealt with, and applications of l a are limited. The statements (1) are ^-theorems and their ^-proofs proceed from ^ c -axioms using ^ c -rules la, I I a only, because (1) is without occurrences of ND or NN. Thus each T ( ^ must contain a variable and the same variable negated. Thus D.^DY^ is a £PCtheorem. By repeated use of II6, c, and of course I a, following the inverse order of the applications we made of their reversibilities, we obtain a ^ o -proof of
DC/HO
Di/rco
—*—• TT He —L— DNNcfxo DNNJ/TO) —rkV,r,
by definition of K.
4. (i) Rule 116' is reversible. (ii) Conjunction is commutative and associative. (iii) Disjunction is distributive with conjunction. (iv) Conjunction is distributive with disjunction. (i) From the reversibility of 116 we can obtain ^ o -proofs of DNN
PROP.
46
Ch. 2 Propositional calculi
(ii) We have
* by the reversibility of 116', hence by 116' we get
) , so conjunction is commutative. We have: to show
rz
, JL , * and
TZT.,
Y
,
*.
* by reversibility of 116' * ditto h
l l v
by 116' Hence conjunction is associative. We have to show
reversibility of 116
D
KDHX , DK
and
DKfaKfx ' KDjfx
'
We have reversibility of 116' Io>n6,
DKfx^DKfxx,
reversibility I J
ofII6
X Hence conjunction is distributive with disjunction. We use the notation (j) = ^ to denote that ^j-A * and ^ji^ *, for any #, we then say that
2.7 Some properties of the remodelling and building schemes
47
PROP. 5
1. D
idem1*. Kcjxj) 'potency commu- 2*. tativity
2. D 0 0 ' = D$'
associ-
3*.
4. BK^>4>'4>" = KD
r
r
_ 5*. NK66 = DN6N6 ganlaws 6. iV'iV^ = $5 double negation.
We have to show: rv. * and }/-> *> for any %, and any pairs (j) and ^ Air /
Air/
listed. x{$} is §S: 1 is 16 and II a, 1* is 116' and its reversibility, 2 and 2*, 3 and 3*, 4 and 4* have already been demonstrated. 5*, 6 come from l i e and its reversibility. And 5, if KN
•
la
48
Ch. 2 Propositional calculi
between this and the base of the ^-proof-tree replace KDcjxj)'^)" by DK
where
then
t
D 2. G for \pp'. DNpp'. Then G is of type ooo, it is called the conditional symbol; in our parenthesesless notation we can define G for DN. We have to show (i), (ii), (iii) and (iv). C expresses material implication. (i) DNCJHJ) is a ^ c -theorem, this is known as tertium non datur. We proceed by induction on the construction of
as desired. If (j) is a disjunction Di/r'i/r" and the result holds for xjr' and for \jr" then we have
DNft'DiJr'iJr"
TLa,
la
DNjr" Df'f"
thus DNcjxj) is a ^-theorem for any ^-statement
C(px
2.8 Deduction theorem
49
(iv) If ^ and %-%- are ^ o -rules then DN(j>x these follow at once from II a, l a and the rule in question by putting into the subsidiary formulae. Thus
la
using the rule y with N<j> in the subsidiary formula. The other cases follow similarly. Thus the deduction theorem holds in 3PC. 2.9 Modus Ponens The rule
X=
™,
DojX
where co is subsidiary and can be absent, x is secondary and must be present, is known as Modus Ponens, the rule of detachment or the cut. The formula $ is known as the cut formula. PROP.
7 Modus Ponens is a derived rule in 0>c.
We have to show: Do) may be absent but x must be present. We have to show how we can obtain a ^ o -proof of DOJX when we are given ^ c -proofs of Doxfi and DNfix- The demonstration is by formula induction on the cut formula
m Ia'IIaorc
50
Ch. 2 Propositional calculi
thus we have by our induction hypothesis: la, I I a or c, exactly as before.
Dcox
Similarly for a two premiss rule 116. If N<j> is in the main formula of the building rule immediately above DNcfrx in the ^ o -proof-tree of DN<j>x then this rule can only be II a since
DN
, or
^s J us ^ X'- Hence we obtain
as desired. (6) (j> is D$'$" and the result holds for ^' and $". We have ^ o -proofs of DcoDfi<})" and of DNDfrfi'x* From the reversibility of 116 we can obtain ^ o -proofs oiDN^'x and o£DN<j>"x- Hence by formula induction: cj)" DNfi'x * ,by tormula i induction • ,1 *•
~z—T,——^
^^
Y
A % by formula induction Prop. 3.
(c) <j) is N(j)r and the result holds for
If w is absent we have
DNfto)
° Dcox
la*
Ha, l a Prop. 3.
2.10 Regularity
2.10 D3
51
Regularity B
for
\pp'
.KCpp'Cp'p.
Thus B
8. £PC is regular.
We have to show
W
i.e. from the ^ o -proofs of Bcfifr and ^{^} we can find a ^ o -proof of xifr}We proceed by formula induction on x{n}- X{n} ^s n> w e have: * reversibility of IIb' * Modus Ponens. X{TT} is NTT we have * reversibility of II b' DNN
N
^
,
_
T Ponens. —^-^r^—- * Modus Nijr
X{TT}
is DTTX we have * reversibility of II b'
—— ^^r^
1
*
Modus Ponens.
Thus SPC- is regular. COR.
(i) BSilr
We demonstrate the result by formula induction on x* If x{4>} i s
52
Ch. 2 Propositional calculi
whence by dilution and permutation
similarly with <j> and ft interchanged, and the result follows. If x{
•}
whence by lie, l a
CxWxW
} DNxWXNx'W CNx'{tf}Nx'{4>]\ CNx'{fi}Nx'{iJr}
which is COR.
as required
(ii)
?* r *
This follows from Prop. 8 since
These are easily established, consider the second one,
DfiNjr
_ TT Ila, la Di/rNft la
TT _ Ila, la DDfrNjrN
D^lrNf r
r
—^-^ l i e and definition of K.
Similarly for the other cases. 2.11
Duality
PROP. 9
1. B
2.11 Duality
53
Using the Deduction Theorem we have 116' = =
Prop. 6, twice
la, l i e Def. of C, K. (i) Again we have
t Ho, la £
Ub, Prop6 . T
*- Ha
w w
II6,
KDWW
Prop.6.
TT
of
^
'
(ii)
4 now follows from (i) and (ii) by 116' and definition of K. The rest are dealt with similarly and are left as exercises to the reader. The dual of a ^-statement
10. If
In the ^-theorem
PROP.
54
Ch. 2 Propositional calculi
where {D,p) is some ^.-statement built up from D and p alone. Then Ncj) may be replaced by (Z),
\ppr,(N,p,p'),
of type ooo,
f
where (N,p,p ) is some ^-statement built up from N, p,p' alone. The only such ^ c -statements are:
p,Np,NNp,...
and
p',Np',NNp',....
Thus D(j)\]r would be independent of one of its arguments. The axiom DpNp would become one of p,Np,NNp,..., one of these would be a ^-theorem. By the reversibility of l i e repeatedly one of p, Np would be a ^-theorem. This is absurd because a ^-theorem must contain at least four symbols. The ^ - a x i o m DpNp is independent because it is impossible to obtain it from the other axioms. Once a variable is introduced into a ^ o -proof it remains in that ^ c -proof from that place onwards. Hence if we are denied use of the axiom DpNp then the resulting ^-theorems will all contain some variable distinct from the variable p. Lastly the <^c-rules are independent. Suppose we omit the rule la then we are unable to obtain the .^-theorem DNpp because the other «^o-rules increase the length of a .^-statement. If we omit rule II a then we are unable to obtain the ^-theorem DpDpNp because the lower formula of l a with a subsidiary formula, IIb, c begin in a different way. In the case of l a without subsidiary formulae we could only obtain DpDpNp from DDpNpp and this is distinct from the lower formula of l a with DpDpNp or DDNppp or DDppNp as upper formula and these fail to be lower formulae of 116, c. If we omit 116 we are unable to obtain DNDppp, this fails to be the lower formula of II a in a «^o-proof, because the upper formula would then be p which is impossible. It also fails to be the lower formula of II c, if it were the lower formula of I a then the upper formula would be DpNDpp this fails to be the lower formula of lie, if it is the lower formula of II a then the upper formula would be NDpp from
2.12 Independence of symbols, axioms and rules
55
which by idempotency we could obtain a «^o-proof of Np, which is impossible. If we omit l i e then we are unable to obtain the ^-theorem DNNpNp by similar considerations. We have just shown that the symbols of 8PC are independent, yet it is possible to define N and D in terms of a single symbol, S or 8'. This can be done as follows: D4.
S
for \pp' .DNpNp'.
D5.
8'
for
Xpp'.KNpNp'.
then 8 and 8' are both of type ooo. We can then define N for "kp. 8pp. D for And
N
for
D for
\pp'. SSppSp'p'. kp.S'pp. -kpp'.S'S'pp'S'pp'.
The axiom scheme would become SSTTTTSSTTTTSTTTT,
S'S'TTS'nnS'nS'Tm.
Thus there can be syzygies between independent symbols. Independence of axioms can also be shown by finding a model J( for the formal system less one axiom and showing that this axiom fails to be 2.13 Consistency and completeness of 12. SPC is model-consistent Consider the applied propositional calculus J(c with the elements t, f of type o called truth-values, t is designated and / is undesignated, and constants Noo, Dooo of the types shown in the subscripts. The rules are (type symbols are omitted) PROP.
and conversely. It is easily verified that *Jifc is a model for 8PC. Thus £PC is a two-valued propositional calculus. An ~#c-valid ^-statement is called a tautology.
56
Ch. 2 Propositional calculi
13. 8PC is complete with respect to J(c. Let ^ be a ^-statement, by repeated use of the de Morgan laws and double negation move negations to the right as far as possible until they act on variables only, this introduces the connective K. Then using the distributive and commutative laws reduce the resulting ^-statement to a conjunction of disjunctions {conjunctive normal form). PROP.
where i/r\ ..., ijr^\ ...,#',..., x{6) &re variables or negated variables. A disjunctive normal form is obtained by interchanging the roles of K and D. A conjunctive normal form is ~#c-valid if and only if each conjunctand contains as disjunctands a variable n and the same variable negated, otherwise by suitable choice of t and / to replace the variables we can make that conjunctand take the value/. Starting from the axiom DTTNTT and using IIa, l a we can easily ^ c -prove any disjunctand of an J(cvalid conjunctive normal form, and hence the conjunction of these disjunctands. If \[r is the conjunctive normal form of
2.13 Consistency and completeness of 0>c
57
T[H', ..., JS^] is t. This is a ^-statement which has J(c-value t just in case F reduces to t. Thus ^ c is functionally complete with respect to J(c, 2.14 Decidability 15. &c is decidable It is an effective process to decide whether a ^-statement is Jtc-valid, hence it is an effective process to decide whether a ^-statement is an ^#c-theorem. Furthermore if a ^-statement is found to be a 0>ctheorem by the test of .^-validity then it is an effective process to supply the ^a-proof. For all we need do is to put the ^-statement into conjunctive normal form, this is easily tested for .^-validity and if the test is affirmative it is a routine matter to supply a ^c-proof. We can obtain this result in another way. We note that apart from l a the length of a ^-statement increases as we proceed down a ^ o proof-tree and la leaves the length unaltered, repeated use of l a will reproduce a previous formula so use of l a is limited. Thus given a ^^-statement <J) we can construct all possible deduction-trees with base 0. We can then decide if any of these trees are ^o-proof-trees of (j) or whether each fails to be a ^ c -proof-tree of (j). Thus we can decide if ^ is a ^-theorem and if so we can find a ^ c -proof for it. For instance DNDDNpp'Npp', call it #, fails to be a ^-theorem. It is of the form DNDcfrtyo) with DNpp' for (j) and Np for i/r and p' for 0). PROP.
This can arise (i) . , or (iii) from v from I a ^ , ^ , , or v(ii) ; from I l a ^^^ DNDcfiiJfG) DND(f>i/ra) v ' —J;
,.
only. Case (i) DcoNDcfriJr can arise from v by l a
which brings us back to where we started or from ^ .-T^ , . I l a only. can arise from
^
- 116 only. Now iV^ is NDNpp' which
can only arise from NNp Npr ^rt^^r—7- Ho. l h e second upper formula fails to arise from any other ^-statement by the ^ o -rules and fails to be a ^ c -axiom. Thus case (i) fails to provide a ^ o -proof of %. Case (ii) o) isp' and so fails to provide a ^ o -proof of x- Case (iii) DNcjxx) can only arise from ^ T .
DN(j)(o
l a or -^AT, I l a this we can reject as before since o) is rp'. J DNcpoj
58
Ch. 2 Propositional calculi
Do)N(j) is Dp'NDNpp' and this can only arise from DNtpoj by la, which we started vwith, or from
NDNpp'
^ — , II a, NDNpp' can only arise from
NNp Np'
, 116 and this we can reject as before. Lastly DNcfra) is
DNDNpp'p' and, apart from cases already considered, can only arise from
DNNpp' DNp'p' DNDNpp'p' DNp'p' can only arise from an axiom by la, but DNNpp' can only arise from Dpp' or Dp'p each of which fails to be a ^-axiom. Hence case (iii) fails to produce a ^ o -proof of x- Case (iii) is more easily dealt with if we consider DNi/roj which is DNNpp'. A system with 16 as an independent rule would fail to be decidable in this way. We can add any ^-statement as an extra axiom without causing every ^-statement to be a ^-theorem, provided the statement contains a connective. A variable would then fail to be a ^-theorem. The resulting system would however fail to have many of the properties of ^ c . 2.15 Truth-tables The simplest way of testing a ^-statement for being a tautology is by the use of truth-tables. Suppose that a .^-statement (j) is written in terms of D, N, K, C and B; we replace D, K, C and B by two place functions, d, k, c and 6 respectively, and replace N by a one-place function n. We then replace the propositional variables by elements t a n d / in any manner, always replacing different occurrences of the same propositional variable by the same element. We then evaluate the resulting composition of functions by using the following truth-tables: p t t
f f
st
dpp' t t t
f
f
p' t
V t t
P' kpp' t t
f
f f
f
t
f f f
V t t
f f
P cpp' t t
f
t
f
f
t t
p t t
f f
P' bpp> t t
P t
f
f
t
f
f f
np
st
t
These tables give the values of the five functions for values of the arguments on the same line.
2.15 Truth-tables
59
For example let us test CCKpp'p"CpCp'p" for tautology. This Restatement is a conditional, let us see if it is possible to make it take the value/ by suitable values, t or/, given to p, p' and p". The only way of making a conditional take the value/ is to make the first component t and the second component/. Thus we want to make GKpp'p" take the value t and CpGp'p" take the value/. Both of these are again conditionals so we want p to have the value t and Cp'p" have the value/, this requires p' to have the value t and p" to have the value/. This then fixes the values of p, p' and p" in order that our original statement take the value/. Putting these values for p, p' a>ndpff in the original statement we easily calculate that it is t, thus it must always be t, and so is a tautology. We put the working down as follows: CCKpp'p'VpCp'p"
f t
f
tf ttf t
f The first line with / under the first occurrence of C indicates that we want to make the whole statement take the value/. To do this we must make the first component take the value t and the second component take the value/, this is indicated by placing in the second line t under the second occurrence of C and/ under the third occurrence of 0. Since the third occurrence of C is to take the value/ then its first component must take the value t and its second component must take the value/; this is indicated by placing t in the third line under the second occurrence of p, and/ in the third line under the fourth occurrence of C. We have now found values that p, p' and p" must have if our statement is to take the value/. Because p' must have the value t and p" must have the value/ since Cp'p" is to have the value /, this is centred in the fourth line. In thefifthline we enter the values of p, p' and jp" under the first occurrence of these symbols. In the sixth line we evaluate Kpp1 andfindthat it is t, in the seventh line we evaluate the first component of our original condi-
60
Ch. 2 Propositional calculi
tional and find that it i s / , but we had found in the second line that it should be t, hence it is impossible to make the original conditional take the value/, hence it always takes the value t and so is a tautology. The working can be put down on one line, since except for the last entry we have always used different columns. Thus: CCKpp'p"CpCp'p"
fttttfftft
f
f 176 555 2334 4 The numerals indicate the order in which the letters t a n d / are put in. The final line puts both t a n d / under K indicating an impossibility. Since SPC is decidable we could have another formulation for it, namely: the axiom-scheme 'a tautology is an axiom5 and dispense with rules. The conditions for being a formal system are satisfied because we have a test for being an axiom. Sometimes it is simpler to evaluate directly for all possible argument values, we would then put down the work as follows: p t t t t f f f f
p' t t f f t t f f
p" t f t f t f t f
BBpp'BBpp'fBp'ptf t t t t t t t t
Thus we have a tautology. Independence of axioms and rules may also be shown by means of models. We find a model J(' for the formal system <£' obtained from the formal system jSf by omitting one axiom, axiom-scheme or rule and such that the omitted axiom or axiom-scheme fails to be valid in the model J(' or the omitted rule fails to preserve JS?'-validity. For instance the following three axiom-schemes and Modus Ponens give a Propositional Calculus 0*x equivalent to ^ c , / is a constant.
2.15 Truth-tables
61
(i) dt
(ii) a
3C6S'C(i
(iii) G( Consider the following truth-tables for the connective M.P. Cpp'
v'
V t t t
g
t
t t
f
f
t
g g g
g
t t
f
f
g
g
f
t
t
f f f
t
(i)
Cpp'
(ii) Cpp'
c. (iii) Cpp'
t
t
t
f f f
g
g
f
f
g
f
t t
g t t t t
t t t
t t
t t t
The constant/ is undesignated. The heading of the various columns denotes that the corresponding rule or axiom-scheme fails for that truth-table, but all the other rules and axiom-schemes hold for that truth-table. In this model we have three elements, t is designated,/and g are undesignated. We leave the checking of this table as an exercise for the reader.
2.16 Boolean Algebra A Boolean Algebra is a formal system with the symbols: nooi a
type I Oii Oii
0 1
i
u n
iii
i
iii
ii
( )
name
variable for an element equality inequality null element unit element union intersection complement generating symbol left parenthesis right parenthesis
62
Ch. 2 Propositional calculi
The axioms are given by the following schemes: We have written (a U /?) for ((U a)/?), etc. a = a 0+ 1 0=1
1=0 au 1 = 1 au 0 = a
a no = o a ni = a
ft cz ftl II >-
na = o a = a
ft cz ft II ft
a na = a (a f) /?) = a II/?
(a u B) = a n B
a n (/? n 7) = (a n /?) n 7
a u (/? U 7) = (a u /?) U y
a n (/? U 7) = (a n /?) U (a n 7)
a U (/? n 7) = (oc U /?) n (a U 7)
Note the duality. We have neglected independence. We have written equalities in the customary manner, a, /?, 7 stand for arbitrary elements. We have omitted parentheses wholesale. The rules are:
We could add the symbols and rules of the Propositional Calculus and so get compound statements, but this is unnecessary. Note that if we replace n by K u by D
—
0 1 = *
by by by by by
N
p&Np py Np B NB
and the variables for elements by propositional variables, then the axioms become tautologies of 0*c. Again if we replace the elements by variables for subsets of a given set X, then the axioms become axioms for elementary set theory, as regards union, intersection and complementation, 0 becomes the null set and 1 the given set X. The simplest Boolean
2.16 Boolean Algebra
63
Algebra consists of the two elements {0,1}. This is called the two-valued Boolean Algebra and corresponds to the pair {/, t}. PROP.
16. We have
(i) a U (a n ft) = a
a n (a U /?) = a
(ii) (any3)U/? = aU/? (iii) aU^ = a if and only if We have
(a U ^) H/tf = a n/? a n /? = /?.
a (J (a n /?) = (a n 1) U (a n A) = an(lU/?) = an 1 = a.
Again
(oc(]fi)[)j3=(oc[)/i)(](^[j^) = (a U /?) H 1
Lastly if a U /? = a then
an/?=(au/?)n/?
= /? by (i). D6.
a - > / ? for aUyff a<->y5 for (a->/?) n ( y ) a ^ J3 for a n /? = a.
Notice the correspondence with ^ c on replacing -> by O and <-> by S . PROP.
17. We have
If
a ^ /? a^cZ ft ^ a
//
a ^ /? a^cZ J3 ^ y 0^ a
//
then
a = /?.
then a ^ y. a ^ 1.
a ^ /? tf^w a n y ^ y? n y.
64
Ch. 2 Propositional calculi
//
a ^ /? then afl/J^a
a u y ^ /? U y. a < all/?
a < /? i / cmd owfo/ ^/ fi ^&>
a < /? if and only if a ->/? = 1 a = /? if and only if a <-> /? = 1. These are very easy and are left as exercises for the reader. 2.17 Normal forms Using the commutative, associative, distributive, de Morgan laws and double negation we can express any J^-statement in an equivalent form, either as a conjunction of disjunctions of propositional variables and negated propositional variables or as a disjunction of conjunctions of propositional variables and negated propositional variables. The first case is called the conjunctive normal form (c.n.f.), and the second case is called the disjunctive normal form (d.n.f.). We push the negations to the right as far as they will go so that they act only on propositional variables or negated propositional variables then use double negation as long as possible and lastly the distributive laws. It is like multiplying out an algebraic formula. We can further ensure that in the first case each propositional variable which occurs in the original J^-statement occurs negated or unnegated in each disjunctand in the c.n.f. and dually in the second case in each conjunctand in the d.n.f. This is achieved by disjuncting KpNp in the first case and conjuncting DpNp in the second case, if p is a relevant variable. Clearly we are left with an equivalent J^-statement in each case. The N. and S.C. that an J^-statement be a tautology is that each disjunctand of its c.n.f. contain a variable and the same variable negated. Dually the N. and S.C. that an J^-statement be refutable is that each conjunction of its d.n.f. contain a variable and the same variable negated. We can further ensure that in the c.n.f. each disjunction contains each variable exactly once either negated or unnegated, unless the c.n.f. is a tautology. For we can omit any disjunction which contains a variable and the same variable negated and obtain an equivalent J^-statement. If the original ^r-statement was a tautology by this removal we would finally remove all the disjunctands. Dually for the d.n.f.
Historical remarks to Chapter 2
65
H I S T O R I C A L R E M A R K S TO C H A P T E R 2
Propositional calculi have a long history. Aristotle's syllogistic, when expressed in modern symbology, amounts to a form of singularly predicate calculus or class theory. This is discussed by Lukasiewicz (1951) in detail. The early history of the classical propositional calculus is given in great detail by Bochenski (1951, 1961), his account covers all the middle ages up to Frege and then on to the present time. Church (1956) gives the history of propositional calculi since Boole. Lewis (1918) gives the history from Leibniz to Schroder in great detail. Other works on the early history of logic are Bochenski (1951) which deals with preAristotlean times, the old Peripatetics, the Stoic-Megara school and the last period after Chrysippus. Another work is Moody (1953) which deals with the Medieval period, and lastly Diirr (1951) which deals with Boethius. The definition we have given for a propositional calculus seems to cover all systems which are usually called propositional calculi. The problem of the independence of symbols, rules, axioms, etc. is of quite modern origin. Huntingdon (1904) and Bernays (1926) were the earliest to discuss these matters. Models for propositional calculi came in with truth-tables. Lukasiewicz (1920, 1941) was the first to formalize a three-valued propositional calculus. Post (1921) also considered many-valued propositional calculi. Independence of axioms, etc. was demonstrated by Bernays (1926) using many-valued models, this is called the matrix method. Extensions of formal systems were studied by Lukasiewicz and Tarski (1930). Modus Ponens first appears in Scholastic Logic. In treating the classical propositional calculus we use the notation of Lukasiewicz (1920). This is easily read in conversational English if one reads as follows: read D(j)f as either (j) or j/r, read N(j) as not <j), read K
66
Ch. 2 Propositional calculi
The form we have taken for the classical propositional calculus is due to Gentzen (1934, 1955) (see also Anderson & Johnson (1962) and J. Dorp (1962)), who used the terms 'remodelling scheme' and 'building scheme'. The type of proof used in Prop. 2 is due to Gentzen (1934). Prop. 3 was pointed out to me by A. H. Lachlan. The direct method has been further developed by Schutte (1950, 1960). The de Morgan laws were stated by de Morgan (1867), but were known long before that, and Prop. 5 probably first appeared in connection with Boolean Algebras. The deduction theorem first appears in Herbrand (1930) and has been much used since, notably by Hilbert-Bernays (1934-6) and Church (1956). Tertium non datur or the law of the excluded middle goes back a long way and was first called in question by Brouwer (1908) in the case of infinite classes. He maintained that rules that apply to finite classes might fail for infinite classes. Thus he invented Intuitionism, a form of mathematics which does not accept T.N.D. It gives rise to a propositional calculus. But it is far more complicated than the classical propositional calculus, and so is his form of analysis, much more complicated than classical analysis (see Heyting (1934, 1955, 1956) and Kleene-Vesley (1965)). But his methods go some way to clarifying ones ideas about effectiveness, finiteness, constructiveness, etc. which are conditions that Hilbert (1904) insisted should apply to metamathematical demonstrations. Prop. 7, the elimination of Modus Ponens, is Gentzen'sHauptsatz (1955). We thought that since it can be eliminated then why have it at all. The word' syzygy' is due to the algebraist Sylvester who used it in connection with non-linear relations between algebraic invariants, it means a yoking together. Many proofs of the decision problem for the classical propositional calculus have been given, notably by Church (1956), Kalmar (1935), etc. A correct truth-table for implication was given by Philo of Megara about 300 B.C. Truth-tables were informally used by Frege in special cases, six years later Peirce stated them as a general decision method for the classical propositional calculus. Much of the recent development is due to Lukasiewicz and Post. The term 'tautology' is due to Wittgenstein (1922). D4, 5 are due to Sheffer (1913). The modern treatment of propositional calculi stems from Boole and
Historical remarks to Chapter 2
67
de Morgan in 1847. This was the algebra of logic. MacColl (1877) was probably the first to deal with a true propositional calculus. Frege gave the first formulation of the classical propositional calculus as a formal system in its own right. But his work was for long neglected and so the propositional calculus developed in the older form as in the work of Peirce, Schroder and Peano. Whitehead and Russell appreciated the work of Frege and gave the classical propositional calculus a formulation with negation and implication as primitives, Modus Ponens and substitution as rules. But substitution was not explicitly mentioned, though this omission was noted later. One of their axioms was found to be redundant by Bernays (1926). Nicod (1916) found a formulation of the classical propositional calculus with only one connective and only one axiom and only one rule apart from substitution. Since then a variety of formulations have been discovered. One by Hilbert (Hilbert-Bernays (1934-6) vol. 1) has 12 axioms in 4 groups of 3 axioms each, this was designed to separate out the roles of the connectives N, C, K, D. If in this formulation we omit one axiom then we get a formulation of the intuitional propositional calculus. Between the two world wars the Poles were very active in reasearch on the propositional calculus, see Jordan (1945), Storrs McCall (1969) and H.Skolimowski (1969). Another study is to formalize a partial system of the classical propositional calculus which has only the implication sign, and the object is to find axioms so that exactly all tautologies which only contain the implication sign and propositional variables are theorems of the system. The chief interest of such studies is to find a formulation of the classical propositional calculus which is an extension of this partial system. For instance CCCpp'p"CCp"pCpfffp, with Modus Ponens and substitution is an elegant formulation of the implicational propositional calculus. If to this we add the axiom Cfp, where / is a constant, we get a formulation of the classical propositional calculus. Implication in the classical propositional calculus allows as true an implication with a false antecedent, this seems in some sense unnatural. This has given rise at the hands of Lewis (1918, 1920, 1932) of various other propositional calculi designed to rectify this. He also considered other connectives such as 'possible' and 'necessary'. These give rise to what are called modal logics. We do not discuss them in this book. Much 3-2
68
Ch. 2 Propositional calculi
work has been done on these systems, finding decision procedures, formulations, etc. Gentzen (1955) used Sequenzen, that is figures of the form
where we have used# and q as variables. This behaves in the same way as CK ...Kp'p" ...jP>D ...Dq'q" -.tf*We have written the sequenzen as p' p"...p{v) qfq"-..q^ but have only used it with one lower formula. Gentzen allows the cases when either upper or lower formula may be void. The idea of using axiom schemes is due to v. Neumann (1925), a substitution rule is then unnecessary. Prop. 8, the substitutivity of equivalent statements, is due to Post (1921). Conjunctive and disjunctive normal forms derive from Boolean Algebra. A large number of examples on the propositional calculus are found in Church (1956). Boolean Algebra was invented by George Boole (1847, 1854, 1916). He noticed the resemblance between the behaviour of K and D and + and x in arithmetic. The history of modern symbolic logic can be traced back to Boole. More will be said about Boolean Algebra at the end of Chapters 3 and 12, there we shall consider Boolean-valued settheory as an extension of the classical two-valued set-theory. Sheffer (1913) gave a set of independent axioms for Boolean Algebra.
EXAMPLES 2
1. Complete the demonstration of Prop. 5 for the cases other than 4*. 2. Complete the demonstration of Prop. 8, Cor. (ii) for the first and third cases. 3. Obtain ^-proofs for BBpp'Bp'p, CCpp'CNp'Np. 4. Give a ^-proof for Cpp. 5. State and prove the Deduction Theorem in £PV
Examples 2
69
6. Obtain a ^ - s t a t e m e n t ^ which has the following table: P
p'
Ptf
t t t t
t t
f
Jt
t
t
f
t
st
f
f
f
f
J f f f
f f
t t
f J
t
t
t
7. Define D, B, K in terms of C, N. Define C, D, B in terms of K, N. Define C, D, B, K, N in terms of S and in terms of S'. 8. Show that SSpSqrSpSSrpSSsqSSpsSps is a tautology, where p, q, r, and s are of type o. 9. A Propositional calculus ^ 2 n a s ^ n e &xiom schemes: (i) (ii) (iii) And the rule Modus Ponens. State and prove the deduction Theorem for ^ 2 . 10. Prove the following theorems of ^ 2 : CNpCpp',
CNNpp,
CpNNp,
CCpp'CNpNp'.
11. Check the table giving the independences of the axioms of 3PX. 12. Give definitions of C and / of SPX in terms of D and N of &c, and give definitions of D and JV of £PC in terms of G and fo{^v 13. Show that theorems of 0*x translate into theorems of ^ c by the definitions found in Ex. 12 but that there are theorems of £PC which are unobtainable in this way, show also that there are theorems of ^ which fail to be translations of theorems of ^ c . 14. Show that the axioms of SPC are theorems of ^ and that the rules of 8PC are derived rules of SPX. 15. Prove BNBNpp'Bpp' in 0>c.
16. Show that if- and t then 4 ^ and f
is a ^-theorem.
70
Ch. 2 Propositional calculi
18. Define t for Cff in the system 8PV Define K of &c in terms of G and / of ^ . Write out DKptNp without definitional abbreviation in terms of C and/of ^ . 19. Write $ = ijr if and only if B^xjr is a ^-theorem. Show that the result is a Boolean Algebra when U 0 are suitably defined for Restatements. 20. Suppose that a Boolean Algebra 88 has an additional functor * of type u with the axiom schemes: a ^ a*, a** = a*,
0* = 0.
Show that the elements of 88 which satisfy a** = a form a Boolean subalgebra «^* of ^ . When the functions in 88* are defined as follows: write a®
for
a**,
then
a&fi
for
a n /?,
a^jS
for
(all/?)®,
a
for
a®.
(If the elements of 88 are sets of points in a topological space then the specified sets are the regular open sets-open sets which have no pin holes or cracks.) 21. Define for ( a n ^ a x /? for a n /?. Show that with these definitions every Boolean Algebra becomes a Boolean Ring, i.e. a ring in which = 0, if
a + /? = 0 then
a = /?,
Examples 2
71
Conversely show that with the definitions all/? for
a + /3+(axjS),
an/?
for
a x /?,
a
for
1 + a,
every Boolean Ring becomes a Boolean Algebra. In both cases the algebraic zero and unit coincide with the Boolean zero and unit respectively.
Chapter 3 Predicate calculi
3.1 Definition of a predicate calculus A predicate or functional calculus of the first order is a formal system J5" which is a primary extension of a propositional calculus SP, it is obtained from a propositional calculus by adding additional symbols for variables or constants of type 1, calledindividual variables or constants respectively, and adding variables or constants of some or all the types 01, oil, out,..., called predicate variables or constant predicates respectively; there may also be variables or constants of types u, u,..., called variables for functions or constant functions respectively, and there may also be constants of some of the types 0(01), 0(011), ..., (ot) 0(01) (on), ... called quantifiers There must be individual variables given by a scheme of generation, and there must be some predicates. If quantifiers are present then the abstraction symbol is required in order to provide arguments of types ot, oil,... for the quantifiers. If propositional variables and constants are excluded the resulting system (which fails to be a primary extension of a propositional calculus) is also called a predictate calculus. If quantifiers are absent the resulting system is called a free variable predicate calculus of the first order with or without functions as the case may be. If individual variables are discarded so that all well-formed formulae of type 1 are constants then quantifiers are useless and the resulting system reduces to a propositional calculus when propositional variables and constants are written in the form
3.1 Definition of a predicate calculus
73
have a diadic predicate calculus of the first order, and so on. Quantifiers of type O(OL) are called simple quantifiers, quantifiers of other types are called compound quantifiers. A predicate or functional calculus of the second order, J^2, is a primary extension of a predicate calculus of the first order obtained by adding quantifiers (simple or compound) for predicates. Simple predicate quantifiers are of types: read' for at least one',' for an unbounded set', 'for all' etc. 0(00),
O(O(OL)),
0(0(011)),...
generally o(oa), where a is a type of a predicate or of a propositional variable, and compound quantifiers are of types: read 'for all pairs', 'for all except a bounded set of pairs', etc. 0(000),
0(00(01)),
0(0(01)0),
0(0(01) (01)),
0(00(011)),
...
generally o(ootfi),o(ooc/3y),..., where a,fl,y,... are types of predicates or of propositional variables. A predicate or functional calculus of the third order, ^ 3 , is a primary extension of a predicate calculus of the second order obtained by adding variables or constants of types 00,
o(ot),
O(OU),
...
generally ooc where a is the type of a predicate or of a propositional variable, and oa/3 where a, J3 are types of predicates or of propositional variables, ..., these are called predicates of predicates, we could also add some mixed predicates requiring for some arguments predicates and for others requiring individuals, these would have types: 001,
010, 0(01) 1,
01(01),
...
and generally oou,otot, ofiyt,..., where a,/?,7 are types of predicates or propositional variables. A predicate or functional calculus of the fourth order, ^"4, is a primary extension of a predicate calculus of the third order obtained by adding quantifiers of various kinds over predicates of predicates or over mixed predicates. And so on. A predicate calculus of the first order without constant individuals or constant functions or constant predicates or constant propositions is called a pure predicate or functional calculus of the first order with or without functions as the case may be. A pure predicate calculus of the
74
Ch. 3 Predicate calculi
second order is defined similarly. A predicate calculus of the third or higher order is called pure if it is an extension of a pure predicate calculus of one lower order and if only variables (for calculi of odd order) or quantifiers (for calculi of even order) are adjoined. For instance a pure calculus of the third order may have variables and constants of type oo, O(OL), because there may be constants of these types in the pure predicate calculus of the second order of which it is a primary extension, but in the extension only variables of these types are adjoined. A predicate calculus of the first order which is without propositional variables and predicate variables is an applied predicate or functional calculus of the first order, with or without functions as the case may be. Similarly a predicate calculus of odd order is applied if it is obtained from a predicate calculus of one order less by adding only constants and is mixed if both constants and variables are added. A predicate calculus of even order with some constant predicates is mixed, it must have variable predicates of appropriate types. A formal system £P is based on a pure predicate calculus of the first order
J^ if the constants of ZF are constants ofSP and if ^ has symbols of each type for which there are variables in SF and has individual variables (we may use the same symbols for individual variables in SP and in 3F) and if whenever
3.1 Definition of a predicate calculus
75
types from among, i\ i",... and possibly individual constants of some of these types, and possibly functions with values and arguments from these types. The order of a many-sorted predicate calculus is defined as for a one-sorted predicate calculus. A many-sorted predicate calculus IF' is based on a pure one-sorted predicate calculus IF when the constants of J^ are constants of IF' and when the axioms and rules of J^ apply to each sort of individual variable in IF'. We shall show later that it is possible to reduce a many-sorted predicate calculus IF' of the first order to a one-sorted predicate calculus IF of the first order by adjoining to IF some constant one-argument predicates. Each such predicate plays the part of saying that its argument is of a certain sort, distinct predicates referring to distinct sorts. Similarly we could consider many sorts of predicates in predicate calculi of higher orders. A predicate calculus is a formal system hence it must be constructive in accordance with the definition of a formal system which was given in Ch. 1. If two predicate calculi have variables of the same types then by a trivial adjustment of notation we may use the same symbols for these variables in both systems. A predicate calculus IF' is weaker than a predicate calculus IF under the following circumstances: (i) IF' is without variables of type oc if IF is without variables of type a. (ii) The constants of IF' can be defined in terms of the constants of IF. (iii) An J^'-theorem
76 3.2
Ch. 3 Predicate calculi Models
A model Jfofa, predicate calculus of the first order &> with only simple quantifiers is a formal system which satisfies the following conditions: (i) Jl has exactly the same constants as !F of types other than i, ii, iii, . . . , O, Oi, OLi,
(ii) Jl is without variables but has constants of types 1 and o and of any of the types a, m, ...,oi,ou,... which occur in 3F. The ^-constants of type 0 are designated or undesignated, at least one is designated and at least one is undesignated, the ^-constants of type o are called elements, the constants of type 1 are called individuals, the constants of types u, in,... are called functions and the constants of types oi, on,... are called predicates. (iii) Jl is without axioms. (iv) The rules of t e n a b l e us to replace any ^-statement by a unique element, called the Jf-value of the ^-statement. (v) If ®{
(
3.2 Models
77
For instance if O ^ X T / . Y ! £,?/}} is an ^'-statement whose sole free variable is £, then J( is to contain a constant of type 01, 3 such that Sa has the same J(-value as 0{oc,A7j.x¥{a,/)]}} for each ^-individual a, where again there is to be an ^-constant H' of type oi such that Y{a, /?} has the same J(-value as S'yff for each ^-individual /?. S' will in general vary as a varies. If J^ contains compound quantifiers then clauses (vi) c, (v) require amendment. Suppose J^ contains a compound quantifier of type o(ott) then if X££'.
78
Ch. 3 Predicate calculi
3.4 The classical predicate calculus of thefirstorder A pure predicate calculus of the first order is classical if it is equivalent to the following predicate calculus of the first order, SFC. Symbols: those of £PC together with: type X
name
I
Voc >Pou>~-
01,011,
E A
o(pi)
individual variable predicate variables existential quantifier abstraction symbol
further variables are obtained by superscripting primes, as for propositional variables. Axioms DnNn, where n is a propositional variable, DTT^NTT^, where n is a one-place predicate and £ is an individual variable, etc., for two or more place predicates. Rules Those for £PC together with: Remodelling scheme \b DDtfxfxi) cancellation, D0G) Building schemes II d D(f>{y} o) existential dilution, DE(kt;. ${£>}) o) £ fails to occur free in ^{I\}, TJ free in lie
DN(p{7j}o) generalization, DNE(k£. {£)) (o, where TJ fails to occur free in OJ, 0{I\} and fails to occur free in Here £, TJ denote individual variables, (f>{r)}, 0) denote ^"^-statements, o) is a subsidiary formula and may be absent, (f>, (j>{7)), N
D7.
(mm ') for
for
E{u-m)-
#(X£.#(X£\$H£, £'}))>
etc.
3.4 The classical predicate calculus of the first order
79
Note that an #~c-proof is direct, the only formulae that can be omitted are duplicates as in 1b. This rule causes the undecidability of 3FC, as we shall see later. 3-5 Properties of the system ^c 1. The schemes la, b, 116, c, e are reversible. In Prop. 2, Ch. 2 we showed the reversibility of la, 116, c for the system 8?c. Similar demonstrations hold for J r c . 16 is reversible II a, l a . PROP.
l i e is reversible. Suppose we have an J^-proof of In this J^-proof corresponding occurrences of N(EE)) ^{£} can only be introduced at II a, e. If a corresponding occurrence of N(E£)(j){£) is introduced at I I a then introduce N
would become Dn^NnTj which fails to be an J^-theorem. This completes the demonstration of the proposition.
80
Ch. 3 Predicate calculi
2. D$N
* *—ii-LI
n e, conditions on variables are satisfied,
Thus the result holds for (E£) i/r{g} if it holds for i/r{£,}. This completes the demonstration of the proposition. Thus Tertium non datur holds in the system !FC. P E O P . 3 . The Deduction Theorem holds in 3^G with the following restriction. If <j>', ...,(f)(s®\-jr ijr, where l i e fails to be applied to any variable which occurs in ftSd\ then f , . . . , ^f^^JDN^f. When lie fails to be applied to any variable in ^ ^ we say that the variables in
f
for
9
""^9
in J V
The demonstration is similar to that of Prop. 6, Ch. 2. We have just shown that (i) (tertium non datur) holds in 3FC, (ii), (iii) follows exactly as in Prop. 6, Ch. 2 and similarly for (iv) except that if the rule used is I I e>
y DN
..., D provided that the variables in D^SK^CJ are held constant.
3.5 Properties of the system
81
We have DND^SK)o)Di/ro) from Prop. 3 whence the result by permutation and cancellation.
UXi6) for x'>
0=1
D8.
SK
n
3= 1
6= 1
d)
tfthenDN] 6
COR. (ii). If<j>\ ...,
X{6)X{SK)-
fr, where (/> is the closure of (j).
1
We proceed as in Prop. 3 except that if lie is used, say — lie, where a X, variable £ free in an hypothesis $ is generalized then we replace it by
lie, condition on variables is satisfied lie. In this way from $\ ...,
lid
Now repeat for
D9.
for
SK
x', K
for 6 =I
X^6)X(SK).
COR. (iii). / / <$> is quantifier-free and is in conjunctive normal form, so that K
(f> is of the form J\ ft6) where ftd) is a disjunction of atomic statements or 6= 1
negations of atomic statements and if C(jx]r is an &c-theorem then We may suppose that in the J^-proof of C^
the free variables in (j>
82
Ch. 3 Predicate calculi
are held constant. Because if one of them, say £, was generalized then this generalization must occur in the J^-proof of C
K
(T{d)
Let ^ be n S 0(*'*°, this distributed becomes £ n fte>r{6)\ call it 0*, 6 = ld' = l
0= 1
r
where 1 ^ r{6) ^ o-{6}, and the summation is over all such r. From 'if N6 DNdilr =?= then , , ' and DN6\jr we obtain DN £ II (j>{e>rmf, whence by ivp* DN(j)*yf T e=i K
the reversibility of 116 repeatedly we obtain DN ]J ^e>T^i/r, for each r. All these J^-proofs are obtained from one tree by replacing part disK
K
junctions of S II
we have })
K
D 2 ^ ^ ' 0=1
6=1 T {
^
for each r. Now consider the places where
enters the J^-proof of D 2 Nftd'T{6})i/r. It will do so either at an 0=1
J^-axiom (T.N.D.) or at IIa or at He only. If it enters by an J^-axiom T{0})^(0,T{0}) replace t h i s b y j]AT?(d 7{e\)M6 TW}\^La' T h i s c o n v e r t s t h e
J^-proof of D S Nfte>TW>ft into an ^-deduction of D S Nftd>T^ft from 0=1
0=1
certain hypotheses <j^e*r^. Once Nfte>r{d)) has entered this deduction it thereafter remains in the subsidiary formulae of building rules because it fails to be governed by a quantifier or by ND or by N in the theorem. We are only considering occurrences of Nft0>T{6}) which correspond to the occurrence of Nfte>T{6}) in ^Nft0'7^. 1
If Nftd>r{d)) enters by l i e then
0=1
is NN(f>(d>T{d}\ where
consider the places where corresponding occurrences of <j>(e>rW> enter the deduction, 00»T0» is an atomic statement so it enters by II a or by (T.N.D.) only. If it enters by (T.N.D.) replace this by
Ila.
3.5 Properties of the system !FC
83
These two cases altogether leave us with an ^-deduction of D 6=1
from hypotheses (jfl>7W\ 1 < 6 < K. In this deduction omit all occurrences of N(jP'r{6)) which correspond to the occurrences of Nft6'7^ we have been considering, so that Nftd>7{6}) fails to be introduced by IIa. We are left with an J^-deduction of ^ from hypotheses 0<^T0», 1 < 6 ^ K, because we have only altered the subsidiary formulae of building rules by omitting from the upper and lower formulae of a rule the same disjunctands and this converts an application of a rule into another application of the same rule, also the parts of upper and lower formulae of remodelling rules have been altered in the same way. These cancellations fail to effect i/r because an occurrence of Nfte>T{d}) in ^N^d'T{6}) fails 6=1
to correspond to any occurrence of Nfte>r{e)) in ^ . The final result when repetitions have been removed, is an ^'-deduction of xjr from hypotheses fto,T{0})i i ^ Q ^ K, this holds for each r we require. an
w
LEMMA. If o),x\ • • • > X^&cfr ^ '> X' • • • > A ^ h ^ ^ where the variables (o, ojr are held constant, then DGJG)', X, • • • > ^ ( l c ) hjr 0 ^. We have Do)G)',x', ...,X{K)\~^CD^O)', by replacing all occurrences of co which correspond to the occurrence of OJ in the first hypothesis by DGJG)'
in
and then putting o)' into the subsidiary formulae, by remodelling, of any rules used: the conditions on variables are satisfied because the variables in co' are held constant. Again we have Di/rco', x'> • • • > y^&JDxlriJr, by replacing o)r by Di/raj' as before in the second hypothesis and then putting ^ into the subsidiary formulae, the condition on variables is satisfied because the free variables in i/r fail to be quantified. Thus we have: (K) D u s i n Ib w e DUG)',/, ...,X ^c ft«>'-> Di/r(o',x'> —itf^rflWr* § §et t h e required result. Returning to Cor. (iii) we have ^ (1>T(1)) ,..., ^(ACjTW)hjr ^ for each r. K
(7(6)
Now the hypotheses arise from distributing n S ^e>e)
so
that we shall
0 = 16' = l
have sets of hypotheses which differ only in their first members; we have already noticed that we may assume that the variables in the hypotheses are kept constant so that we may apply the lemma repeatedly (T(l)
and obtain 2
C
result when the hypotheses ft2>TW for all possible r are the same,
84
Ch. 3 Predicate calculi o-(i)
whencebythelemmarepeatedlyweget £ 0 (MO , S Continuing in this manner we finally arrive at 0', ...j^H-jr^, as desired. C O R . (iv). / / ^ is £Ae closure of
a
^ d where
6= 1
fte) is a disjunction of atomic statements or negations of atomic statements, and if C^xjr is an &c-theorem then $',..., <^K>>V&C ft-
Consider the proof-tree ofC^i/r, if lid is applied to a variable in <j) and if this variable is held constant in \[r or is absent from i/r then omit that application of lid, the result is an J^-proof-tree of C$*^r where $* differs from (j) by omission of generalizations. Every other variable in (f> is first restricted in ^ and later on generalized in i/r. Variables in (j) fail to get generalized. Now consider the portion of the J^-proof-tree of C
—-—"K b/ —>—>-— I I c , e t) for i]',...,r D(Et)) (Eg) N<j>DN{Eri) Nf'o) replace the first piece by <J>',...,
as before but without I I d on N,
fr"
ft
as before but without lid on
3.6 Modus Ponens 4. Modus Ponens is a derived We have to show
PROP.
Given the J^>-proofs of the upper formulae we have to show how to obtain an J^-proof of the lower formula. The demonstration is by formula induction on the cut formula $. With one of a), x non-null.
3.6 Modus Ponens
85
(a)
D 2 N(j>x (this is to account for all the cancellations of N
formula immediately above D 2 N
mula. We have already shown that the result holds if the right upper forSK
mula is an axiom. If 2 N<j> is in the subsidiary formula of a rule and if the 6=1
result holds for the upper formula or formulae of that rule then it follows at once that it holds for the lower formula of that rule. Thus if we have SK D S N<j>X' 6=1 SK
then we have SK
Daxf> D 2 =r—j^^1 £
* by induction hypothesis
deduction as before,
and similarly for a two-premiss rule. SK
If the whole or part of 2 N<j) is in the main formula of a building rule 6=1 SK
then this building rule can only be I I a since <j> is atomic. By la D 2 Ncfrx 6=1
could become DNifriJr, which since ^ is atomic is different from any of the forms of the lower formulae of building rules other than IIa. Or, by l a SK
D 2 N
an(
SK
*S ^
occurs in i/r,
6=1
this could be of the form of the lower formula of the building rules SK
IIa, 6, c,d,e, but then 2 N
contrary to the case considered. Thus the rule can only be II a. In this
86
Ch. 3 Predicate calculi SK
case the formula immediately above D 2 Ntfix is either x i*1 which case 6= 1
the result is trivial or is of the form we are considering. Thus the only SK
non-trivial case is when 2 N(f>x is in the subsidiary formula. By our 0= 1
supposition 16 with N<j> as main formula fails to be the rule immediately SK
SK
above D 2 N<j>X* If the rule is II a with part or all of 2 N6 in the main 6= 1
formula then we have
6=1
f
^
N,
,
6= 1 6=1
where there may be more Nfts in the lower formula so that we have V
diluted with a formula of the form D 2 N<j>x" (x" m a y be null) or a per6=1
mutation of this, and if the result holds for the upper formula then: Dco
^
Dux' then by dilution we easily get DGJX* This completes this case. It is impossible for x! to be null, otherwise 2iV0 would arise from an axiom, which is absurd. (b) <j) is D(f)f<})" and the result holds for $' and $". We have ^-proofs of DOJDC})'(j)" and DND(j)'(j)"x,fromthe reversibility of II b we then have ^Qproofs ofDN
1 DDGJ6'6"
* by induction hypothesis, la, 6. (c) (f> is N
3.6 Modus Ponens
If a) is absent we have
87
N(j)' 16
X (d)
Do)f{7j}
K
D N < f >w ' { 7 ) } X , . , . . , . . . - * by induction hypothesis;
we wish to demonstrate
It suffices to demonstrate
because from the reversibility of l i e we can obtain an ^>-proof of DN(j)'{rj}x for any variable r\ which fails to occur free in DN
If 2 (Eg) $'{£} fails to occur in the left upper formula then the result 0=1
follows at once by dilution. Otherwise we use theorem induction on the left upper formula. If the left upper formula is an axiom then (Eg) '{£} is absent. If the left upper formula is the lower formula of any building rule other than the introduction of (Eg)
same rule because D 2 (Eg) $'{£} must be in the subsidiary formula 6=1
of that rule. We had an almost similar situation under (a). But if the rule is an introduction of (Eg)
bymductxonhypothesis, by formula induction,
From the demonstration of Prop. 1 we see that the variable 7} in DN
88
Ch. 3 Predicate calculi
variable £ which is restricted by lid! in the upper left formula might be any variable. If we change all free occurrences of £ in the J^-proof of the upper left formula to a new variable then we obtain an J^-proof of a formula which differs from the upper left formula in that now £ is a new variable, but o) and {EE,) <j)'{£] may have suffered a change of variable to this new one. In this case we should end up with DGJ'X, where co' differs from o) in that one free variable in o) has been changed to a new variable (which fails to occur in o)). By change of variable in the J^-proof of Dcox we can get back to Dcox- This completes the demonstration of the proposition. Note that we have given an effective method for eliminating the cut. This consists in taking a highest cut in the proof-tree and either eliminating it outright or replacing it by a cut higher up the proof-tree or by a cut or cuts with simpler cut formulae. Thus applying the process a highest cut ultimately gets replaced by cuts with atomic cut formulae and these can be made to disappear altogether. 3.7 Regularity PROP.
5. !FC is regular.
We have to show
— , /f , ,, *.
We show from this the result and
B6xlr y{6\ r ;C *
follow, the first by the reversibility of 116' and the last follows easily by Modus Ponens which can be eliminated. We proceed by formula induction on x{n}' The cases when ^{77-} is n or is Nx'{rr} or is DX'{TT}x'i71} are dealt with as in Cor. (i), Prop. 8, Ch. 2. If x{n} is (E£) x{n> £} a n ( i ^ n e result holds for ^{TT, £} then we have
whence by IId, la we get
, £} (Eg) xtt,
3.7 Regularity
89
and by lie DN (Eg) X{, £} (Eg) X{f, £} DN(Eg) X{f, Z) (Eg) X& as desired. The variable £ can occur in (j) or in i/r or in both. CoR.(i).
DB^o) Dx{
We proceed by formula induction on xi17}- The details are left to the reader. D10 (^ §)#{£} for N(Eg)N${Q. A is called the universal quantifier. PROP.
6. 1-5 and l*-5* and 6 of Prop. 5, Ch. 2 hold in ^c, also
7. B(Eg)Df{QW!£)t{&fc
7
*-
8. B(^g)D^{g}^2)(ilg)^{g}^; 8.* distributivity of quantifiers. In 7-8* incl. the variable E, fails to occur free in i/r or 0{I\}. 9. BN{E£)
\r\lfn
9 where £ fails to occur free in co or free in
' is a derived rule in ^c. The rule l i d ' is reversible. For 7-9* consider 7. We have from Prop. 2 T TT, la, Ha, DND
DftQtDftQ^
on d a t u r r e v e r s i b m t y
1 a, 11 a, in
DNf{E£) Dm f the result now follows by life and definition of B.
90
Ch. 3 Predicate calculi
7*, 8, 8* follow similarly and are left as exercises to the reader. 9. We have DN(Eg) N(j>{^} (Eg) N
DN(E£)N{£}(E£)N
O>
Tertiumnondatur, ° b
the result now follows by 116' and the definition of B. 9* follows similarly and is left to the reader. 10, 10* are trivial. 3.8 The system ^"c The system 8F"Q is like the system !FC except that we add the symbol A of type 0(01) instead of E and replace the building rules II d, e by I I d' D(fi{y} (o rj is absent from 0) and <j) {I\}, D(A £) (j){£) o) £ is absent from
DN(J){rj] OJ
g is absent from
we then replace D 10 by:
D10'
(mm
for
The systems !FC and tF"G are equivalent via the definitions D 10 and D10'. A convenient system which is equivalent to SFC and to 3f"c is the system IF'Q where we use the building rules lid, l i d ' and one of the definitions D 10, D10'. All our results so far hold for &"c. 7. / / §5 is an ^c-iheorem then so is N<j>, where (j> is obtained from
o b t a i n B
COR. (i). If= without use of l i e then ——.
3.8 The system &"c
We verify that if % is a rule other than He then -~
91
is a derived rule.
This fails for He. P R O P . 8.
Consider the first, we have Tertium non datur
Ila, la
IIdIa
Again DN
Tertium non datur
Ila,la
TT,T 11 a, la
DN(E£) Pm
TTp
the result follows from these by 116'. The remainder follow in a similar manner and are left to the reader. 3.9 Prenex normal forms An e^-statement is said to be in prenex normal form if it is of the form (Qic)x{%}y where (Qj) is a sequence of quantifiers, existential or universal or both, and x{%} is an J^-statement void of quantifiers. (Q%) is called the prefix and x{%} is called the matrix. By repeated application of 7-9* incl. of Prop. 6 and Cor. (i) of Prop. 5 we see that each ^,-statement
92
Ch. 3 Predicate calculi
closed J^-statement in prenex normal form with prefix (Qj) and matrix X{ic}. Let y. be the sequence of distinct variables £',..., Qe) in t h a t order and {Q%) a sequence of quantifiers on these variables in the same order. A variable QK\ 1 ^ K ^ 0 is called general if (AE,M) occurs in (Qj) and is called restricted if (#£<*>) occurs in (Qj). I f l ^ y < / c < # then £<"> is called superior to QK\ and £(/c) inferior to £(y).
An ^-statement ^ without bound variables is said to be tautologous under the following circumstances: ^ is built up from other ^ - s t a t e ments joined together by N and D; we replace each of these part statements b y / or by t, but make the same replacement at each occurrence of a variant, we then have a formula built up from/and t (regarded as of type o) by N and D, we then calculate the value of the statement as follows: replace Nf by t, Nt by/; Dffbyf, Dtf, Dft, Dtt by t. The final result is either t or/. If the final result is always t however we make the initial replacements of the part statements then the ^-statement is said to be tautologous. It is easily seen from Ch. 2 that a ^-statement is a ^-theorem if and only if it is tautologous. An J^-statement
K
We shall find that the J^-statement 0 is a disjunction 2 fr{d\ where 01
3.9 Prenex normal forms
93
the disjunctands are of the same logical structure but merely differ by choice of individual variables, they are variants of a common form. l a is used to bring one of the disjunctands \Jrf, ..., i/r^ to the left when we apply lid or lid' to it. 16 is used to discard duplicates as they occur. The final ^ - t h e o r e m is of the form (#'£')... (Qf*>gM) jjr{£',..., £<*>} where each QM is either E or A and ^{£',..., £(7r)} differs from any of the disjunctands ft',..., i/rM merely by change of individual variables. Suppose we have an ^"^-proof of x m prenex normal form. We first modify applications of II a, if necessary, so that they fail to introduce quantifiers. Suppose that Dx'x" i s introduced by I I a then introduce X' x" o n e after the other, suppose NDx'x" *s introduced at II a then introduce Nx',Nx" separately and apply 116, (this forms two branches), if NNx is introduced then introduce x a n ( i a PPty Hc> suppose that (2?£) x! {£} is introduced at II a then introduce #'{£} instead and apply II d, suppose that {A£) #'{£} is introduced at I I a then introduce x'{v} instead, where 7/ is new, and apply II d'. Repeat this process as long as possible and we shall have used II a only with atomic statements or negations of atomic statements as main formulae. Thus we suppose that the ^>-proof of (Qi) ^{j} only uses II a with main formulae which are atomic formulae or negations of atomic formulae. Note that we do this without using 16. Instead of using the system tF'c we shall use an equivalent system which has the rules IId*, d'* in place of rules lid, d\ where D % {&*>}
lid'* in lid'* a) is free for £',..., Qd) and so is 0{I\}, and £',..., £(^ are distinct. An ^t-proof will use rules lid*, d'* whenever possible, so it will be without a sequence of applications of II d followed by 16, etc. We now want to show how to modify an J^-proof of a prenex formula so that applications of 16, IId*, d'* come after applications of IIa, 6, c. Clearly the systems tF'c and ^"c are equivalent. We define the rank of an J^-proof of an J^-formula (Qj) ^{j} in prenex normal form, where % stands for £', ...,£(77), as the ordered Sntuplet {v, v',..., v(n)}9 where v is the number of occurrences of applications
94
Ch. 3 Predicate calculi
of rules II a, 6, c beneath applications of rule 16, v* is the number of applications of rules IIa, 6, c beneath rules IId*, d'* which bind a variable standing in the first argument place in ^{r.},..., v^n) is the number of applications of rules Ila, 6, c beneath applications of rules lid*, d'* which bind variables standing in the 77th argument place in ${%}. These are calculated as follows: take for instance rule 16. Mark applications of 16 in the J^-proof o f (Qj) 0{j}, let these be denoted by K\ ...,#<*>, let /i', ...,/^M be respectively the number of applications of rules Ila, 6, c beneath K', ...,K(n\ then /i' +/i"+... +JLC{K) = v, similarly for the other cases. Ranks are ordered lexicographically. We now show how to modify the J^-proof of the prenex ^-formula {Ql)
where ^r is atomic or the negation of an atomic formula. Replace this by
j. a.
DDDjfr
DNi/ra)
DND
3.9 Prenex normal forms
Replace this by DDN
95
116, II a, to introduce N(j) in right upper formula, DNi/rco TTl^ TT . , _T_ . , . . , ^ 1 A l ' ' " +.o introduce ND
Call this case (i). Case (ii) is
Replace by Lj DDNi/rx«> ^— 116, II a to introduce another x i n right
upper formula,
Again we are left with an ^e-proof of (#£) ^6{j} of lesser rank provided we are using a highest case of 16 above 116. The use of rule II a, as already observed, is without any use of 16, so that if we have a highest case of 16 above 116 then the rank has fallen. Suppose we have DD^f 10,
^
He,
DNN<j>(o. Replace this by
DD
Again we are left with an J^-proof of (Q%) <j>{%\ of lower rank if we are dealing with a highest case of 16 above lie. Call this case (i). Case (ii) is DD<j><j> Dijra)
DD
96
Ch. 3 Predicate calculi
Replace this by
DDcjxjyD^o) lie, DD
Again we are left with an J^-proof of (Q%) ^{j} which is of lower rank. Now consider rule lid*. Suppose we have
Replace this by
2)2^{9/} o)
TT
Again we are left with an ^"^-proof of (Q%) ^{j} of lower rank. Rule l i d ' is dealt with similarly, but may require a change of variable. Suppose we have
where o)' differs from co by having NN placed over a disjunctand. The other case where NN is placed over (E£) <£{£} is impossible because the theorem is in prenex normal form. Replace this by
Again we are left with an J^o-proof of (Qj) {$.} of lower rank. Rule lid'* is dealt with similarly. Suppose we have
) ${§ Nfco
3.9 Prenex normal forms
97
Replace this by
where (E£)${£} is introduced into the branch above the right upper formula in (a) at applications
Nowwehave
JTO^^-DTO^Q^ DD(E£)f{£\N/
where (i?g) 0{£} is in the subsidiary formula. Hence we shall have the same figure with these occurrences of (EE) <£{£} everywhere replaced by
Now add applications of I I a as follows:
Now from (6), (c) and (d) we obtain _
IIa,etc....
r c
J
IIa, etc.
Use this as the right upper part of (6), then finish up as in (b) and we have placed the application of II & above the application of IIeZ*. In doing this we have had to introduce various variants of <£{£}, this is done without using 16 or any applications of II eZ*, d'* that bind variables earlier in the list than the variables we are binding in (a). The effect of this is that in the rank {v,v',..., v^} the first component is unaltered because we have made our alteration without use of 16, and if the variables we are binding is £f®9 then v',..., j/^-1) are unaltered while if® is decreased by one, the other components may be increased. The total result is a reduction in rank.
98
Ch. 3 Predicate calculi
Suppose we have
a*
Replace this by DDX<j>{V}NXG>
DDXttQNfu '
'
'
by the reversibilityof lid'*,
In the reversibility of lid'* we may take the variables £ to be new and distinct from the variables r\. This allows us to apply lie?'*. As in the case of 16 below II d* we have decreased the rank. The reversibility of II cZ'* is performed without use of 16. For completeness we add: Rule lid'* is reversible. By this we mean that if we have an J^'-proof of D(Ag) ^{£} o) then we can find an J^-proof of JD2^{?/} OJ for some 2^{?/}. In the J^-proof-tree of D(A£,)(J){E]G) note the places where corresponding occurrences of (A£,)^{^} are introduced by lid'*. These will be of the form j>s^}ft/ DXtfrf»}<,fi» LEMMA.
J
In the JF^-proof from these places to D(AE>)^>{£)}(0 the part will remain in the subsidiary formulae everywhere. Hence we may replace all these occurrences of (A£) <j){£] by the disjunction of These can enter by IIa, etc., applied to the upper formulae of (/). In this way we obtain an ^"^-proof of instead of one of D(AE,)(f>{^}o). IIa, etc., as before observed, has been done without use of 16, and the only use of lid*, d'* has been on variables later in the list £',..., g(7r) than the variable £, so that the rank of the J^^-proof of (e) has decreased.
3.9 Prenex normal forms
99
To conplete the demonstration of Prop. 9 we make the alterations discussed above starting from the highest available places. Each time the rank is reduced, and as long as the rank is greater than the lowest rank we can always reduce it. This completes the demonstration of the proposition. 3.10
Let
H-disjunctions
(QV)..>W"Vn))n?>-''>^
(!)
be a closed ^^-statement in prenex normal form where the matrix i/r{£', ...,£(7r)} is quantifier-free. Here each Q^ is either A or E. If Q^d) is E then Qe) is called a restricted variable, if Q^e) is A then £(^ is called a general variable. Let there be n' restricted variables in (1) and let there be n" general variables in (1), then n' + n" = n. Form a list of all ordered nrtuplets of natural numbers {*/,..., *>(7r)} ordered by the sum v' + ...+ v^r) and lexicographically for those of equal sum. Take the initial segment consisting of the first K members. Now write down the list
£'
(2)
£(ff) )
b / o • • • ? Z>K
•
J
where the restricted variables in the yth line are x^v'\..., x^n}), {v*',..., v^} being the vih Tr'-tuplet in our list of Tr'-tuplets, and where the general variables in the first line are in order from left to right if £' is general in (1) or
SiT f)
x",..., x^ '
if £' is restricted in (1).
Suppose that exactly the first A restricted variables in line 6\ 6' < 6 are from left to right the same as in line 6, then the general variables are the same from left to right in these two lines up to and including the general variable immediately following the Ath restricted variable. The remaining general variables in line 6 are in order from left to right the next new variables in the alphabetical list x,x',x",... of variables. An example will make this clear. Let (1) be: (Ex') (Ax") (Ex'") (Ex*) (A&) DNpx^x"x^px'x'"x^,
(3)
where p is a three-place predicate. Let K be 12. For greater clarity we write xv instead of x'...' with v superscript primes. 4-2
100
Ch. 3 Predicate calculi
H-scheme of order 12 variables r
g
r
r
x2 ~xx x3 JX±
xx x2
x2
XQ
_x3 x13
x2 _x3
xx x2 _x3
g x3 x±
x8 x15 x5 x9 Xu
x10 ~xx x7 V'X1 _x2 xl± \_x2 xx x12 X,
x
l
xr
xu
line triplet sum line r 1 a? 1 [1, 1, 1] 3 2 [1, 1, 2]i 2 x 5 [1, 2, 1] 4 3 # 11 3 6 12
[2, 1, 1]. [1, 1, 3]" [1, 2, 2] 7 [1, 3, 1] 4 [2, 1, 2] 5 8 [2, 2, 1] 9 [3, 1, 1] [1, 1, 4]" R 10 [1, 2, 3L u
4
x2
g x2 x2 x2 XQ
x2 6 #! x2 7 # x2 5
iCj
8
tf2
9 10 11 12
XQ
x2 XQ #3 x13 #! x2 x. x2
r xY x± x2 xx xx x2 x3 xx x2 xx xx x2
r
xx x2 xx xx x3 x2 x1 x2 x1 x
l
g x3 x4 x5 x7 x8 x9 x10 xxl x12 xu X
15
x3
x1Q
The first twelve triplets have been written down in the prescribed order. In the column headed ' variables' the restricted variables occur in the first, third and fourth places and the general variables in the second and fifth places. The suffices of the restricted variables agree in order from left to right with the members of the ordered triplet in the same row. The general variables are then put in, x% and xz in the first line, x2 and #4 in the second line, since the first line begins with xx and is followed with x2 and the second line begins with xx then the general variable in the second place in the second line is also x2. Generally the second variable, which is a general variable, is x2 whenever the first variable, which is a restricted variable, is xv In the fourth line the second variable (the first general variable in that line) is x6 because this is the first available new variable in the alphabetical list of variables and this is the first time that x2 has occurred in the first place. Generally whenever the first variable is x2 then the second variable is x6. The H-scheme is obtained by writing down line 1 followed by those lines whose initial segment is the same as in line 1 for as long as possible. Thus lines 1, 2, 5 and 11 agree in having initial segments xxx2xx\ These are followed by lines 3, 6 and 12 which agree in having initial segments x1x2x2. This in turn is followed by line 7 which agrees with the above in having initial segments x±x2. The agreement of initial segments is denoted by bracketing. The 17-seheme of order 12 for (1) is the list (2) arranged by bracketing together lines with equal initial segments and ordering lexicographically within the brackets.
3.10 H-disjunctions Write
<&£2&«5
for
qe
for
101
q&°>&°> &»&>&*>
where ($», Qfi, QpQp and #»> are variables in line 6 of (2). We note that the H-disjunction of order 12 namely: 12
(4)
P
n is a tautology, in fact Dq±q12 is a tautology. 2 #0 fails to be a tautology, be0=1
cause we can only have a tautology when fflffiffl is the same as £f)£f )£f) a n d for 1 ^ e,df < 12 this only occurs when (9=1 and #' = 12. Thus the i?-scheme of order 12 for the statement (3) makes the ^-disjunction of order 12 a tautology. Now consider (Ex±) (Ax2) (Ex3, a?4) (Ax5) DNpx5xAx1px1x2x3. (5) Any £T-scheme for the statement (5) fails to make an iZ-disjunction a tautology because Qd) i s alphabetically later than £i0) while ^2e) is alphabetically later than $*\ hence &e)QP£e) fails to agree with g f ) ^ ' ) ^ ' ) for any 6, dr. Consider again the statement (5), an i7-scheme of order K for (5) gives rise to an if-disjunction which fails to be a tautology for any numeral K. A disjunctand of the if-disjunction of (5) is ,3> V4] xVl xVipxVi xa[vi] xVa,
(6)
where (T[v^\ > vx and p\yx, vs, v^\ > vly v3, v±. Consider the 2-valued model e/T in which the individuals are the natural numbers. We can make (6)
take the Jf-value/ by taking: pv1v2v3=f
for
v1
and pvxv2v3 = t
for
v1 > v2,
the values for v1 = v2 are immaterial. Consider the negation of (5) (Axx) (Ex2) (AxSi xA) (Ex5) Kpxhx±xxls[pxxx2xz.
(7)
Thus we can obtain a satisfaction of (7) over the model *A^ We shall demonstrate later the general proposition that if the Hdisjunctions of (1) all fail to be tautologies then there is a satisfaction of the negation of (1) over the 2-valued model in which the individuals are the natural numbers. Note that we lack a method for deciding whether
102
Ch. 3 Predicate calculi
there is a numeral K such that the if-disjunction of order K is a tautology. If we had such a method then the system &c would be decidable, we show later on that the system ^c is undecidable. From the tautology (4) we may obtain an ^^-proof in normal form of the statement (3). Apply universal quantification to the variables x5, x7, x8, xQ, x10, xll9 x12, xu, x15, x16 successively, these variables occur at one place only in (4) so the condition on variables in the rule for universal quantification is satisfied. Delete these variables from the H-scheme of order 12 this leaves: line 1 2 5 11 3 6 12 7 4 8 9 10
x6
The tautology (4) has become 12
(4.1) where q'd is qe if d = 1,2 otherwise q$ is (Ax5) q£{d)&>^Q0)x5, Now apply existential quantification to the fourth variable in every disjunction of (4.1) except q1 and q2 and delete those variables from the J?-scheme of order 12. The disjunction (4.1) becomes 12
(4.2) 0= 1
where qf{ = ql9 & = q2 otherwise q" is {Ex^){Axb)q^^^x^xb.
In
(4.2) ql is the same as q[x so cancel qu, also ql, qfQ, q[2 are the same so cancel ql and ql%, also q% is the same as ql so cancel q%. Thus (4.2) becomes (4.3) D...Dq1q2qlqlq';qlqlql0. 7-times
3.10 if-disjunctions
103
The deleted H -scheme of order 12 has become: line 1
X2
xz
x±z
x2 xz "x x2 x^
3 7 4 9 10
In the disjunction (4.3) the variable x4 occurs only in q2 hence we may apply universal quantification to it, we can then apply existential quantification to the fourth variable in q2 this makes q2 the same as ql so cancel ql and (4.3) becomes: (4.4) 6-times Delete these variables from the ZT-scheme. This is indicated above by a stroke through them and through 5. Now apply existential quantifiers to the third variable in each disjunction of (4.4) except qx and cross these variables out of the ^-scheme. The disjunction (4.4) becomes Z>...Z>
(4.5)
A = 2, 3, 4, 7, 9, 10.
f
In the disjunction (4.5) q 2", q%, q? are the same, so are q± and qfg. Cancel duplicates and we obtain the disjunction (4.6)
Uio and the deleted i/-scheme: r
g
xx
x2
xz
xlz
r
r
g
JUX
JUX
JUZ
L
line i
2 10
In the disjunction (4.6) the variable xls occurs only in q±Q and the variable XQ occurs only in q% hence we may apply universal quantification to them, we can then apply existential quantification to the first variables in q'l
104
Ch. 3 Predicate calculi
and q^Q. Cross out these variables from the deleted //-scheme. q*l and qx0 have now become the same, so omit q'i0. The disjunction (4.6) has become: Vxj) {Ax2) (Exs, x4) (Ax6)
qxxx2xzx4xh,
and the deleted //-scheme is: r
g
r
r
g
X-^
X%
I X^
X^
Xg
L
line JL
2 4
The variable xz occurs free only in qx so we may apply universal quantification to it, we can then apply existential quantification to the third and fourth variables in qv This makes q± the same as q2, so cancel q2. We then obtain the disjunction D(Ex3, x4) (Ax5) qQV^x3x4xb{Exx)
{Ax2) (Exs, x4) (Ax5) qxxx2xzx4xb. (4.7)
In the disjunction (4.7) the variable x2 occurs free only in the first disjunctand, so we may apply universal quantification to it, we can then apply existential quantification to the first variable. This makes the disjunctands the same, cancel one of them, and we are left with (3). The //-scheme for (1) of order K can be written down on a fixed plan for any numeral K, hence the place number of a general variable is uniquely determined by the place number of the superior restricted variables. Thus for the statement (3) the place number of the second variable is uniquely determined by the place number of the first variable, and the place number of the fifth variable is uniquely determined by the triplet of the place numbers of the three superior restricted variables. P R O P . 10. If for some numeral K the H-disjunction of order K of a closed ^Q-statement
The method of demonstration is the same as that given in the worked example. We apply quantifications to the various disjunctands of the //-disjunction of order K and delete the corresponding variables from the //-scheme, and cancel duplicates as they occur. At any stage in the proceedings a variable in the //-scheme is available if it is at the end of its line in a deleted //-scheme and is a restricted variable or is a similarly
3.10 H-disjunctions
105
situated general variable which fails to occur elsewhere in the deleted H-scheme. If there is always an available variable until all the variables are deleted from the H-scheme then we obtain an J^-proof of (j). Suppose that at some stage there fails to be an available variable, then in the deleted if-scheme at that stage the variable at the end of each line is a general variable and each such variable occurs elsewhere in the deleted H-scheme. The lines in the deleted if-scheme are always distinct because identical lines get deleted as soon as they arise by cancelling duplicates. A variable can only occur once as a general variable in a deleted if-scheme but it can occur again as a restricted variable. For instance the variable x2 occurs once as a general variable and six times as a restricted variable in the complete if-scheme of order 12 for the statement (3). The restricted variables which precede a general variable in a line of an if-scheme are alphabetically earlier variables. Thus if there fails to be an available variable then each general variable £ at the end of a line occurs again as a restricted variable in another line which ends in an alphabetically later general variable TJ. In turn there is another general variable alphabetically later than TJ and so on without end. This is absurd because the if-scheme of order K is displayed. Thus there is always an available variable and we may continue to quantify and remove duplicates until we obtain an J^-proof of 0. This demonstrates the proposition. P R O P . 11. If cj) is an ^'c-theorem in prenex normal form then there is a numeral K such that the H-disjunction of order K is a tautology. According to Prop. 9 the J^-proof of (j) can be modified to one in normal form. We then have a tautology (8) 9= 1
where each i/r^ differs from ${£,',..., £(77)} by change of individual variables. From (8) we can obtain 0 by l a , 6, IId, d'. Let 3F'Q be the same as ^'Q except that the individual variables are x*, x*', #*",..., and let ^'c (J 3F'* be the same as the system tF'c except that the variables are those of !F'C and those of 3F'Q . We now change the individual variables in (8) to those of ^'Q by superscripting an asterisk to each variable. In this way let ^(A) become ^(A)* and (8) become (8*). We will give a method of changing the individual variables in (8*) to J ^ variables in such a way that (8*)
106
Ch. 3 Predicate calculi
is changed into part of an //-disjunction. Let (8*) become (9) by this change. Clearly if we change an individual variable at all its occurrences to another one then a tautology remains a tautology. Thus (9) will be a tautology and part of an //-disjunction. Thus there will be a numeral K such that the H-disjunction of order of K is a tautology. Let (j> be
where if ft is zero the initial set of universal quantifiers is absent. Let the variables in ijr'*,..., ^JrOO* be (10) We now replace the ^f-variables by J^-variables in (10) from left to right. We first replace through (8*), (10) the first [i variables in each line of (10) by x',..., x^ respectively, that is ££*, ...,££* are replaced by x' at all their occurrences,..., ^ * , . . . , ffl* a r e aU replaced by x^ at all their -statement, occurrences. Let (8*) then become (8'), it is an ^'Q^^'Q clearly it is a tautology and we can obtain (8) from it. This is because all the other general variables in (8*) are distinct from x\ ...,x^\ By this change (10) becomes (10'). Secondly if the lines V and v" of (10) agree in having the same initial segment and if the next variable is different and is a general variable then we may alter this general variable in one of the lines so that they are both the same. Suppose these general variables are $*>* and $?A>* so that the segments £*, ...,£* and £*, ...,£<*>* are the same. In passing from (8') to
3.10 ^-disjunctions
107
initial segments of the two lines are the same) can take place immediately after the generalization of £££A)*, any such variable is distinct from £*£A)*, again because the initial segments are the same. Having made these modifications so that we generalize £^A)* immediately after (except for permutations) generalizing £*?A)* we now everywhere replace £$?A) * by £
108
Ch. 3 Predicate calculi
variable £p[0] is the same as a general variable in line p[p[0]], say p2[d], and so on. Since the scheme (10) is displayed we must have for some fi' < ju,"'.
Thus we have a general variable in line p^Xff] is equal to a restricted variable (which is available) in line p^'^ld] and so it is impossible to generalize the former until the latter has been restricted, but this restricted variable is inferior to a general variable which must be generalized first, this in turn is the same as a restricted (available) variable in line pfl"'1'2[6] which must be restricted first, and so on until, a general variable in line ps^'[d] is the same as a restricted (available) variable in line p^'id], but this variable is £a itself. Thus finally, we are to restrict a restricted variable £a in line a before we generalize an inferior general variable. This is absurd. Thus there is always an available restricted variable distinct from any general variable. Replace the alphabetically earliest such restricted variable (which is an 3F'* -variable) at all its occurrences by the first as yet unused 3F'Qvariable. This leaves the general variables unaffected. Thus an available variable (which is an J^^f-variable) can always be replaced by an SF'G- variable without upsetting our build-up of an //-scheme by renaming of general variables. Finally each 3F'£-variable is replaced by an tF'cvariable in such a way that the resulting disjunction is a part disjunction of an if-disjunction, because we have chosen the general variables so that this should be so. This completes the demonstration of Prop. 11. 3.11 Validity and satisfaction An J^-statement
3.11 Validity and satisfaction
109
(d) similarly for many-place predicate variables, (iii) We replace the free individual variables by numerals. (iv) We replace a part (A!;) i]r{£) of $ by t if and only if i/r{v} is replaced by t for v = 0,1,2,... otherwise we replace {AE) ^{£} b y / . (v) We replace Dff b y / and Dft, Dtf, Dtt by t. (vi) We replace Nt b y / and Nf by t. (vii) $ reduces to t however the replacements (ii) (iii) are carried out. We lack a test for general validity. This will be demonstrated in Ch. 7. A closed J^-statement is said to be satisfiable over J^ if it can be shown to reduce to t for at least one replacement under (ii), (iii). For example consider: DN(A^DN^}x{QDN{A^^{Q(A^x{^ ( n ) We show that it is impossible for (11) to reduce to/when the above process is carried out. If (11) reduces t o / then N(AE)DNi/r{!;}x{£} and DN(AE) i/r{Q (Ag) x{£} must both reduce t o / . In order that this happen DN\lr{v}x{v} a n ( i ^ M must both reduce to t for each numeral v, and x{v] must reduce to / for at least one numeral, say A:. Then DNJ/T{K} X{K} reduces to t hence ^{K} must reduce t o / , this is absurd. Thus (11) always reduces to t no matter how the replacements are carried out. Consider (3), give t, / to jpX/jiV in any manner, if pv'v"v'" is t for some set v', v'\ v'" of numerals then (3) is t, iipv'v"v'" is always/then (3) is t. 12. A closed ^c-statement is an ^c-theorem if and only if it is generally valid over JV. The J^-axioms are generally valid over JV*. The J^-rules preserve general validity over JV*. Thus J^-theorems are generally valid over Jf. Now suppose that the J^-statement
110
Ch. 3 Predicate calculi
the value / for some assignment of values t, f to each TTV' ... y(A) for each predicate variable which occurs in <j) and for each set of arguments which occurs in an H-disjunction. If a set of arguments fails to occur in any if-disjunction then we give nv'..MX) the value t. Let i^K be an assignment of values t, f to each TTV' ... y(A) which occurs in HK, the JBT-disjunction of order K. irK will only give values to TTV'' ...v^ for those argument sets v' ...v^ which occur in HK. Let J(K be the set of assignments i^K. Now HK is a part disjunction of He for K < 6, hence one i^e will contain all the values given by at least one i^K. We can express this by saying that at least one irK can be extended to become a ^ or that a *Ve with domain of definition restricted to that of i^K becomes a i^K. Each *JlK contains at least one ^ Hence there is a valuation i^ which defines TTV'.. Mx) for each argument set which occurs in some HK and which gives the value / to (j). Now this valuation says that for any values given to the restricted variables there are values that can be given to the general variables, which values depend on the values given to the superior restricted variables, in such a manner that ^ takes the value/. Then N<j) takes the value t and for any values given to the general variables in N
dity over J^K is decidable, we need only replace (J57£) ${£} by 2 0{#} 0=0 K
(and (A £)${!;} by n
and evaluate by truth-tables. COR. (ii). An ^-statement which is valid over JVK for each natural number K but which fails to be valid over JV can be found. Consider the conjunction P of the following J^-statements: {Ax) Npxx (Ax, x', x") CK
3.11 Validity and satisfaction
111
It is clear that P fails to be satisfiable over any JVK. Hence NP is valid over every JVK. But P is satisfiable over Jf (let pxx' be x < x'), hence NP fails to be valid over J^. Another example is the negation of the conjunction Q of the following ^-statements: (Ex)(Ax')Npx'x {Ax, x', x", x'") GKKpxx"px'x"pxulxpx'"x' (Ax) (Ex1) pxx'. Again it is clear that Q fails to be satisfiable over any ^ fiable over JV (let pxx' be Sx = x').
but is satis-
3.12 Independence 13. The symbols, axioms and rules of J ^ are independent. We have to show that N, D, E, p^x\ #(A) are independent. Clearly p is independent otherwise ^-theorems would be without occurrences of p, similarly for x and the other variables. But note that if we omit p we get an equivalent system, similarly for x and the other variables. The only closed J^-formulae of type 00 formed from D, E, X and variables are: PROP.
\p.p, *kp.E(\x.p), \p.E(kx.DpE(kx'.p)),
Xp.Dpp, Xp.DpE(Xx.p), etc.
but these all using A-rule (i) give J3A00, where A stands for any one of the above formula. Hence if we took A as a definition of N then BNtfxfi Whence
DpNp
* 16 P and this is absurd because an ^"o-theorem must contain D. Thus the symbol N is independent. The only closed ^-formulae of type 000 without occurrences of D are: Xpp'.
112
Ch. 3 Predicate calculi
E is independent because the only J^-formula of type 0(01) that we can construct from N, D and variables is Xp0L. (j>, where (j) is of type o and fails to contain E, but this fails to be closed and so violates the conditions for a definition. The demonstration that the ^-axioms are independent is the same as for &>c. The ^,-rules are independent. First, rule l a is independent, because if we omit rule la then we are unable to obtain the J^-theorem DNpp. Any J^>-proof of DNpp fails to use IId,e, because once E enters an .^-proof then it remains in that #^-proof from that place till the base. Thus any J^-proof of DNpp will be a ^ c -proof possibly using 16. Any J^-proof of DNpp will fail to use 116, c because it is without occurrence of NN or of ND, hence an J^-proof of DNpp which omits la proceeds from the axiom DpNp using 16, II a only. We are unable to apply 16 to DpNp so we can only apply II a obtaining DtfiDpNp where (j) is an ^cstatement built up from D, N and p only, other variables and E must be absent because if they were introduced into the ,^c-proof by II a then they would remain in the ^ o -proof from that place to the base. We can only use 16 on Dcj)DpNp if
D(Ex)pxNpx and
DN(Ex)px(Ex)px.
For the rule 16 we note that if we are denied the use of rule 16 then we are unable to obtain: (Ex') (Ax") (Ex" Any J^-proof of (3) which fails to use 16 also fails to use IIa, 6, c (these
3.12 Independence
113
lengthen the formula) and must start with the axiom px'x"xmNxfx"xr" it must then use la, lie, lid twice l i e and lastly lid. But it is impossible to generalize only the first occurrence from the left of x'". 3.13 Consistency 14. The system &c is model consistent. We show that fFc has a model with a sole individual a and two elements t,f of which t is designated a n d / is undesignated. It has the constants N, D, E. N and D obey the same rules as in the model J(c for 8PC. The PROP.
rulefor2?is
It is easily verified that these rules give a model for ^c. 15. The system !FC is consistent with respect to negation. We have to show that if ^ is a closed ^-statement then at least one of
3.14 ^Q with functors The system J ^ with functors or constant individuals or both can be dealt with as the system ZFC. We have the additional rule: v free in
^'
a b s e n t
from
114
Ch. 3 Predicate calculi
This is called the rule of substitution. Here a is a term of type i and is free in ^{a}, i.e. if g is a variable which is free in a then corresponding occurrences of £ are free in
Now suppose that
of atomic statements or negations of atomic statements, from the J^>theorem C(J)DOJX we obtain by Prop. 3, Cor. (iv) <j)r, ...,ftK)\r#r>Da)x, thus is a ^"-theorem. 3, COR. (V). Modus Ponens can be eliminated from a theory whose axioms are disjunctions of atomic statements or negations of atomic statements. Suppose that we have 6',..., ^Y^' Doj^Jr and ^',...,ftK))r&'DNftx then we
PROP.
c
c
have C$DG)X as above whence we have 0', ...,^K)\-^DO)X,
by Prop. 3,
3.15 Theories
115
Cor. (iv). A theory whose axioms contain free variables would normally be based on &"'c rather than on ^c. The rule for substitution of variables merely converts an axiom scheme (as used in ^c) into a set of particular axioms. A theory whose axioms are disjunctions of atomic statements or negations of atomic statements is called a theory in free disjunctive form, free variables are allowed. COR. (vi). If jJ.
is a rule in a theory in free disjunctive form then
^— *,
where co is subsidiary.
We have ^Sffi taking N<j) for to. Thus Cffl, now is a ^-theorem, whence by Modus Ponens twice we get DNtfxo if we have DNi/ro). But Modus Ponens can be eliminated. COR. (vii). The deduction theorem holds in a theory whose special rules are without restrictions on variables. The demonstration is the same as before, the extra rules of the theory behave just like the e^c-rules other than l i e . COR. (viii). Modus Ponens can be eliminated from a theory without axioms and whose special rules are without restrictions on variables or iutroductions of E. We proceed as in Prop. 4. The case when $ is atomic and DN
3.16 Many-sorted predicate calculi A K-sorted classical predicate calculus of the first order is formed from the symbols: type
xc • XL{K)
C : L(K)
pOL>,...,pol(K)
OL', ..., OL(K)
name
individual variable of the first sort individual variable of the /cth sort one-place predicate variable
116
Ch. 3 Predicate calculi
type name es as Poi'i > Voi'i"> - • • 9 PoiWiW tyP shown two-place predicate variables generally Poe\..eW> where e^, types as shown A-place predicate variables for 1 ^ 6 ^ A, is one of *',..., *M E',..., EM O(OL'), ..., O(OL{K)) existential quantifiers of types shown X abstraction symbol ' generating symbol N oo negation symbol D ooo disjunction symbol ( ) parentheses The axioms are T.N.D. for all atomic statements. The rules are those of fFc with restriction and generalization for each type of individual variable, denoted by l i d ' , ...,IId{K\ He', ...,IIe (lc) . We denote the /csorted classical predicate calculus by ^CK. The situation is just as if in $FC we labelled the variables as x^-K+d\ 1 ^ Q ^ K, and stated lid, e separately for each 6,1 < 6 ^ A:. But the main difference is in the argument places of the predicates. Let J r ( J ) be !FC plus constant one-place predicates 8\ ...,$ (AC) and additional axioms DNS'^S'Z,..., DNSMgSMg. We give a method for translating J^-statements into J ^ - s t a t e m e n t s in such a way that 3^CK' theorems translate into jF^-theorems. The translation of an ^CKstatement is obtained as follows: f
(a) <j) is atomic, say P01{B')^L^)X%I). (A)
K n S^x^'^+^p^e')
. .a^&o), the translation is l(0 (A))^-/+^...^^
(A)
+«.
v=l
If two of the individual variables are the same then we omit an occurrence of S followed by that variable, (6) ^ is Ni/r, its translation is Ni/r\ where ^r' is the translation of ^ , (c) ^ is J D ^ X its translation is Di/r'x', where i/r', %' are the translations oii/r\X respectively. (d) $ is {E0g0)f{!;e)9 its translation is (Eg') f'{£'}, translation of
where ^'{g'} is the
3.16 Many-sorted predicate calculi
117
(e) (j)r is the translation of
if and only if its transla-
tion into 3~ is a 3~ -theorem, (ii) If 3~Kis consistent with respect to negation then so is 3T. (iii) If ^ is consistent with respect to negation then so is 2TK. (iv) There is an effective method whereby given a 3~K-proof of a &~Kstatement (j) we can find a & -proof of the translation of
3.18 The predicate calculus with equality and functors
125
Let $ be the translation of
the translation of this is We have
8<*
Prop ? 8? M R
* revers lbl htyofll6'. For the rule I(ii)
D
the translation is
We have
D(E^Ka{^{Q(o
D(E£,)Ka{£}b{g}a) t
as desired. For the substitution rule this becomes
(
D(E£)Ka{® ${
Prop
126
Ch. 3 Predicate calculi
We have the tautology CxCDcfri/rDKxfift, hence we have
5>
)> M p
as desired. Thus the ^-rules and the J^^-rules translate into or derived ^"-rules, and the result follows. 3.19 Elimination of axiom schemes P R O P . 20. A theory ^ without functors and based on I^c and with axiom schemes can be replaced by an equivalent 2-sorted theory £f with a terminating sequence of axioms. We are thinking of axiom schemes involving arbitrary statements; for instance Mathematical Induction. Axiom schemes which contain arbitrary terms but are without arbitrary statements can be replaced by axioms provided we add the rule of substitution 11/. To demonstrate the proposition we introduce two new symbols: s and e of types at and 0(01) t respectively, where 01 is the type of a second sort of variable. The construction will be clearer if the reader interprets
Jx for N(Ex', x") (x = sx'x") read (x is an individual', sxx' is the ordered pair xx', sxsx'x" is the ordered triplet xx'x", etc. The first sort of variable is a variable for sequences of individuals. We frequently write (xeX) instead of exX and read it as 'x has the property X\ where X is a variable of the second sort, we call them properties. We now translate the theory 2T into a two-sorted theory £f as follows. 7r£ by 7T& by by
(£e77-), where n is atomic and of type 01, (sggen), where n is atomic and of type ott, («£«£TOT), etc.
3.19 Elimination of axiom schemes
127
We shall frequently write j ^ for sgsg's... sg^K)g^8K\ £nt) is the sequence made up from the members of the sequence £ in their proper order followed by the members of the sequence t) in their proper order. We frequently omit the suffix 8K in %^K) and just write £. We now give some definitions: D12 D13 D14 D15 D16 D17 D18 D19
(£e3 U 3') (£eS n 3') eZ x V)
(geSE)
KJglg^g'eldS)
complement
for for for for for
K(£eE)(£eS)
for
(rieS)
for
iL/£(£n£/ngeS)
D(ieS)(ieE)
(E£)(KJ£'{£"&•£))
union intersection direct product domain inverse permutation identification.
We now show that by means of these definitions a statement ^{£',...,
£^SK)}
becomes
(jr^eA),
where A is constructed from atomic properties by means of complement, union, intersection, domain, direct product, inverse, permutation and identification. Our second sort of variable is a variable for predicates or properties so we must ensure that we can perform operations on these variables corresponding to the operations we can perform on predicates by means of logical connectives. To do this we adopt the axioms: (1) D20
B(geE)N(geE)
(V)
r
X =X
(EX){Ag)B(geX)N(geE).
for (Ag) B(£eX) (geX')
(sxyel)
for x — y,
then (1') becomes (EX) (X = E), i.e. S is a thing of the second sort. We want similar axioms for (2)-(8) below. (2) (3) (4) (5) (6) (')
(8)
£jJ^J\.J c, J c, (£ £
£ EKjnV^Hi) r^J\.o (^ J t, (
128
Ch. 3 Predicate calculi
We want ~ U fl x 2 Cnv1 Cnv2 Id as operations on properties satisfying the above axioms, then we can show that we have J3(£eA) ${£} for any statement §J and some property A. We proceed by formula induction on 0. (a) (j) is atomic, say fi is TTE,' ... gK\ replace as described above. (b) (j) is N(j>' and the result holds for 0', so $'{£,',..., QSK)} has been replaced by fec^A'), we replace N
(3(/^A").
We first replace (t)(A)eA') by (fe^eA^) and (j^eA") by (E^eA^) where £',..., £ ( ^ are the variables 9/',..., 9/(A) and £',..., £(^ in some order, say in alphabetical order without repetitions. We have for individuals £, £' and sequences V:
Any permutation of a sequence can be brought about by repeatedly interchanging a pair of consecutive members. Thus a sequence of natural numbers can be brought into a sequence in order of magnitude by repeatedly interchanging consecutive members. Repeated application of Cnv1 will bring any given member of a sequence to the front then applications of Cnv2 will interchange the first two members then repeated applications of Cnv1 will bring all the members back to their original places except that two consecutive members have been interchanged. Thus by repeated applications of Cnv1 and Cnv2 we can bring any sequence into a given permutation. Thus by applications of Cnv1 and Cnv2 we can bring 7i'...^ into gr>mmmgm where 6' < 6" <...<&# if 80™ < 6^ apply Cnv1 until g^(7r)> is at the end then apply the direct product d^-Sd^times. This will give us the sequence
Now apply repeatedly until we have filled in all the gaps. Then apply Cnvx repeatedly until % is in front. The new variables introduced can be
3.19 Elimination of axiom schemes
129
K
relabelled so that the whole sequence is £'... Q \ If there were any duplicates in 7]'... rffi then permute until one is in front and its duplicate at the rear then apply Id to eliminate the rear one. Finally replace ^ by (ite^e/i!" U Aiv). This completes case (c). (d) is (E£)^{g,t)} and we have replaced $${£, §} by !?t)eE, we replace {Eg) 0'{£, t)} by tye^E. This completes the description of the replacement. For the full development we require some more axioms: (9) xey Xex
XeY
(10) {Ax,y){Ez){z = sxy) (11) GKJxJyLxy (12) CKKLzu(x = szv) (y = suw) BLxyLvw (13) GK(x = szu) (y = suz)Lxy (14)
(AX)(Ax,y)CK(xeX)(yeX)Lxy,
where Lxy expresses that the sequences x and y are of equal length. For instance the last of these axioms says that two sequences having the same property have the same length. This should suffice to complete the demonstration of Prop. 20. In any system the rules are normally rule schemes. Our method allows us to replace rule schemes by single rules provided we have a substitution rule. For instance the / ^ - r u l e s become: xeU' u l u f u P ~xeU'\)Y\)X\)U' TT
LLa
xeX —==—=; xeX\jY xe@(X u U)
__lib
xeX[jX\jU xeX[)U~xeX U U xeY u U —— ; xeX[)Y[)U xe2X
TT lie
xeX U U —= ; xeX u U
U U
la, ...,IIc give the rules of Boolean Algebra, if we add IId, e we get a Boolean Algebra with a projection operation superimposed on it. In the 2-sorted theory © quantification is allowed over one sort of individual only. A predicate calculus of the second order is obtained from a predicate calculus of the first order by adding quantifiers over predicate variables. It can be shown that a theory ST based on the pure predicate calculus of the second order with axiom schemes fails to be replaceable by a 2-sorted theory without axiom schemes.
130
Ch. 3 Predicate calculi
3.20 Special cases of the decision problem We are as yet unable to show that J ^ has an unsolvable decision problem, namely the problem of finding a uniform method of deciding of any ^cstatement whether it is an ^-theorem, because we are as yet without a precise definition of what we mean by 'solvable'. Later on we shall identify 'solvable' by 'calculable by a Turing machine'. We shall then show that there fails to be a Turing machine which will tell us whether an c^-statement is an ^^-theorem. We have to postpone the undecidability of !FC (which is entirely due to I b) until we have given the theory of Turing machines. Meanwhile we can give some methods for deciding certain types of J^-statements. When we come to the theory of Turing machines it will be seen that these methods can be set up on a suitable Turing machine. P R O P . 21. The monadic !FC is solvable. More explicitly an ^-statement containing only one-place predicates is an ^c-theorem if and only if it is valid in a domain of 2K elements, where K is the number of one-place predicates in the statement. Let ^ be a monadic ^-statement containing exactly K distinct oneplace predicates
22. The ^-statement
is an fFc-theorem if and only if it is valid over a domain with A elements (one element if A = 0). Thus it is solvable.
3.20 Special cases of the decision problem
131
First consider (Erf,..., T/W)
Thus if (Erf,..., 7/W)
If the disjunction 2 ^ w is a tautology then ^ is an ^-theorem. We have:
By taking w to be 2 * ( ^ (A) , where the asterisk indicates the omission of fte), and repeating and then discarding duplicates by 16 we obtain
but the upper formula is an ^r-theorem. Apply l i e ' repeatedly to the lower formula and we obtain the ^ o -theorem i/r. K
If jr is valid over a A-element domain then the disjunction 2
takes the value t no matter how the values t, f are given to its atomic formulae n^... £(d(v)), 1 < 6', ...,6^v) ^ A because one disjunctand will then take the value t, namely that disjunctand that gives the satisfying set £', ...,£(/M). Thus the disjunctand is a tautology and so jr is an !FCtheorem. Thus if jr is valid over a A-element domain then jr is an !FCtheorem. If jr is an .^,-theorem then it is valid over any *WK in particular it is valid over a A-element domain. This completes the demonstration of the proposition. The decision problem for #~o can be framed in several ways: (i) To decide if an J^-statement is valid over JV', this is the same as to decide if an ^.-statement is an ^.-theorem. 5-2
132
Ch. 3 Predicate calculi
(ii) To find the natural numbers K such that an ^-statement is valid over JfK and to decide if it is valid over Jf. (iii) To decide if an .^-statement
COR. (i) A closed &c-statement in prenex normal form with a prefix of the form can be decided as regards validity. We first reduce the problem to the case of a closed ^-statement in prenex normal form with a prefix of the form (Aij, ijf) (E£'9..., £(A)). We give the demonstration for the case ju, = A = 1 and where the matrix is built up from a single binary predicate variable n. (EO(AV,V')(EQ{n;£,V,V',Q.
(1)
The case when A > 1 is dealt with similarly, the case when ji > 1 follows by repetition of the case fi = 1. We show that (1) is satisfiable if and only if
{AT,, r,') m,
r, r, n
KKK^-,
'?,V,V', Q 4,^- 'e,v/e,
n
#r;T,T,9.n#»;T,T,T,n (2) is satisfiable, where the variable g has been eliminated by writing TT'V for ngv, n"v for nut; and a for TT££, a is a propositional variable. This we have indicated by writing '£' instead of £. We have, as it were, taken the resolved form as far as the first restricted variable is concerned. First suppose that (1) is satisfiable over «^#, where J( is JfK or JV*. Then we show that (2) is similarly satisfiable. L e t / b e a function over c^#2 which satisfies (1), if g is in J( then for arbitrary ij,ijf in ^# we have for suitable
; g, ij, g, H
3.20 Special cases of the decision problem
133
all take the value t. Hence the conjunction
(EC, r, r, n KKKW, Z, v, n1, Q M Z, % z, n w> £> z, v, n
v,z,z,z,n (3) also takes the value t. Now define/'f for f£,v,f"v forfvg, a for/££ so a is t or/. Put these in (3) and we have a satisfaction of (2) over JK, satisfying functions being We now show that if (2) is satisfied over ^#, where JK is JfK or df, then (1) is satisfied over a domain obtained from Jl by adjoining a new element. Suppose then that the functions/,/',/" and a satisfy (2) over J( and that £ = g[v,v'], £' = g'[vl T = g'Wl C = * give the values of £, £', ^", ^7// in the model over J(. It is easily seen, by distribution of quantifiers, that £, £', £" and Q" depend only on the variables shown. Now let a be a new element and define
g[aoc] = a.
Then g is defined over <J( u {oc}. The conjunction (3) takes the value whence so do: (EQ ji{/, a, ?/, 9/', £} = ^ for arbitrary 7/, r/f in ^ , /, a, ?/, a, £'} = t oc, a, V', £"} = t
for arbitrary 7/ in JK, for arbitrary if in uT,
but this says that (1) is satisfied over ^ u {oc} as desired. We now show that there is an effective method of deciding whether a closed ^.-statement, (A£9 £') (EV', ...9^)^{n; £, g',^, . . . , ^ } (0) containing exactly one predicate variable and that binary, can be satisfied. The method with more predicates and with various place numbers is similar. A table of order v for a binary logical function / is a v x v array of fs and/'s giving the value offA/i for 1 ^ A, ju, < v. A table T^ of order (ju, + 2) for a binary logical function/, is said to satisfy (fi{n; £, £',y f ,...,T/^} if $5{/;l,2,3,...,(/* + 2)} = *. 7
(4)
Denote by [I //c/c'], 1 ^ K, K' < /£, that table of order 2 which is obtained
134
Ch. 3 Predicate calculi
from the table T^ when we extract the 2 x 2 principal minor (AC, K'), i.e. the intersection of the K, K' rows and columns. We show that N. and S.C. that (0) be satisfied is that there is a nonempty set S of tables of order ([i + 2) which satisfy (4) and which have the following properties: (A) If TQ is a table of 2 and 1 ^ K, K' < [i + 2, K =f= K', then there is a table T of S such that [T/12] = [TOJKK']. (B) If To is a table of S then there is a table 21' of 2 such that [T'/l] = [T0/l] and T' = [77//1134 (ji + 2)]. i.e., the first two rows are columns are the same. (C) If Tl9 T2 are tables of S then there is a table I7 of S such that If (0) is satisfied then clearly there is a set S of tables of order which satisfy (4) and which have the properties (A), (B) and (C). If (0) is satisfiable then it is satisfiable over Jf and there will be a table over JV x ./K giving the value of//c/c' for all 1 ^ K, K' . In this satisfaction we replace £, £' by any two members of Jf and there will be fi other members of JV for if,..., 9/(^, call these 3,..., (ji + 2), where £, £' are 1,2; there will be other /i members of JV*, if £, £' are K, K', 1 ^ AC, K' ^ /^ + 2 which gives (A). Similarly (B) if we replace £, £' by the same element ofJ^, namely 1. If we replace £, % by d, 6' and again by zr, TT' then there will be a satisfaction when we replace £, £' by #, TT which gives us (C). Let ^ be a table over ^ which satisfies (0) then we choose any two members of N for £, £', say K, K', and there will be fi other members of JV for ?/', ...,9/ ( ^, say y', ...,v ( ^, then the (/^ + 2) minor of &" formed from the K, K\ V', ..., v^ rows and columns is a table I7 which satisfies (4). The set S of tables which satisfy (4) and are of order (/i + 2) is bounded because there are at most 2(/*+2)2 of them and these can be written down and tested to see if they satisfy (4). We now show that if there is a non-empty set S of tables of order (/£ + 2) which satisfy (4) and have properties (A), (B) and (C) then (0) is satisfiable over JK It is an effective process to decide if a set 2 of tables of order (ju, + 2) which satisfy (4) also satisfy (A), (B) and (C). Let
co[v9 K] = |(/c 2 + 2KV + v2 - 3K - v + 2), this is the place number of the ordered pair
3.20 Special cases of the decision problem
135
Let 2 be a set of tables of order (fi + 2) which satisfy (4) and (A), (B) and (C). Denote by S(TV T2) a table of 2 such that
such a table exists by (C). By some numbering of all possible tables we can easily arrange that S(Tl9 T2) is uniquely defined. Let 2X be the set of all tables of order 1 which are minors of the tables of 2. Let 2 2 be the set of all tables of order 2 which are minors of tables of 2. Then 2X has at most 2 members and 2 2 has at most 16 members. We now define a set Z0,O = 0,1,2, ...,of tables of order 1 + 0./i as follows: Zo is an arbitrary member of 2X. If P2 is a member of 2 2 we denote by H(P2) a table of 2 such that [H(P2)/1, 2] = P2, by (A) there is such a table. When ZQ has been determined in such a way that \_Z0\K, K'] is in 2 2 , 1 ^ K, K' < 1 + O.JLC, then we determine Z0+1 as follows : (for Zo this condition becomes [ZQ/1, 1] is in 2 2 , this is satisfied by (B)). Let 0 = OJ[K, K'], then [Ze+1jl, 2,..., 1 + 0./i] = Z6, i.e. Z0+1is the same as Ze whenever Z0 is defined. [Z0+1/K, K', 2 + 0./I, ..., l + (d+ l)./i] = E[Z0/K, K'], this follows from (A) and the hypothesis that \Z0\K, K'] is in 2 2 . Altogether this gives the values for Z6+1 at the points (y, v') where v,v' = K,K',2 + 0./i,...,l
+ (0+l).fi
or
v,v' =
1,2,...,1+0./
We also want this is possible by (C), here A = 0,1,2,...,1 + 0. fi except K, K',
This gives the values of Z0+1 at the points for 1 < v,v' ^ l + (0+l)./i. It remains to show that [Ze+1jv, v'] is in 2 2 for 1 ^ v, v' < 1 + (0 + 1) ./i, so that we can proceed as above to determine Z0+2 from Z0+1. Clearly this already holds for 1 < v < 1+0.ju, and 2 + 0 < v' ^ l + (0+l)./i by our construction. It also holds for 2 + O.ji ^ v ^ l + (0+l)./i because then [Z0+1/v, vf] = [£([Z0IK, K'])/V, V'] which is in 2 2 . Thus it holds generally.
136
Ch. 3 Predicate calculi
We now see that (0) is satisfied over Jf, for let AC, K' be a pair of natural numbers, 6 = G)[K9K']. Then 0{/;#c, Kf,2 + d./i,..., 1 + (# + l)./i} = t, where / is the function such that /[>,*/] is given by
E([Z0JK9 K*]).
This completes the demonstration of the proposition. The same procedure can be applied to
or we can treat this as (Ag, g') (EV',...,rjW)Ktfn;
£,?', ...,^>) (Dng'g'Nng'g'),
to which it is equivalent. But if we apply the procedure to
then it fails, because though we can find conditions analogous to (A), (B) and (C) yet some are necessary and others are sufficient. We are unable to find necessary and sufficient conditions. 3.21 The reduction problem Since we are unable to solve the decision problem for 3FC it is of some interest to find types of J^-statements to which any J^-statement is equivalent as regards satisfiability or as regards validity. We have just shown that any J^-statement of the form (0) can be decided as regards satisfiability, in this paragraph we show that any J^-statement is equivalent as regards satisfiability to an effectively constructible J^-statement of similar form to (0) but with three universal quantifiers at the head of the prefix instead of only two. We thus have achieved some finality in the decision problem as regards satisfaction in the sense that statements of the form (0) are solvable as regards satisfaction, but those like (0) but with 3 universal quantifiers are unsolvable, (in fact equivalent to the general case). Again if the binary predicate is replaced by a unary predicate then we again have solvability. Thus we have gone as far as we can in this direction. There are a variety of other directions in which we could try to get similar results. These come about by considering different types of prefix.
3.21 The reduction problem
137
P R O P . 24. A closed ^c statement $ is satisfiable over J^ if and only if a certain binary 3Fc-statement of the form
(Ag, g', g") {Ei,', ...,ifi»W{p; g, g', g',v', ->V(li)}
(5)
is satisfiable over JV, where p stands for nf, ...,7T^K)9 where 7Tr,...,7T^K) are binary predicates and there is an effective method of finding ft given <j>. (i). If (f> is a closed ^c-statement then there is an effective process for finding a closed binary ^-statement ft such that $ is an £Fc-theorem if and only if ft is an ^-theorem. A binary ^Q -statement is an .^-statement which contains only binary and singularly predicates. Let n', ...,7r(A) be exactly all the predicate variables which occur in the J^rstatement 0. Let n^ have /i^ argument places. Let /i = Max \ji\ ..., /^(A)]. Let n[,..., n[X) be new and distinct oneplace predicate variables, let 7r'2, ...,7r^) be new and distinct two-place predicate variables. Let v be a new individual variable. We form ty from $ by replacing uP>g...gW9f> by LEMMA
(Av)K n TT^^vn^v
(6)
6' = l
throughout (p. Clearly i/r is binary. If (j> is an J^-theorem then cj> is generally valid over Jf. If we replace n'2,..., n^6^, 7r[e) by logical functions over Jf then (6) gives a logical function over JV but if in
138
Ch. 3 Predicate calculi
ing. We define one-place logical functions q', ...,g(A) and two-place logical functions r\ ...,r ( ^ over JVP as follows:
v' = t if and only if vx = v'% (1 < 6 < A). Since ^ is generally valid over Jfp (since «/f^ is enumerated) then it is satisfied over «#> by replacing n^ by g(0) and n^ by r(^. Let \jr become \Jr after this substitution. In particular (6) becomes:
(Av)C U r^^vq^v.
(7)
0'i
Let AA/£ be that logical function over J^ which has the value t if and only if A = [i. Then (7) becomes (Av',...,itf*(n>)C II
0'1 0'=1
this has the same truth-value as p<®£'... g(/*(0)), hence
An ^ o -statement is said to be in Skolem V-normal form if it is closed and in prenex normal form and has a prefix in which each existential quantifier preceeds each universal quantifier, i.e. if it is of the form
where the matrix
Let <j) be a closed binary J^-statement, let $' be a prenex normal form of it. Then ^ is satisfiable if and only if
3.21 The reduction problem 139
J^-theorem. If $' begins with an existential quantifier then conjunct to it (AQDNTT&T^ and reduce the result to prenex normal form beginning with the universal quantifier (AQ. Call the result $". Then B<j>'<j)" is an J^-theorem so $' is satisfiable if and only if $" is satisfiable. Thus 0 is satisfiable if and only if <j)rr is satisfiable. Let <j)" be (Q) % where (Q) is (A1c1)(Et)1)(Qf) and (Qf) is (A$2)(Et)2)(Q"), here (Q), (Qf), (Q") denote sequences of quantifiers. Write: <»[j,tj] for ( ^ ^ n ^ ^ n ^ ^ w ^ . 0=1
(8)
0'=1
Also write (E£)(o0[£, £,t)] for (8). Here n\ ...,TT
Let ^ i v be
(AQ (Eh) (E\)2) (E£) (Q")Ka>0 K', Sl , where the sets of variables j 1 ? t)l9 j , t), J2? ^2 a r e distinct. If ^ i v is satisfiable then so is (Q) #, i.e. $" is satisfiable. We have B(j)"'(])iw is an ^-theorem, because 0 i v is a prenex normal form of <j)'". Also
because we have the J^^-theorems: C(At)) CG>[
whence by Modus Ponens: C(Ai) (At)) COJ [jr, whence, using which is
C(j)'"(Q)x
(10)
as desired, hence C
140
Ch. 3 Predicate calculi
such that (Q) x takes the value t when the predicates in x are replaced by the logical functions just mentioned. Call the set of natural numbers vr, ...,v(A), K',...,KW a T-set, since (Q)x is satisfiable then there are T-sets. Enumerate the ordered (A +/^)-tuplets and let the 0th be <#l>--->#U+/*)>-
Define logical functions £l±[v, K], ..., Q(A+/t)[>, K] such t h a t an[69Onl = t
for
I^TT^A, (A+/0
if and only if 0 X ,..., du+/l) is a T-set. Then (EQ JJ Qw[£, 0 J takes the value 7T=1
t for T-sets otherwise the value/. Thus for these logical functions 0 l v takes the value t because (A%)(Et))(EQ fl o)[£,g®] then takes the 0=1
value taken by (Q)x
so
^ na ^ ^0)[%^](Q)X takes the value f. Thus
K(Ai)(Et))oj[^t)](Ai){At))Co)[i,t)](Q)x takes the value t. Thus 0 iv is satisfiable as desired. Repeat the process until all the existential quantifiers succeed all the universal quantifiers. This completes the demonstration of the lemma. LEMMA, (iii). If (pis a closed binary ^-statement then there is an effective method of finding a closed binary ^-statement i]r in Sholem V-normal form such that $ is generally valid if and only if \jr is generally valid. The demonstration is similar to that of lemma (ii). These are similar lemmas omitting the word 'binary'. L E M M A (iv). If (j) is a closed binary ^-statement in Skolem S-normal form then there is an effective method of finding a closed binary !Fc-statement i/r in Skolem S-normal form and containing exactly three universal quantifiers such that $ is satisfiable if and only if \[r is satisfiable.
Let 0 be (4^...,gW)(^.---.^ ) )^{^--^£ ( A ) »^---,^ ) } where $' is binary and is without quantifiers and
..., ZCn'%%'",
(AC, C, £"') KOK7r%'CTr'CC'K%"'OKn"CC7r"CC'ICC',
(12) (13)
3.21 The reduction problem
141
where £', £" are new. Let the predicate variables which occur in $' be nm9 ...,7T^K+2\ then these are binary. We show that if
F
and (Ag)m, (A£,V)(CU7)Cn£n7i, (A£,V,QCI&iCnKmi£ and (A£tV,QCI&iChrgn&i,
for all singularity and binary predicates which occur in $'" including the equality relation itself. The prenex normal form of
25.f^4 binary ^-statement
{A?, £",£'")(W, ...,*«)#*', -Mx\ g', ...,y^} in Skolem 8-normal form is satisfiable if and only if a binary <^c-statement (A £', £", £") (Ey',..., rf») ir{n; £', £", £", y'9..., ^ \ also in Skolem 8-normal t Prop. 25 follows closely the proof given by Kalmar and Suranyi (1947), though the symbolism has been changed to that of the present author. It is reproduced by permission of the publisher, the Association for Symbolic Logic.
142
Ch. 3 Predicate calculi
form and containing only a single binary predicate, is satisfiable. Also there is an effective method for finding xjr given (j).
Let
(A?, £", nW, ...,^ ) )^K....,»«; £', i", Z",y', ..
be the given binary ^ - s t a t e m e n t in Skolem ^-normal form, where
(15)
where (j) is what ^ becomes when n', ...,TT^ are replaced by p', ...,p (A) respectively and the propositional connectives are replaced by their respective truth-functions.
(17)
where T denotes the logical function which arises from
which consists of the natural numbers J¥', triads of the forms (y{6'\ v«n9 0> and <^'>, v^, 1> and2/, ...,^ A ) . The triads (v^, v^"\ 0> will play the role of the pairs (v^'\ it0"*} while the triads (y^\ v^e"\ 1> serve merely to express the coincidence of the first and second components of pairs. Thusfor
T
= <„>',()>
3.21 The reduction problem
143
we shall define Qrcr = t if and only if v' = v", QCTT = t if and only if K' = AC". v is the first component of r is expressed by QVT, K is the second component of r is expressed by QTK. p' is characterized as the only element of J for which Qp'p' = t, p" is characterized as the only element of J for which NQp"p' = t, pW is characterized as the only element of J for which Qp^pW-V = t, 3 ^ 6 < A. The triads r = (y\ K', 0) are characterized by QrpW = t, we distinguish between the natural numbers and the triads (y", K", 1} = cr by Q^V = * and Qp'cr = / , then the triads or can be characterized by' o* is different from p', ...,#(A) and o-is different from a triad (y'9 K',0) and Q^'cr = / ' . Now if (15) holds then, for numerical functions />(iv), ...,/o(/*+3),
Let
^" denote the set of triads (J/, /c', 0>, ^ denote the set of triads (vf, K',1), & denote the set of {p\ ..., #(A)}, ,/K* denote the set of natural numbers,
We define a binary logical function Q over f by the following table, in this Q has the same truth-value at an entry in the table as the statement standing at that entry.
Qxx'
x' = K xf = X' = {K\ K", 1 >
x' = pW 1 ^ 6f < A [K(d =t= 2)(6>r = 1)
x = pi®
x =x
X*X
D
\ [K(0 = d'+l)(d'* 1 <6>
1) x 3F x v = K' v" = K x 4= a
v — K' */ = /c'
144
Ch. 3 Predicate calculi
Consulting the table we see that (a2) (a6) (6) (c)
Qxx = t if and only if (iff) x = p\ NQxp' = t iff x = p", QxpW-» = t iff x = pW (3 < 6 < A), QxpM = t iff xe^, KKNQxxNQxpWQp'x = t iff
(d) KQxp'Y[NQxpWNQp'x
=t
iff
0= 2
Further supposing that r = (yr, v", 0), a = then (e) (/) (g) (h) (i)
Qvr = t QW = ^ QTV = t Qcrv = t QTO- = t
iff iff iff iff iff
(j)
Q(TT = t
iff
v = v\ V = K\
v" = v, /c" = p, / = K', ACr/ = ^ .
Also s u p p o s i n g t h a t rx =
if
(Z) for
KKKQT1orQT2orQ(rT1Qo'r2
1 ^ 6 < A,
/[^;2/',2/(A)]
Write
y[>;*/',...,*/
for for
= tf t h e n
rx = r2,
KKNQxxNQxy^Qy'x' KQxy' \{
KNQxy^NQy'x.
e=2
Then for arbitrary elements a/, x", x'" of ^ / and if y' = p',...,y(x) we have (a)
by
KKKQy'y'NQy"y'
U
(ae),
= p{X)
1 < 6> < A,
Qy{e)yV-1)CDDKQx'x'Qx"x"KNQx'y'NQx"y'
6= 3
2 KQxfy^-^Qxffy^-^KBQxfxf"Qx"xmBQx'"xfQx'"xfr.
(18)
This says that if #' = a" = #<*>, 1 < 6 ^ A, then BQx'x'"Qx"x"' and BQxmx'Qx'"x\ and that ^ = ^ , 1 < 6 ^ A. (6) By(c),(d),(/)and(A), CKI[x';y'9
y™] I[x"; y\ y™] (Eu) KK7[u;
y',...,
y™] Qx'uQux".
(19)
3.21 The reduction problem
145
This says that if x', X"EJV then there is a triad (xr, x", 1 >. (c) By (c), (b) (d), (e), (/), (»), (fir), (A) and (j)
CKCKQx'x"Qx'x'"Qx"x'"CKQx"x'Qx'"x'Qx'"x". (20) This says that iix'ejV, x"e$~, x'" e <%, y' = p',..., y™ = p™ then if x' is the first component of x" and also is the first component of x'" then x" and x'" have the samefirstcomponents, and similarly for the second components. (d) By (b), (d) and (k), CK... KQx'^Qx"^y{x'";y',
...,yw]
6 times
Qx'x'"Qx"xr"Qx"lx'Qx"lx" n BQy^x'Qy^x".
(21)
0=1
This says that if x\ xfre^ and o^^e^/" and if x', x" have the same first and the same second components then Bp^x^p^xlxl, 1 < 6 ^ A, where Finally by (c), (6), (e), (g), (I) and the definition of T the fact that for x',x\xmejr and x* = p^x'xW, ...,^+ 3 ) = j W / / (15) holds, 3 C
0 = 4:
n
7 [
^)
y 9
^
( i 7 % i s B m myU
)
0 1
fi+3
n 0,0=1
(22)
This says that for any #', a;", X'^JV there are a;iv,..., x(^+%/T such that if ^ ^ r =
(23)
We have just shown that if (14) can be satisfied then so can (23), namely by the logical function Q given by the table. Now suppose that (23) can be satisfied over JV*, we wish to show that (14) can also be satisfied over JVI
146
Ch. 3 Predicate calculi
By hypothesis we can find a binary logical function over JV*, say Q, and ternary numerical functions over JV
such that on replacing q by Q, y(0) u
tftox'x"^,
by
by a)xrx"x'\ "x'"
(4
in (23) and #', x'\ x'" by ^', ^ , v1" respectively we obtain the value t. Consequently we have CKI[x';y',x'x"x'";y',...,y™] QX'O)X'X"X'"Q
(20') and (21') are obtained by writing p^x', x"x'" for xf-e\ 1 ^ d < 3 p+3
3
/t+3
e=i
0=1
e,6'=i
K n I[p^x'x"x'";y', t/W]0 n I&»;y',ym]K
n
QpWx'x"x'"Te, e, x'x"x'"QTe> g. x'x"x'"pWx'x"x'"W{Q ru^"
KKQT6 e,x'x"x"Y»
;y',...,
y™, (22")
V3,,«^1
for any x',x",x'"ejV and for y«» = ^x'x"x'", 1 < 6> < A. Choose a fixed member of Jr, say 1. Write a(B) for ^ 1 1 1 . Then by (18) for any x',x",x'"eJr,
Qx'x'x"x'"x'x'x"x'",
(23')
NQx"x'xV'x'x'x"x"',
(23")
'/f-iVa;"/
(23^)
(3 ^ 6 < A)
CKQv'v'Qv"v"KBQv'v"lQv"v'"BQv'"v'Qv"'v",
and
CKNQv'x'v'v"v"lNQv"x'v'v"v"'KBQv'v'"Qv"v"lBQv"'v'Qvl"v", e l)
l
6 1)
l
GKQvY - v'v"v" Qv'Y - v'v"v'"KBQv'v'"Qv"v" BQv'"v'Qv'"v".
(24') (24")
3.21 The reduction problem
147
From (23<*>), 1 ^ 6 ^ A, we obtain for v' = v" = v"' = 1, Qa'a',
(25')
NQa"a\
(25")
QaWaP-*,
3 < d < A.
(25<«)
w
Now we show that for any v\ v", v eJV* and 1 < 6 ^ A,
<9 = 1 in (24') put
^WV"
for
i/,
a'
for
/',
^
for
^,
detach (23') and (25'), and we get (26J) and (262). For 6> = 2 in (26^) put X"v'v"vw for i; and use (23") we get %WW, In (24") put
/WV"
(27") for
v\
a"
for
J/',
j;
for
^w,
and replace I W V " by a' in virtue of (26^), then detach (27") and we are left with (26J) and (262). Supposing that we have shown (262(^~1)) for some 3 ^ 6 ^ A then in (262(^-1)) put F W V " for v use the resulting equivalence in (23(^) and we get
in (24<*>) p u t
a«» v
for
v\
for
v\
for
v"\
replace ^-^v'vnvm by a^"1* in virtue of (262^(-1)) then detach (25<*>) and (27^) and we are left with (26f >) and (26^). Consequently we may replace y^v'v"^" by a(^ for 1 ^ 6 ^ A whenever it occurs as an argument of Q. In particular (18'), (20'), (21'), (22'), (23") hold for any v', v\ v^eJf and y^ = dd\ 1 ^ 6 < A.
148
Ch. 3 Predicate calculi
Let JV' denote the set of elements of N for which I\y\ a', a(A)] holds. In virtue of (22'), Jf' has a member. Choose a fixed element b o{J^\ say b = p l v l l l (see (22')). We define predicates^?', ...,^ (A) overJ^' by pWVK = Qddh1>2VKb
(1 ^ 6 ^ A, v9 KeJT').
We now show that these satisfy (14) over^T'. Let y,/ce^'and^ = r1}2VKb. By (22") we have ^(A) ^ = ft)> (28)
LEMMA.
Qvp
(v = v',d=l),
(29)
Q^
(/c = v",0' = 2).
(30)
For any element qe^V' for which Qqa,
(28')
Qvq,
(29') (30')
, we have BQa^qp^vKfor v, mjV', 1 s? 6 < A, i.e. Indeed, for r = U>VKI we have by (19') y[r;a',... ,a«],
BQa^qQa^p. (28")
Q^r,
(29")
Qr/c.
(30")
In (20') put v for )/, ^ for v", r for i/" detach l[v; a', oW], (28) (28"), (29), (29") and we obtain ^ (31) In (20) put v for v', q for v", r for v'" detach i[v; a', a
(31'")
3.21 The reduction problem
149
Finally in (21') put p for x\ q for x\ r for x"' detach (28), (28'), (28"), (31), (30'), (31"), (31"') and we obtain which is the lemma. Now let v1', v", v1" be elements of J^\ and let
(this holds also for 1 ^ 6 ^ 3 by definition). By (28') v*, ..., i^+3) also belong to JV'. Let te e, = Td^v'v"v"\ 1 ^ 0, 6' ^ ji + 3. By (28") we obtain,
i
'
^
(32)
>,
(33)
\
(34)
e
{
A,
...,1^^}.
(35)
The lemma now gives from (32), (33), (34) BQa<%,t r: p<W>">,
1 ^ 6 < A, 1 ^ 6', 6" < /i + 3.
Thus by the definition of T we infer from (35)
i.e. the binary logical functions p',..., p^ over ^T' satisfy (14). This completes the demonstration of the proposition. 3.22
Method of semantic tableaux
We shall show in Ch. 7 that ^Q is undecidable, that is to say that any proposed method of deciding whether an J^-statement is an fFctheorem or otherwise will fail to give a result in some cases. We now give a method for deciding whether an J^-statement is generally valid or whether its negation can be satisfied. Of course the method will fail to give a result in some cases. The method consists in trying to find a counter example by the use of semantic tableaux. We try to make a given J^statement
150
Ch. 3 Predicate calculi
Consider an J^-statement (j>. We form two columns and call one an f-column and the other one a t-column, each column will be called opposite to the other and they will be said to correspond. We place
3.22 Method of semantic tableaux
151
f. The method always comes to a definite conclusion, if the first or the last case holds, but fails if the second case arises. If all the atomic formulae are singulary we can obtain a counter example, if that case arises, over a bounded domain. But if there are binary predicates then we have the general case (see Prop. 25), and unbounded domains may be required and the functions introduced in dealing with a general statement in an /-column may introduce such complications that we are unable to tell whether certain atomic formulae occur in both of two opposite corresponding columns. Consider GKN(Ex) Kpxp'x(Ax) Cp"xpxN(Ex) Kp'x
(i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix)
KN(Ex) Kpxp'x(Ax) Gp"xpx N(Ex)Kpxp'x (Ax) Gp"xpx (Ex)Kp'xp"x Cp"apa we have substituted for a variable
Kp'ap"af p'a p"a Np"a 1
:
(x) pa
2
:
(0) (xi) N(Ex) Kprxp"x (xii) (Ex) Kpxp'x (xiii) Kpap'a we have substituted for a
variable
(xiv) (xvi)
pa
1 : (xv) p'a
2
p"a
Below entry (xiii) the/-column splits into two columns indicating the two ways in which (xiii) can be made/. The second of these two columns can be closed because (xv) agrees with (vii). Now there is only one/-column and the f-column splits indicating the two ways in which (v) can be made t. This gives us the pair of opposite columns labelled 1, 1 and the pair of opposite columns labelled 2, 1. Both these columns can be closed because the last member of t-2,1 (x) agrees with (xiv) and the last member of/-1,1 (xvi) agrees with (viii). Thus our attempt to make (0)/has failed. Hence (0) is generally valid and so is an ^^-theorem. From the tableau we can find an ^,-proof of (0), as follows: We start with the hypotheses: (ii), (iii), (iv) and (vi)x from these we deduce pa as shown in the tableau but we had already produced p 'a so that we get Kpap'a and hence (xii), thus we have: N(xii), (iii), (iv),
152
Ch. 3 Predicate calculi
where (vi)^ is (vi) with x instead of a. By the Deduction Theorem we have (iii),(iv)
\-C(vi)xCN(xii)(xii),
(iii), (iv) f-C(iv) (xii), Prop. 6 (iii) (iii) (iii) \-CN(xii)N(iv), hC(iii) CiV(xii) N(iv), which is equivalent to (0). Now consider CK(Ex) KpxNp'x(Ex) Kp'xNp"x{Ex) KpxNp"x
(0)
Our tableau is: t (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (xv)
(Ex)KpxNp'x (Ex) Kp'xNp"x KpaNp'a pa Np'a Kp'bNp"b p'b Np"b p"a
f (ix) (x) (xi) (xii)
(0) (Ex)KpxNp"x KpxNp"x KpaNp"a pa 1 : (xiii) Np"a 2 : (xiv) KpbNp"b
(xvi) pb 21 : (xvii) Np"b 22 (xviii) p'a (xix) p"b
In this case the tableau terminates without the tableau being closed, and we are able to read off a counter example, namely the domain consists of two elements a and b, pa, p"a and p'b are t while pb, p'a and p"b are/. We take pa to be t because as far as making the entry (ix)/we need only take (xiii)/independently of the value of (xii), but to make (i) and (ii) t we must have (xii), i.e. (iv) t. In the ^-column (i) gives rise to (iii), (iv) and (v) and (ii) gives rise to (vi), (vii) and (viii). In the/-column (xii) and (xiii) are the two alternative ways in which (xi) can be made/ of these (xii) closes with (iv). We still have (xiv) and this gives rise to two columns 21 and 22, of these 22 closes with (viii). This column consists of (ix), (x), (xiii), (xiv), (xvii).
3.22 Method of semantic tableaux
153
But column 1 consisting of (ix), (x), (xiii), (xiv), (xvi) can be continued with (xviii) and (xix) from (v) and (viii) respectively, and this column is still open, we could go on with KpcNp"c but it is pointless to do so. The final ^-column consists of (i), (ii), (iii), (iv), (v), (vi), (vii), (viii), (xv) and thefinal/-column consists of (0), (ix), (x), (xi), (xiii), (xiv), (xvi), (xviii), (xix). We read off the counter example by the values given for the predicates p,p',p" at a, 6, in these two final columns. Now consider: C(Ax)(Ex')KDpxx'p'xx'Dpx'xp'xx(Ex) {Ax^BKpxxp'x'x'Kpxx'p'xx' (0) t Dpxap'xa Dpaxp'xx
(0)
1 pxb 11 pxx paa pab pbb
f Kpxxp'bb Kpxbp'xb 2p'xb 12 p'bb 21 pxx pab pbb
paa pbb
22 p'bb p'ab
p'ab p'bb
This gives 4/-columns each of which correspond to the single ^-column at present consisting of only two entries. Now the ^-column similarly splits up into 4 columns, thus 1 pxa LI pax paa pba pab
I2p'xx p'aa
2 p xa 21 pax paa
paa pba
pab
p'bb
p'aa
p'ba
22 p'xx p'bb p'ba p'aa
Each of these four ^-columns corresponds to each of the four/-columns. To make (0) take the value / we require that a pair of corresponding columns be open. We have entered in the columns all the results of substitution over a two-element domain. Thus we require to test 16 cases, viz. any ^-column with any/-column, we tabulate the results:
154
Ch. 3 Predicate calculi /-ll /-12 /-21 /-22 /-ll /-12 /-21 /-22 /-ll /-12 /-21 /-22 /-ll /-12 /-21 /-22
t-ll t-ll t-ll t-ll t-12 t-12 t-12 t-12 t-2l t-2l t-2l t-2l t-22 t-22 t-22 t-22
closed by closed by closed by open closed by closed by closed by closed by closed by closed by closed by open open closed by closed by closed by
paa pab paa paa p'bb paa p'bb paa pab paa p'bb pfbb p'bb p'bb
Thus we have found three open pairs Each of these gives a way of making (0)/over a two-element domain. Namely /-22 and *-ll t paa pab pba
/ p'bb p'ab
S-22 and t-2l
/-ll t
p'bb p'ab
p'bb p'ab p'ba
t paa
p'ba
/
pab
and *-22 / paa pbb pab
p'aa
the values omitted can be chosen arbitrarily.
3.23 An application of the method of semantic tableaux As an application of the use of semantic tableaux we demonstrate the following proposition: 26. Let Cifiijr be a closed ^-theorem where neither N
3.23 An application of the method of semantic tableaux
155
both are constructed from atomic statements as described in the K
n 0=1'
enunciation of the proposition. Suppose also that (f>0 is of the form
while i/rQ is of the form 2 ^o6) a n d that $f} is of the form 2 $0% while 0 = 1
7T = 1
$ f * is of the form n ^ofl where 0O^ and i/rff], are atomic statements or 77=1
negations of atomic statements. Now form the semantic tableau for C
156
Ch. 3 Predicate calculi
Thus each side of the final tableau has l ( ^x ... x A<^x 1<">X ... x i6v) columns which correspond in pairs, we continue these columns with substitutions on the free variables, then if we make all the atoms in one / - column/and all the atoms in the corresponding ^-column t then we shall make C
3.23 An application of the method of semantic tableaux
157
Similarly the entry t can be replaced by Dy'Ny', this in an/-column becomes y' with Ny' below it; this gives y' in that column and in the corresponding column, hence the column is closed as before. Thus columns which were originally closed by a's or /?'s are now closed entirely by y's. We have the closed tableau for Cx* ^o which is an ^.-statement free of quantifiers. We call Xo the sentential power of (p. Similarly if we interchange the treatment of the two sides we get the tableau for C(j)*x*X* is the same in both cases, the structure is the same in both cases, if one divides so does the other and for the same reason. We want to put the quantifiers back so as to get the quantificational power of
Note that if x* i s / then N<j) is an ^-theorem because the tableau for Ncp is closed on its own, and if Xo is t then \jr is an J^-theorem because then the tableau for i/r is closed on its own. Note t h a t / and t can be omitted from x because in the formation of x we can replace KSf and KfS by / , DSt and DtS by t, DSf and DfS by 8, lastly KSt and KtS by S. S o / and t will disappear unless the final result is / or t, but we have discarded this case. Note also that only those y's are used in x which occur positively in both (j) and ^r or occur negatively in both (j> and i/r. Where an atom is said to occur positively in co if it is on the same side of the tableau as a), otherwise it is said to occur negatively in o). Thus only some of the y's are used in x* We restrict a variable which replaces a term which originated from treatment of (f> because terms which at their first occurrence arise from 0 do so by treatment of existential quantifiers. We generalize a variable which originated from treatment of T/T because terms which at their first occurrence arise from i/r do so by treatment of universal quantifiers. Let us return to case (ii) where we were unable to decide whether the tableau closed or was open. A tableau might show that certain compositions of functions were the same, i.e. that f...gx = l...mx for all x, we
158
Ch. 3 Predicate calculi
only need on the t side an .^-statement which has this as an interpretation. By substitutions for variables we can arrive at great complications as regards composition of functions. A composition of singulary functions can be written as a word, i.e. a string of letters or primed letters from a given set called the alphabet, then if we have/. ..gx = Z.. .mx for all x a consecutive part/...
CKKDNy'yDay'DKyNay'DDKPyKyy'KNyp. The tableau is given on facing page. There are 64 pairs of corresponding columns, viz. v, v' where v, v' — 1,... ,8. In the next table we show that the tableau is closed by giving the value of the associated ^-statement.
In many cases corresponding columns can be closed in several different ways. Thus 55' can be closed in three different ways, viz. by oc and NOL being in column 5 which gives/, by y and Ny being in column 5' which gives us t, and lastly by y being in both columns. We have now associated an ^-statement with each column. As we go up the tableau these columns unite by treatment of the *-side. Accord-
3.23 An application of the method of semantic tableaux
159
KKDNy'yDay'DKyNay' DNy'y Day' DKyNay' Ny' a
7
a
KyNa
/
7
KyNa
7'
7 Na
Not 1
2
7
KyNa
r'
4
7'
7 Na
7 Na
3
KyNa
5
8
6
7
/?
Ny
n
V
8'
DDKfiyKyy'KNyfi Kfiy Kyy' KNyfi
7 Ny
r
7
y i
/?
12'
Ny
Ny 4'
5'
Y
ing to our rules for finding x we take the disjunction of the ^-statements associated with lines: 8. lvf and 2v\ 3 / and 4J/, 5v' and 6 / , 7^' and 8J/, The results we show in the following table: ll'/,',7 21/, * 31^,y 41'*, y
12'/,y 22/ 32'y 42'y
13'/,/ 23/, y' 33y 43y
14'/,yf 24/, y' 34'y' 44y
I5y,tfy 25/, t 35/,t,y 45't,y
16'/,y 26/ 36'y 46'y
lVf,t,y,y',Dyy' 18/,y,/ 2f7J,t,y/ 28/, y' 3Vf,t,y,y',Dyy' 3&'y,y'9Dyy' 47% y, y', Dyy' 48'y, y', Z>yy'
Here ^ v' indicates the disjunction of (2v— 1) v' and (2p) p'. In some cases
there are several different ways in which the disjunction can be taken. Thus 47' can give either y, y', Dyy' or t according as to which entry in 77' and 87' we use. The next stage is the disjunction of the pairs (2v— 1) v' and (2y) v'. The result is put down in the next table: 12/,yl3/,y' 14/,y' 15/,t,y 16/,y 17/,t,y,y',Dyy' 26'y 2Tt,y,y\Dyy' 28'y,y',Dyy' 22'y 23'y' 24'y' 25'r,y
160
Ch. 3 Predicate calculi
Again we take the disjunction of the two members in each column. l't,y
2'y 3'y',
4'y' 5%y9 6'y Tt,y,y',Dyy'
%'y,y
We now form the conjunctions of consecutive pairs corresponding to the junction of columns on the/-side. This gives
y; y'; KyDyy',y;
KyDyy',Ky'Dyy',Kyy',y,y',Dyy'.
We have to do this a second time, getting
Kyy;
y, KyDyy', KKyy'Byy', Ky'Dyy', Kyy1
and lastly we require the conjunction of these, however we do this we get something equivalent to Kyy'; this then is the sentential power. We can easily verify that
CKyy'D*K/3yKyy'KNyj3 and CKZDNy'yDay'DKyNay'Kyy' are tautologies. 3.24
Resolved ^Q
The resolved !FC or the resolved I^c is obtained as follows: 7} is a new symbol of type t(ot), the axioms and rules are the same as for ?FC or I^c which ever is the case, except that the symbol E is omitted and the rules lid, e are discarded and the following rule replaces them: H where we have written D21 nM&
for
Here f\^(j>{^) plays the part of a thing which has the property ( It is a term of type 1, it can only be used after {^\ can be used. The theorems of H^c are the ^-theorems in resolved form. By this we mean that if
3.24 Resolved ^c
161
tion for these new functions. The system IHFC is without quantifiers but it is sometimes convenient to have a system using both lid, e and rule H, we then have a system like SFO with functors and constant individuals. A related system is the following system based on I^c\ it is together with the symbol i of type t(ot), we write as usual D22
h
m
for
We have the rule
is read 'the thing with the property (X£^{£})'. The formula is of type i and can be used only after the premisses of J have been proved. The resulting system, which again fails to be a formal system, is denoted by J^Q. From J we obtain r,
From the hypotheses of J' we obtain by !FC (Ag, n) CKK<j>{g)
whence from J and (Eg) K
whence
162
Ch. 3 Predicate calculi
this is called an e-term. £ is called the bound variable of the e-term }. The symbol E and the rules lid, e are omitted and replaced by E where co is subsidiary and can be omitted, this is called the e-rule or the E-rule, we say that this application of the e-rule belongs to the e-term
§ fails to occur free in o) and in If we allow the rule of substitution then rule E' becomes a case of the rule of substitution and so can be dispensed with. If we write D24 then rules E and E' translate respectively into rules lid, e. If we write D25 then we have a definition for the universal quantifier. Thus in the systems E3FC and EISFQ we have definitions for both quantifiers, if we have the rule of substitution then we need only rule E. These two systems are formal because the e-term e^{£} can be used without restriction. The e-term, €
£0{£}> i s r e a d 'the most ^-like thing'; then
un-0-like thing is §J-like', if this is the case then everything is ^-like. The term e^{£} can be used when {A^)N(j){^ is an J^-theorem. For instance if r is a variable for a rational number then er(r2 = 2) is read' the rational number whose square is most nearly equal to 2' but we have (er(r2 = 2))2 =f= 2, so that e(r2 = 2) might be any rational number (see Ch. 12). It is readily seen that if ^ is an i?J^-theorem without occurrences of any e-term then the jE/«^-proof of
3.24 Resolved &c
163
particular term for which 0 holds. If however
Suppose that we have j^\) {
U(p\oCj 0)
in an iJ/J^-proof-tree, where £ fails to
occur free in D0{Tt} o). In this tree replace all corresponding occurrences of £ by a. Axioms remain axioms, applications of rules remain applications of the same rules or repetitions and we are left with an EI^FCproof-tree of D${(X}G). In other words, substitutions may be pushed back to the axioms. Note in particular that rule E is preserved. 27. Rules 116, c, e are reversible in 116 is reversible. We have to show PROP.
Consider the iJ/J^-proof-tree of DNDfrfi'a) and note the places at which a related occurrence oiNDficj)" enters this tree. This will occur at IIa, 6 only. If a related occurrence NDtfi^cfil enters the tree at I I a then replace it by Nfa and if at II6 then strike out the upper right formula and the branch above it and replace ND^>[ (j>[ byN
If in this we strike out related occurrences o£D—<j>" then it becomes
164
Ch. 3 Predicate calculi
this fails to be a case of any rule. We shall have to call the occurrence of ND—(j>" in the c-term related to the occurrence of ND—$" in the upper formula. By this device we preserve the proof. This completes this case. l i e is reversible. We have to show —^ . *. Dj In the U/J^-proof-tree oiDNN^o) consider the corresponding or related occurrences of NN<j>. These will be introduced at II a, c only, omit NN at each of these occurrences throughout the proof-tree and we are left with a tree with Drfxo at its base. Applications of rules remain applications of the same rules or repetitions except that rule E can become upset if a related occurrence of NN
DN
±
Again in the U/J^-proof-tree oiDN(Eg) ${g\ o) consider the corresponding or related occurrences of N(Eg) <£{£} and in these omit (Eg) or the part related to it. These occurrences can only be introduced into the tree at II a, e, when we omit (Eg) applications of rules remain applications of the same rules except that II a, e can become repetitions, axioms remain axioms, but rule E can be upset if (Eg) occurs in the main formula of that rule. We use the same device as before, and the result follows. 28. Modus Ponens is a derived rule o We have to show
PROP.
D(j)(j) DN<j>x
*' Prom the 2?/J^-proofs of the upper formulae we require to find proof of the lower formula. We proceed as in Prop. 4, by formula induction on the cut formula. There are only three cases, namely when the cut formula is atomic, a disjunction or a negation. The last two cases are dealt with exactly as before. The first case is also dealt with as before but the theorem induction
3.24 Resolved ^c
165
requires the additional consideration of rule E. Thus suppose that in our theorem induction we have a case of rule E:
by the reversibility of l i e we obtain D
Here {e^N${£}} is 7ra'*...a(/c)s|:, where a(A)* is the result of replacing all free occurrences of £ in a(A) by a, where a is egNnot'... a(A). Thus 0{£} contains fewer occurrences of e than
i
I{U)
'
where a contains e-terms but J3 is without e-terms. If o) and 0{I\} are without c-terms then Dcj){^}(i) is without e-terms. Consider the highest such case of rule I(ii), follow the related occurrences oflocfi up the tree until we come to the highest case of/a'/?' where a! contains e-terms but /?' is without them. Then in the tree above this each IyS must be such that y, S both contain e or neither do. Equations of the type of IyS just mentioned are incapable of producing equations of the type of Iccfl above. Hence the equation IyS must arise from an axiom T.N.D., but this introduces NlyS as well. The descend-
166
Ch. 3 Predicate calculi
ants of NlyS will be in the subsidiary formula of the application of I (ii) which eliminated the e-term This leaves an e-term in the lower formula of that application of I(ii) which will have to be eliminated lower down the tree. This, in turn, in the same manner, will introduce another eterm, which in turn will have to be eliminated still further down the tree and so on without end. This is absurd. Hence the deduction must be without e-terms altogether. Prop. 29 enables us to show the consistency with respect to negation of theories based on EI^C whose axioms are without the e-symbol and which are verifiable. That is to say: a closed statement of the theory without €-terms can be decided using a suitable definition of truth for the theory. If the axioms ^',..., <j>^ of the theory are valid then so is every theorem (j) of the theory which lacks e-terms because a proof of
(£ = ?) for
K(AQBpttp£V(AQBp&p£V.
= is a symbol of type on, we use the more familiar way of writing equalities. More generally equality can always be defined in a theory 2T
3.25 The systems 3&!FC
167
which contains a terminating sequence of predicates, but is without functions. We define (£ = v) for the conjunction of
taken over all the predicates contained in the theory ^ . It is more usual to define D26
(£ = ?,) for
(AQBpZZp&i,
(g * V)
for
and take C(AQ Bp^pv^E, = v) as an axiom, or to define D26'
(£ = ?) for
{AQBp&pr,^
(£ * V)
for
N(£ = V),
and take O(AQBp^p^v (£ = v) as an axiom. In either case if (£ = TJ) we have {AQ Bp^ptyj and (AQBpg^pw^ so that by 3FC pat; may be replaced by par/ and#£a by pyoc in any ^ - s t a t e ment, in other words we have -a) (a (a-a)
aand nd
*>(
Dcj>{fi}oj
by regularity <j) being built up from the sole predicate p, this is the same axiom and rule as for the equality symbol / . D27
(asyff)
for
(A£)Cp£ap£fi9
(ac/?)
for
K(oc c /?) (a # fi).
Read (a c y?) as 'a is contained in /?'. The things which stand in the relation p to oc are less extensive than the things which stand in the relation pto/3. <= is called the inclusion symbol. D28
Smoi for
{A£)Np£a.
'oc is without ^-predecessors', or oc is empty. We can regard the predicate p as being an ordering relation. If pocfi we shall call oc an immediate p-predecessor to /? and /? an immediate psuccessor to oc. We can then identify fi with the class of its immediate ppredecessors, and so identify the relation# with the membership relation. This amounts to reading pocfl as 'oc stands in the relation p to /?' or as c a is a member of the class of things which stand in the relation ptofi'. Now /? is of type i and (Agpg/3) is of type (01). Let A be a symbol of type t(ot) then A(A£p£/?) is of type £ and this term is uniquely fixed by fi. Thus
] 68
Ch. 3 Predicate calculi
if we identity /? with A(Xgp£/?) then we have formalized the above informal exposition. To make this identification we require (/? = A(X|p£/?)) more fully:
BptfpgVkZpgfi)
and J3p/?gp*(Xgp&ff) £
In conformity with previous uses of (X£^{£}) we define D29 If we have a set ^ of things oc, /?,..., and are given a truth-value for each oipococ, pa/3, pfiot, pfifi,..., then for fixed /? we can form the class of things oc for which pafi istf.This gives us another set, say Sf*, whose members are classes of the members of £?, and there is a (1-1)-correspondence between £f and 5^*. Every member of ^ * is a class of members of SP and there is a (1-1)-correspondence between £f and ^ * whereby oc of S? corresponds to the member a* of £?*, where a* is the class of members of £f which stand in the relation^ to oc, call this class (£pz). The relation poc/3 translates into p*oc*fi* or (oc*e/3*), read 'a* is a member of the class /?*'. With this interpretation ^ * is a set of classes whose members are also classes whose members in turn are again classes, and so on, stopping, if at all, only at the null or empty class which is without members. The things a*, /?*, as defined, are subclasses of £f*, and between these subclasses we define the relation p* which we interpret as the membership relation. The members of ^ * are classes of members of «$^*. Classes can be combined by certain operations to yield new classes. Thus two classes can combine to form their union or their intersection. From a single class we form its complement, and so on. Thus we can extend ^ * if necessary so as to contain these and other combinations. We define D30
(#0/?) for
gKpgocplzfl intersection,
D31
(<*U/?) for
^Dpfop^p
union,
D32
oc
^Np^oc
complement.
for
We shall require as axioms or theorems
Bpg(a<\fi)Kp&p£P, Bp£(a{jp)Dpfrpgp,
Bp&Npga.
This suggests that we have in general but this leads to an absurdity. Take NpijT} for ${>)}, then the suggested
3.25 The systems 0^^c
169
axiom becomes: Bpr/^Np^NpT/i], now substitute the term £Np£t; of type i for y and we get: Bp(£Npgi;)(£Npgg)Np(£Npgg)(£Npgli) which is absurd. If instead we take: C(EQpy^Bpy^{^}<j){rj}, and proceed as before we merely arrive at (AQ Np(£Npg£) £, i.e. the term £Np£l; fails to stand in the relation^ to anything. It will be a maximal element in the ordering given by the relation p. D33 @£ for D 34 *g£ for iV(3£,
£ is a proper class.
Thus if we adjoin the symbolA we obtain constant terms of type t of two kinds, sets and proper classes. Let us then change the notation for variables of type t and take them to be: X, X', X",.... Then we define another sort of variable, called set variables, by relativization, thus: D35
(Ex)
D36
$
These give
(Ax)
(EX) K(SX
for
XK(SX(f>{X}. (AX)C&X
We shall want some information as to whether a given class is a set or is a proper class. For convenience we use the letters x, y, z, u, v, w with subscripts or superscripts as variables for sets and X, Y, Z, U, V, W with subscripts or superscripts as variables for classes. We define: D 37
{X, Y}
for
uD(u = X)(u=
Y), pair class,
T> 38 SX
for
il(Ev) KpuvpvX,
D 39 PX
for
H(u c X),
power class,
D40
{X}
for
{X,X},
unit class,
D41
(X,Yy
for
{{X}, {X, F}},
ordered pair,
D42
(X, Y,Z) for (X, (Y,Z)), ordered triplet, etc., in vtuplets, (xr, ...,x^v)}, pointed brackets are put back by association to the right,
D43
V
for &(u — u),
universal class,
D 44
A
for H(u 4= u),
null class,
union class,
170
Ch. 3 Predicate calculi
D45 D46
for p(UV)X, U stands in the relation X to F. 2 u)(Eu, v) K (w = (u, v}) puXpv Y, direct product. 2 X (XxX), 3 X (X2xX),etc, / ®x H(Ev)p(v, u) X, domain, %(Eu)p(y, u)X, range, mx u X or or CnvxX u)(Eu, v) K(w = (u, v}) p(v u) X, converse, X BelX (X c F2), relation, X"Y H(Ev) Kpv Yp(u, Vs) X, transform of Y by X, (Au, v, w) CKp(u, v} Xp(w, v) X(u = w), oneUnX valued relation, $ ib(Eu,v)K(w = (u,v})puv, membership relation, / fi)(Eu, v) K(w = (u, v}) (u = v), identity relation, %(Eu, v, w) K\z =
D47 D48 D49 D 50 D50 D51 D52 D53 D54 D 55 D56 D57 D58 D59 D60 D61
D62 D63 D64 D65
(UXV) (XxY)
3.26 Set theory
171
3.26 Set theory The system £8!FC consists of the binary !FC with exactly one predicate p and the class-forming symbolA of type t{ot), so far we have only given definitions, if we wish to use the system as a set theory we shall have to add some axioms and rules. For instance we want some information as to which classes are sets, and conditions when a member of a class satisfies the defining statement of that class, i.e. for which ^ ^ - s t a t e m e n t s <j) do we have: Bp7/£
(X=Y)
for
(Au)B{ueX) (ueY), and (aeyff)
for N(aej3).
This is the extensional approach, common in mathematics. Two classes are called equal if they have exactly the same members. We add the rule: D(AU)K(XeU)(YeU)o) D(X= T)co " As axioms to tell us which classes are sets we take: Ax. 1. @2#, Ax. 2. <S{x,y}, XX.X.. O.
\£)± JU)
Ax. 4.
CUnX(SX"y,
these say that the union class of a set is a set, the pair class of two sets is a set, the power class of a set is a set and lastly the transform of a set by a one-valued relation (which may be a class) is a set. The free set variables in these axioms mean that we may only quantify with set quantifiers. We add the following rules to tell us when a class satisfies the defining statement of that class: D(xey)(o Jrv ^-1 . •——~~
~——~ .
DD(ueX)(ueY)(o D(ueXuY)oj '
DN(xey)o) xv & .
•
DND(ueX)(ueY)(o DN(ueX{jY)
172
Ch. 3 Predicate calculi
DN(ueX) o) D{ueX)a) '
D(ueX) co ' DN(ueX)co'
D(Ev)((v,u)eX)G>
DN(Ev)((v,u)eX)o)
D(ueX)u> KiO . -^rr,
r
T^z
DN(ueX)a>
i^r—.
±4
D((v,u)e(VxX))(oD«u,v)eX)(o
0.
D((v,u)eX)o) D((u,v,w}eCnv2X)o)'
R8".'
D((u,w,v)eX)
'
DN«v,u)e{VxX))o)DN((u,v)eX)(o DN«y,u)eX)a) DN«v,w,u)eX)a) DN((u,v,w}eCnv2X)o)m DN((u,w,v)eX)(o DN({u9v,wyeCnv3X)co'
30. The rules R2', ...,R9" are reversible. Take for instance rule R 4'. We have an ^-proof of D(ueX) o), we wish to find an^-proof of DN(ueX) co. In the y-proof of D(ueX)o) note the places where related occurrences of (ueX) enter the e9^-proof. These will be at R4', I l a or T.N.D., viz. DN(ueX) (ueX), in the first two cases replace (ueX) by N(ueX) and similarly for all related occurrences, in the third case add above T.N.D. DN(ueX)N(ueX). This is an ^-theorem, from R4" with N(ueX) for o). Now replace related occurrences of (ueX) by N(ueX) and we have an ^-proof of DN(ueX). Similarly for R4" and the other rules. PROP.
D67.
U{i}
for
p(Ex',...,xM)K(y=(1c))
where £ stands for x'9..., a^^ and <j> for (x',..., x^v)}. 31. / / all the bound variables in $ {t)} are set variables and ift) contains the complete list of free variables in
PROP.
and
DN
and conversely.
Prop. 31 says that we can replace the suggested general rule (*) applied to ^-statements without bound class variables by 8 particular cases
3.26 Set theory
173
R2', ...,R9". If all the bound variables in
(xex) by
(Ey){x = y)(xey).
Suppose t) is <#',...,x^} and 0{t)} is x^X)ex^\ We have: R 5', 5" (a) (z#i/> cX ~ (xy) e@X with 2; for v and <#?/> for w, R 9', 9"
<#z2/> cX ~ <#2/z> eCnv3X,
R8', 8" R 5', 5"
~
R 8', 8" R5', 5"
- <#2/> e@ Cnv2CnvsX, (xyz) eX - (zxy) eCnv2X.
(c)
- <#2/> e@Cnv2X.
From these we get: R 5', 5"
(d) <2/z'... BM> eX - <»'... ^ > e^X,
from (a) with ?/ for z, x' for a; and (xr ...x(p)) for 1/, hence by repetition:
(( /<-times
(/) (x'j/Y... *W> eX ~ <«/'a;V... «w> df 2 ?,.Z, from (6) with ?/' for a;, a;' for z and (x" ... x w ) for «/. (/') is equivalent to <x' ...xfir>)e@'gi&sX, (writing ^ 2 for Cnv2 and ^3 for Cnv3), hence by repetition: (g) <»y ... 2 3
(
/t-times
< z V y ... 2/^> 6X ~ {x'x") € ^ ^ 2 X , from (c) with x' fora;, x" for«/ and ($'... t/^) for 2.
> el Lastly
(j) (i7x') {(x'x"... z(">> eX) ~ >
174
Ch. 3 Predicate calculi
(i) (j) is atomic. Thus (j> is either #(A) ex^ or #(A) €x
If A ^ JLL we have If /£ < A we have By (A)
Hence altogether:
(x'... ^ > cX is equivalent to ( ( ( (
2
/i—A — 1- times
=X
Nowwrite
3
2
A —1-times
and
= X.
7 x ("^3 1 ••• (^3^11 x ( F x . . . ( F x « f ) ...))•••)) for X and we obtain: fi—A — 1-times
>
A —1-times
l x( (
3
| (
/t—A —1-times
Since
V x (Fx...(Vx«f)...))•••))• A —1-times
it follows from the definitions of F, x , <^2,
Thus any two ^-statements may be put into equivalent forms with exactly the same free variables. (ii)
?? as desired, (iii) ^{x} is
By induction hypothesis we have and
3.26 Set theory
175
as desired. (iv) ^{r.} is (Ey) (j)'{y, £}. By induction hypothesis we have:
We have-
Wlfr.S}
yP>
This completes the demonstration of the proposition. 3.27 Ordinals The system £? is very useful for model making. For instance we can define the natural numbers thus: 0 for A,
1 for {0}, 2 for {0, {0}}, 3 for {0, {0}, {0{0}}},...
so that Sv is defined as the set of lesser natural numbers. The first transfinite number co is then defined as the class of all natural numbers. To carry this through we have to^-prove that the natural numbers defined as above are sets, otherwise they are debarred from membership of other sets and the process breaks down. @A is easily ^-proved so are: (&(xn y), &(x u y), ©(# x y), (S&x, (&&x, (&x, etc.
and these suffice to show that the natural numbers as defined above are sets. But to show that 0) is a set so that the process can continue into the transfinite we require another axiom. We could take So itself as an extra axiom, but it is usual to take: Ax. 5.
(Ex) KN^mx(Ay) C(yex) (Ez) K(zex) (y c z).
This ensures the existence of a set containing an unending strictly increasing sequence of sets. It is called the axiom of infinity. An ordinal is defined as the class of lesser ordinals and is well-ordered
176
Ch. 3 Predicate calculi
by the membership relation #. The successor of an ordinal x is then x U {#}, and the limit of a class of ordinals X is 2X. D68 Xi^eY
for K(X* c 7 u Y U I)(AU)CKN
i.e. X is well-ordered by Y; for any two members #, x1 of X we have DD(xYx') (x' Yx) (x = x'), and every non-empty subclass of X has a least member in the ordering Y. The official definition of an ordinal is D69
OrdX
for KX1Te£{X c PX),
i.e. X is well-ordered by the membership relation and members of members of X are members of X. It is easy to show Ord 0, i.e. OrdA. On for ftOrdx. On is the class of all set ordinals. D70
X< Y
for
XeY,
X^Y
for D(X < Y) (X = Y).
We shall use a, 6, c, d, a',..., as variables for set ordinals. We have _ ,Tr// l w , x , aea, NK(aeb) (pea), etc.
Any member of an ordinal is an ordinal. In fact an ordinal is the class of all lesser ordinals. The tricotomy holds DD{a < b) (a = b) (b < a). a c: On any ordinal is a subset of and a member of any larger ordinal. On itself is an ordinal but is a proper class SPr On. Thus we are unable to form the successor of On and so the antinomy of the greatest ordinal is avoided. An ordinal is either a member of On or is On itself. D71
LimX
D72
X+l
D73
1 for 0 + 1 , 2 for
or MaxX for
Xu{X}.
1 + 1, etc.
for SX.
3.27 Ordinals
177
We have G(X c: On) OrdHX. The limit of a class of ordinals is an ordinal. We have
C(X <= &n) KOrd I>X(Aa) C(aeX) (a ^ SZ), C(X c &n)KOrd?:X(Aa)C(X
c a)(2Z ^ a),
the properties of the limit ordinal of a class of ordinals. W e haVe
(Ax) B(x + leffn) (xe&n), N(a < b < a+1).
D 74 Kz
for
&(Ea) D(x = a + 1) (x = 0), ordinals of the first kind,
D 75 Kn
for
0w — iTj,
We have
ordinals of the second kind.
C(aeKn) K(a = Sa) (a 4= 0), C(aeKj)D(a = S a + l)(a = 0).
D76 w for the members of w and the members of the members of co are all of the first kind. We have Ord OJ, SOJ and a) e Ku. o) is a set ordinal of the second kind. Members of a) are called intergers. We use i,j,i',... as variables for integers. The principle of Mathematical Induction can be obtained in the form
provided that (j) is without bound class variables. D77
(X ~ Y)
for
(Ez)KKKUn2zRelz(@z
= X)(0tz = Y).
X is similar to Y and both are sets. D78
Sq
for
&$(x ~ y).
The similarity relation for sets. Then
D 79
Fin
for
^(JJ/i) (i ~ x)
InFin
for
S(^4i) j^(i d^ x)
the class of finite sets. D 80
the class of infinite sets.
178
Ch. 3 Predicate calculi
Having defined the integers we can then define rational numbers as triplets of integers, then real numbers as Dedekind sections of rational numbers and lastly complex numbers as ordered pairs of real numbers. This is further discussed in Ch. 7, § 2 8. We are then ready to develop analysis and as explained in § 32 of this chapter we can introduce all topological concepts. An ordinal is either of the first kind or of the second kind or the ordinal is On. If X is a non-void class of ordinals then IIX is the least member of X. Thus for I c f e w e have TlXeX and £m(X{\ IIZ). 3.28 Transfinite induction The principle of transfinite induction is (Aa) C
(Aa) provided that
NimX X^On, (Ev)K{veX)£m{X(\v)9 X^On
N(Ev) K(veOn - X) Sm((0n -X)(]v) _____ ^ X^On
(Aa) C(a c X) (aeX) X = On
We also want to define functions by transfinite induction. It makes for easier reading if we use F, G, H, F',..., as variables for functions, corresponding small letters if they are sets, and B,S,T,R',... as variables for relations, corresponding small letters if they are sets. We want then to define F'a by means of the behaviour of F for arguments less than a. Now F [ a is the function F with arguments restricted to a. Hence the induction should have the form
3.28 Transfinite induction
179
where G is a previously defined function. Thus we shall have (AG) (E\F) (KF^n6n(Aa) (F'a = G\F [a)). The method of demonstrating this is to take the union of all partial solutions. Thus //
for f(Eb) (Kf^nb(Aa) C(a < b) (f'a = G'(f [ a)),
then show that 2 i / has the required property. Something like this is done in detail in Ch. 11. The addition, multiplication and exponentiation of ordinals are defined by transfinite induction thus: D81
+6
for
(
(
)( (Ea) K(v = a = 2a) (v = 2 +1'a),
D82
xb
for tiv(DDK(u = O)(v = 0)(Ea)K(u = a+l)(v
=+bx'ba)
(Ea) K(u = a = 2a) (v = 2 x b'a), D 83
expb for
uv(DDK(u = 0) (v = 1) (Ea) K(u = a + 1) (v = x b exp'ba) (Ea) K(u = a = 2a) (v = 2 e ^ ^a).
The first of these defines the function +bi the second the function x b and the last the function expb. They are all of the form F(a = Gl(F [a). Weusuallywrite (6xa) a
b
for
x^a
for exp'b a.
(a + b),(axb) and ab are all set ordinals. They satisfy some of the usual rules of addition, multiplication and exponentiation. But some rules fail, e.g. l+(o = a). It can be verified that the ordinal (a + b) is isomorphic as regards order to the order type obtained when we stick the order type b at the end of the order type a, and that the ordinal (a x b) is isomorphic as regards order to the order type (c,d),cea, deb ordered by last differences, i.e. (c,d) < (c',df) ifd < d'ovd = d'andc < c\ The ordinal ab is isomorphic to as regards order to the ordering by last differences of functions over b with values in a, but with only a bounded number of non-zero values, i.e. if/and g are two such functions then/ < g if f'd < g'd, where d is the greatest ordinal for which
180
Ch. 3 Predicate calculi
f'd =f= g'd. In fact we could have taken these properties as definitions of addition, multiplication and exponentiation of ordinals provided we have shown yTT m xWeT where
D84
RIsom(*'^\
for
KKKKUn2RRelRSJR = X01R = Y(Au,v)C(u,veX)B(uSv)(RcuTR'v), i.e. R is a (1-1 ^correspondence between X and Y such that two members of X stand in the relation S if and only if their images in Y by R stand in the relation T. Any class of ordinals is well-ordered by £, hence a decreasing sequence of ordinals terminates, otherwise the sequence would be without first member and so would violate the condition of being well-ordered. 3.29 Cardinals A cardinal number is frequently defined as the class of classes similar to a given class. We shall define a cardinal as an ordinal which is dissimilar to any lesser ordinal. This is less general than the usual definition because there may be classes dissimilar to any ordinal. D85
f
for
t(Aa) (C{a ~ x) (bea),
then f is the least ordinal which is similar to the set x. f is called an initial ordinal or the cardinal integer of the set x in case x is finite. The class of ordinals similar to a given ordinal is non-void, because a is similar to itself. Hence the class of ordinals similar to a given ordinal exists and being a class of ordinals has a least member U. But if x is any set then set theory as we are developing it may turn out to be so poor in modes of expression that the (1-1)-correspondence required to show that x is similar to an ordinal may be missing. Thus the concept of cardinal is relative, that is relative to the set theory used. We have U = ti. We divide ordinals into classes, the members of one class being similar to each other, except that the first class is to consist of all the integers. The ordinals of the second class are similar to co, they are the denumerable
3.29 Cardinals
181
ordinals. The least member of class III is denoted by ti, it is the least nondenumerable ordinal. Q for 26(6 ~ o)). D 86 Note that this use of the word' class' is distinct from ' class' as opposed to 'set 5 . D87 JT for &(Eb)(a = $), JV* is the class of integers and initial ordinals. D88
JT'
for
JV-G),
J/*' is the class of initial ordinals. Jf' being a class of ordinals is wellordered, hence there is an isomorphism between the initial ordinals and the ordinals. Let X be this isomorphism. D89 G)a or Ka for K'a. Then GJ0 = G) = No, fi = K1? etc. These cardinals are called alephs. We can define addition, multiplication and exponentiation for cardinals. If otj9 je J is a class of cardinals then 2 % is the cardinal of the union of classes of cardinals otj for jeJ. In forming the union we require that the representative classes be distinct. This is achieved by using ordered pairs {a,j}, aeAjSOCj. Then the ordered pairs are distinct for different jeJ. If the cardinals are initial ordinals & then 5 itself is a representative class of that cardinal. The cardinal of the product of the class of cardinals ocj,jeJ is the cardinal of the class of functions / over J such that/'Jea^-. This amounts to picking out one member from each otj and doing this in all possible ways. This raises the question as to whether there is any such function at all. The statement of the existence of such a function is known as the Multiplicative Axiom, Axiom of Choice (A.C.) or Zermelo's Axiom, If it failed then the product of an unbounded set of cardinals would be zero. The axiom is: Ax. 6.
(EF) (Ax) KF^n VBSmx(Fixex\
This is a very strong form of the axiom of choice because it allows for the simultaneous choice from each set of an element of that set. The axiom of choice occurs frequently in mathematics, sometimes it is possible to avoid it by a more elaborate proof. At the end of Ch. 12 we sketch a demonstration of the independence of the axiom of choice from the other axioms of set theory.
182
Ch. 3 Predicate calculi
Using the axiom of choice we can show that every set can be wellordered, conversely if every set can be well-ordered then A.C. (Ax) (EaJ)KK(f^na) Un2f{x = /"a), so that f'b for b < a well-orders x. By transfinite induction we define a function G so that G^nOn and (Aa) (G'a = F'(x-@(G [a))), where F is the function postulated in Ax. 6. Then G'O = F(x, the member chosen from x by F, G'l = F\x — {G'O}), the member chosen from x-{GiO) by F, etc. Then G'b for b < a well-orders x. If we use Ax. 6 then every set can be well-ordered, hence every set is similar to an ordinal and so all cardinals are alephs, and hence the tricotamy will hold for cardinals. But without the axiom of choice there may be cardinals without any order relationship with any aleph. The exponentiation of cardinals is defined by: a^ is the cardinal of the class of functions over /? with values in a. Thus 2so is the cardinal of the real numbers. The equation 2**o = Nx is known as the Continuum Hypothesis (C.H.). It is now known to be independent of the other axioms of set theory and a brief sketch of this is given at the end of Ch. 12. It can be shown that the sum, product and exponent of alephs is an aleph. The equation 2Na = XSoc is known as the Generalized Continuum Hypothesis (G.C.H.). Again it is now known to be independent of the other axioms of set theory and a brief sketch of this is given at the end of Ch. 12. Many statements about cardinals are now known to be independent of the axioms of set theory. But there are some important theorems about cardinals. D90 <x
We shall show £ < Px. Each member y of x gives rise to a subset {y} otx, hence x can be put into (1-1 ^correspondence with a proper subset of Px, thus f ^ Px. Note that x and Px are sets. Suppose that % = Px, then there is a (1-1)-correspondence between x and Px. Let o-{y} be the correlate of y by this correspondence. Let a be the class of members of y such that y1cr{y\. Let a be the correlate of z so thata = cr{z}. If 2;ea then 2:ecr{2:} by definition of a and Prop. 31, but cr{2;} = a
3.29 Cardinals
183
andsozea. Again if zea then by definition of a ze(x{z} and Prop. 31,i.e. zeoc. We have an absurdity in either case, hence X #= Px. 33. If a ~ /?' and ft c ft and J3 ~ a' and a,' c a, then a ~ /?. Let / map a (1-1) onto /?' <= y? and gr map /? (1-1) onto a' c: a. We can clearly assume that oc(]j3 = A. Now (oc\Jj3) is the disjoint union of sequences PROP.
cr':(b,g%rg'b,...) (be/S),
LEMMA
(i). For any set a there is a well-ordered set w, such that w ^ P4a
and w ^ U.
Consider the class w of well-orderings of a and subsets of a. A well-ordering of a is a class of ordered pairs hence is in P 3 a. Thus w c= P^cc.w is isomorphic to a class of ordinals and so is well-ordered and is isomorphic to an ordinal. If w ^fiethen w would be order isomorphic to a well-ordered subset of a and so w would be order isomorphic to a proper subset of itself, which is impossible. Hence lemma (i). For the moment we assume that 2 P a = Pa. LEMMA
(ii). If y and 8 are disjoint sets such that y u S = P(2y) then
$>Py. 2y denotes the union of two disjoint copies of y, say yx and y2 If/maps y U S onto P{yx U y2) ^ Pji x Py2, then the image of y projected into
184
Ch. 3 Predicate calculi
Pyx is only a proper part of Pyv since y1 < Pyv and hence if £ is outside the projection, / must map some subset of S onto £ x Py2, which means that 8 ^ Py. Whence lemma (ii). Now we have P 3 a ^ w + Psa ^ P 4 a + P 3 a ^ P 4 a by the assumption. Thus by G.C.H. either: w + Pzoc = P 4 a
or
Consider the first case. w + Pzoc = P 4 a = P(2P 3 a), by the assumption. We have, by lemma (ii), w ^ P 4 a, but w ^ P^oc, hence w = P±oc. Thus there is a (1-1 ^correspondence between P^oc and w, thus jF^a is wellordered, but a can be embedded in P^a, hence a can be well-ordered, and we are done. In the other case we have w + P>a = P,a then we have w ^ P 3 a, hence w = P 3 a, and we are done, as before, or w < P3oc, and whence by G.C.H. w < P 2 a, but then ^ = P 2 a, and we are done as before, or W < P x a, now by lemma (i) w ^ a, whence by G.C.H. P = P x a, and we are done as before. It remains to show 2Pioc = P^, for 0 < i < 4. If we put /? = P(a U &> where af]o) = A, then easily 2iy? = Pip, for 0 < i < 4. Let y be new. 2/? = P(au^u{y}) = P(a U w) = /?. Also yff^ y^uy ^ 2/?, so /?Uy= Now 2.2/* = 2/*uW = 20, and similarly 2P^/? = P ^ , for all i. Hence our argument can be applied to /?, and so ft can be well-ordered. But a can be embedded in /? in a natural manner, hence a can be well-ordered. Thus Prop. 33 is demonstrated. 3.30 Elimination of the e-symbol The choice effected by A.C. can also be effected by the e-symbol, thus (Ax)CN£>mx(ey(yex)ex), then tivCNdfrnviu = ey(yev)), is the required function that picks out a member from each set. We could also have the rule. C
provided that
3.30 Elimination of the e-symbol
185
The system Sf with the e-symbol and rule C is called the system CSP. We could also, as in D 24, define the existential quantifier in terms of the e-symbol. The system SP with rule II d replaced by rule E and rule II e' replaced by rule E' is called the system ESf. We could then dispense with rule E' in favour of a rule of substitution. The system CE£f is the system ESf plus rule C. 35. If an £f-statement is a CSf-theroem then it is an £f-theorem. We shall show that if the e-symbol is used in a C^-proof of an Sfstatement ^ then it can be eliminated leaving an e-free proof of ^ , which is thus an ^-proof. Thus (J) is an ^-theorem. This proposition says that if CSP is inconsistent with respect to negation then so is Sf. For if we can C^-prove the ^-statements
this, consider rule C, say, _ ,,
,}J^
, and consider the places where
related occurrences of (EE) <j){£) entered the C^-proof, these will be of the form
n/grrx^/m /> w n e r e 0'{£} i s
a
variant of ^{g} by I(ii), replace
the lower formula by D^'{e^'{^}} GJ', and similarly for all descendents. Entrance by II a can be replaced by entrance by II d, the special ^-rules fail to introduce D(E£) $${£} OJ. The original Cc^-proof-tree becomes an E£fproof-tree of the original statement, because these related occurrences of (E£)(j){Q all occur in the subsidiary formulae, except I(ii), from their introduction to the application of the C-rule under discussion. Thus we require to eliminate the e-symbol from an E^-proof of an ^-statement, free from the e-symbol. We next replace the special ^rules by axioms, thus: if _ ,
is an
Dyrco
^-rule replace it by CD^o)Dj/rco, we recover the rule by Modus Ponens, which can be eliminated from a theory in free disjunctive form. To make use of this result we replace rule R1 by its free variable form viz. DK(XeU)(YeU)a>. ,,, .,.r, ,TT,., —Y^i— because by the reversibility of lie if we can ^-prove n v
186
Ch. 3 Predicate calculi
the upper formula of R1 then we can ^-prove the upper formula of its free variable form. We similarly replace R 5 ' and R5" by D((oc,u)eX)oj —^7—'v.
and
D((v9u)eX)o) " \'v,
,. . respectively.
This assures the reversibility of l i e ' because A can then only be introduced at l i e ' and so the demonstration of reversibility of l i e ' goes through as before, The axioms can be put into resolved form, viz.: Ax. 1. (LxeiXx}), Ax. 2. ({x,y}e{{x,y}}), Ax. 3. (Pxe{Px}) Ax. 4. CCK{(u, v)eX) ((w, v)eX) B{yeu) {yew) (X"ye{X"y}). From these the original axioms may be recovered. The system SP can now be put into free disjunctive form, so that Modus Ponens can be eliminated. If we retain the axiom of infinity then we replace it by: (£2e{Q}), (Aefi), C{ueQ)({u}e£l), then Q, contains the unending set: {A}, {{A}}, {{{A}}},.... Call the resulting system Sf\ then Modus Ponens can be eliminated from Sf'. Let ^ be an E^-theorem then 0 is an ESP'theorem, let x be the closure of the ^'-axioms used in the ES^'-"proof of <j), then by the Deduction Theorem G^r^> is an iJ^-theorem. From its .EJ^-proof we wish to eliminate the e-symbol. By hypothesis the e-symbol is absent from <j> also the ^'-axioms are without the e-symbol, hence Cx
,*
is an E-rule used in the l£^'-proof of 6,
where \jr is e-free and closed; in this replace the e-term e^{Q by a, we are left with a tree with (f> at its base because this is €-free. The above E-rule becomes a repetition, all applications of other rules are preserved except other applications of E-rules, which by supposition belong& to the same e-term. For instance -^ , , ,1^—, E becomes Z)^{e^{£}}6/ n
,\{
, which fails to be a case of any rule. But if we add ^{a} as an extra
axiom we can obtain Di/r{x} o)f by II a, each case of rule E in the j&^'-proof of $ can be replaced by a case of II a; hence we obtain an ^-deduction of (j) from the hypothesis ^{a}. Similarly we get an ^-deduction of
3.30 Elimination of the e-symbol
187
for all d for which i/r{a{6)} occurs as the main upper formula of an E-rule. Since we are supposing that all E-rules belong to the same e-term then ^r is the same in each case. Thus we get an ^-deduction of (j) from the hypotheses ^{a'}, ifr{oc"}, ...,i/r{oc(p)}, where this is the complete list of main formulae in the upper formulae of E-rules used in the V
proof of (j). Hence we obtain an J^-deduction of $ from 2 ^{#(7!r n= l
On the other hand we have Dft{aP>}at*> -i.V_L. _L •
)
TT
But Modus Ponens can be eliminated, and so we get an .^-deduction of (f> from the hypotheses Ni/r{ocf}, Ni/r{ot"},..., Ni/rfaM}, where we have the same set of a',..., a^ as before. Thus by the Deduction Theorem we obtain the ^-theorems CM<x'}$,...,CMoiM}$, and ON 2 {<*<«>}0. Whence 0= 1 v
d)
(e)
C 2 ^{o^ }(j> and CNHi/r{x }^) are J^-theorems and so by Modus Ponens 0=1
is
188
Ch. 3 Predicate calculi
in the €-term j3 then the replacement of oc by another e-term lacking the binding variable of fi has no effect on the rank of/?. And if a is an e-term occurring in the e-term /? and containing the binding variable of /? then a is of lower rank than /?. Also the rank of an e-term is unaltered when we change a free variable to a new variable. We propose to eliminate the e-terms by replacing e-terms by other terms as we did in the simple case when all E-rules belonged to the same e-term, thus in the E-rule
., *. j ^
we replace the e-term e^t/r{Q
wherever it occurs by a, then the above application of the E-rule becomes a repetition; the application of the E-rule ^ , c , i>>^—; becomes Dj/r{e^{Q}ojf -r^. \ ^I0) , which fails to be an application of any rule if (a 4= fi), but we can obtain the lower formula from the hypothesis i/r{(x}. The subsitution of a for €^{£} may alter the end formula
, f* —^,
1 < 0 ^ K. By the
Di/r{ei/r{^}}(i)^)3
J
deduction theorem and &c this would give us: C 2 ft{ote)} 2 ^=1
6=1
without using the E-rule belonging to the e-term e^{^}. On the other hand we have by Modus Ponens
but Modus Ponens can be eliminated. Thus if we introduce Ni/r{oc'}, Ni/r{a"},..., Ni/r{a^K)} as hypotheses we obtain an iJ^-deduction of
the e-term e^{^}.
of C Yl Ni/r{oL(e)}
This gives us an E^c--proof
6= 1
C 2 ifr{ot^} 2 ^{cii^} and 6=1
CNYii]f{a!&)} ^{ci}
6=1 K
we obtain by Modus Ponens 2
can be eliminated so we obtain an JS/J^-proof of 2 ^{cti^}. The reason for
3.30 Elimination of the e-symbol
189
replacing rule II d by rule E is to avoid the existential quantifier binding an e-term, thus:
will fail to occur instead, we shall have in the lower formula
which has an e-term of higher rank. But if there are other applications of the E-rule in the jE/J^-proof of 0{ct} then the substitution of af® for €^{^} may destroy them. The following cases can arise for the E-rule
(i) eMB occurs at most only in fi, the E-rule is then 1 this becomes _ } *• / {. which is the E-rule.
Dfai}} y (ii) €j^{^} occurs in x{v} a n d possibly in fi as well the E-rule is then DM*},*}* this becomes w h i c h is t h e E . r u l D a a x{*x{v h ) ^ here y stands for (iii) One or both of/?, # where S = €£#{£} is contained in y = (so that in the first case y is of the form y'{/?}), and x{v} * the form X'{y'{V}}; the E-rule is then comes •=- /f
/r
^./ / r ^ , ^
D
^
{
^ ^ ^
this } } } a >
be
"
which fails to be an E-rule, f is new to
Dx{?M{v{Q)}}<* X'{y'{^i}}' Similarly if y is of the form y'{8}, and x(v)^s of the form ^'{y'{9/}}, the E-rule is XxrxPss
^ j c h f a ii s to be an E-rule. Similarly if y is of the form
y'{(3,8} and x is of one of the forms x'{Y{y, *}}5 X'{7'{P> y}}> X'fy'fy* V}}>etc-
In all these cases the e-term ^x{j'{Q} ^ °f higher rank than the term y'{/?} for which a substitution is being made.
190
Ch. 3 Predicate calculi
(iv) The only remaining case is when one or both of /?, S are contained in y so that y = y'{/?}, etc., but the variable 7/ in x{v} fau"s to occur in a part y'{ij} of x{v}- I*1 this case the E-rule remains correct as can be seen from case (iii) by omitting y' in xij'iv)} m the lower formulae. In case (ii) the e-term belonging to the E-rule is of higher order than y but is of the same rank as y, but in case (iii) it is of higher rank than y. Thus if we make the substitution for an €-term of highest rank and among those of highest rank we choose one of highest order then cases (ii), (iii) will fail to arise; we can proceed as in the first case where all the E-rules belonged to the same €-term. The result of the elimination of one K
€-term is an UJ^-proof of a disjunction 2 ^ {ct(^}. A second application 0=0
will produce a similar disjunction of these disjunctions which is merely a longer disjunction of the same kind. Finally we can eliminate all the V
E-rules and are left with an ^-proof of a disjunction 2 <j>{oSe)}. If there e=i
are any €-terms left in this disjunction then replace them all with the same new free variable, the result is an J^-proof of a free variable disjunction which is €-free. This is possible because all substitutions had been pushed back into the axioms before we started so that an e-term V
in the J^-proof of the disjunction 2
into the J^-proof at an axiom and if it is replaced by a new variable axioms remain axioms and rules are preserved. From the free variable disjunction we easily obtain (E%) ^{j}. This completes the case when $ is without universal quantifiers. We may suppose that (f> is in Skolem F-normal form, i.e. is of the form: (E$) (At)) fr{£,t)} where the matrix ifr{%,t)} is quantifier-free. Since
3.30 Elimination of the e-symbol
191
hypotheses dc into an J^-deduction of (j) from hypotheses X. Thus a C^-proof of (j) can be converted into an ^-proof of
C{E$ (At)) <%, t)} (Ed flj, fj}.
Thus from the iJJ^-proof of (E%)(Ati))
a'9a",...,a«>
are sequences of terms of type t, €-free but whichmay contain/',/", ...,/(7r). Moreover the ^ c -proof of this disjunction is a free variable ^ c -proof free from the substitution rule, because all substitutions had been pushed K
back into the axioms before we began. If in 2
proof we replace each atomic formula by a propositional variable, distinct formulae by distinct propositional variables, identical formulae by the same propositional variable, we are left with a ^-proof. Thus K
2
mulae of a theory for propositional variables. The same will hold if we replace the terms f^a(fl) by new free individual variables using distinct variables for distinct terms, the same variable for different occurrences of the same term. Consider the term / ^ a ( ^ and the number of distinct occurrences of /',/", ...,/(7r) which are contained in it. Call this number the complexity of f(d)a{fl). Then the complexity of/'a ( ^,... ,/(7r)a(^ are all the same. We associate this complexity number with K
Now arrange the disjunctive terms of 2 (f>{a(e\ \a{6)} in order of increasing 0=0
192
Ch. 3 Predicate calculi
complexity from left to right. We suppose that duplicates have been omitted from the disjunction. Then f(6)a(/l) is distinct from f^aW if and only if 6 4= d' or JJL =)= /JL'. Also f^a(fl) can only occur in a(/O if [i < /if. Now replace /'a', ...,/ (7 %',/a", ...,f(n)a(lc) by new individual variables £',..., £<™>, so that /% is replaced by £«/-«*+«>, then £ 0{a<*>, fa<*>} 0=0
becomes: 2 ^{b^, ^'
7r+1)
, ...,^'
7r+7r)
}. In this disjunction the variables
0=0
, ...,Qv'n) are absent from the first (v—1) disjunctions because can only be a member of the sequence a(A) or be contained in a member of the sequence a(A) when JLC < A, also f^ai/l) fails to occur in or be any member of the sequence a^\ Now we can generalize the variables QK~1)rr+1, ...,£,K'n then apply lid repeatedly to the terms in b(/c) this converts the last disjunctand to (E%) (At))^^,^}. We can proceed similarly with each disjunctand obtaining a disjunction of K disjunctions all being (EAjc)(At))
3.31 Complete Boolean Algebras In some Boolean Algebras any subset has an l.u.b. that is, if 9£ is a subset of a Boolean Algebra 88 then there is an element a of 88 such that /? ^ a for each element /? of 9C, and if /? ^ y for each element ft of 9£ then a < 7. We denote a by l.u.b. 9£. Similarly a g.l.b. can exist; in any case if the l.u.b. exists then we call the Boolean Algebra complete. PROP.
36. If 88 is a complete Boolean Algebra then l.u.b. & = g.l.b.3?,
where 2£ is the set of complements of members of SC. We sometimes write U at for l.u.b. {aj, and similarly f|
PROP.
iel
35. l.u.b. l.u.b. a^ = l.u.b. aij9 iel
jej
iel,jej
«nu A = UMA), iel iel
iel
A
iel
iel
iel
3.31 Complete Boolean Algebras
Uai
=
0 iff
cx>i = 0 for
193
iel,
iel
fl oci = 1 iff
a* = 1 for
ie/.
The V is the membership symbol. A Boolean Algebra is said to satisfy the countable chain condition if every disjoint set of non-zero elements is countable. Two elements of a Boolean Algebra are said to be disjoint if their intersection is zero. The l.u.b. acts like the union of an unbounded set and the g.l.b. acts like their intersection. These correspond to the Existential and Universal Quantifiers respectively. Distributive Laws In some complete Boolean Algebras there are extensions to the distributive laws corresponding to unbounded sets. Thus: PI U ay = U n aiT(i) and ieljej
re J1 iel
(J D % = fl U ocT(j)j. jej
iel
lfjj
Here J1 denotes the set of functions with domain / and range in J. But these laws can fail in some complete Boolean Algebras. 3.32 Truth-definitions for set theory A truth-definition for a formal system can be given by formula induction. First a truth-definition is given for closed atomic statements, then the truth-definition for closed compound statements are obtained in the usual manner by truth tables, if we are seeking a standard two-valued truth-definition. If closed atomic formulae are lacking as in ^Q then we usually give a definition of validity. In set theory the closed atomic statements are of the form a closed statement of this form will be true if and only if this in turn will be true if and only if t{x<j){x}} & ^ { a ? } } & (Ey) (x
for some x> but this will be true if and only if which is what we had before, so we must abandon this method.
194
Ch. 3 Predicate calculi
Another way of giving a truth-definition is to construct a model. By this we mean a class of elements V and truth-values for all atomic statements of the form (aeb), where a and b are two elements of F. But we can easily be more general because we can take the truth-values to be members of a Boolean Algebra ^ . Let then ||ae6|| and \\a = b\\ be the members of the Boolean Algebra associated with the statements (aeb) and (a = b) respectively. From the Boolean values of these statements we can find the Boolean values of compound statements thus:
= ¥\l aeV
from these we obtain
\\D
U
||##
aeV
We write \=$ for ||^j| = 1, and h^ for §1 is an ^-theorem. We want to show that, if h^ then |= <j>, i.e. all ^-theorems take the Boolean value 1. First of all we easily show that f= DNcjxj), and, if \= i/r then = |
It is impracticable to introduce all the members of V at once. We proceed by a transfinite process. We start with Vo = A. The elements to be added to Va to produce Va+1 will be functions over Va with values in 38. These will correspond to new ' sets'. Thus/{#} defined over Va with values in £8 will be the 'characteristic function' of a new set. Clearly Vx = {A}. If a is a limit ordinal Va= \J ¥#. p<*
The step from Va to Va+1 is defined as follows: we assume that the
3.32 Truth-definitions for set theory
195
members of Va have been defined and that Vp ^ Vy for fi < y ^ a and that for a, b e Va we have
\\aeb\\=JJJ\K(xeb)(a = x)\\,
(1)
||a = 6||= n \\C(xea)(xeb)\\n fl \\C(xeb)(xea)\\, II
II
' ' I I
V
/ V
xeBa
' II
' '
II
*
' *
(2)
' II '
^ '
xe&b
and that for a, b, c, eVa we have^ a = a,
(3)
(a = b)(b = c)(a = c),
(5)
(a = 6) (bee) (aec),
(6)
eb) (b = c) (aec),
(7)
and we also assume that every member of Va is a function whose values are in the Boolean Algebra 3$. aeVfi+1, aeVp, fi < ot then
if if
aeT^,a;e^a then x,ye@a then
2a = Vfi9
(8)
\\xea\\ = a{o;},
a{#}n |a? = y\\ ^ «{i/},
(9) (10)
a function which satisfies (10) is called extensional. We first put into Va+1 every member of Va. Next we generate each function / from Va to 0$. As @f — Va for each new /, the value of \\x = y\\ has been determined for each x,ye3>f. Hence for each x,ye@f the value of/{#}n ||» = y|| is determined, we discard all functions/for which this value is ^ f{y}. Thus we restrict our choice to extensional functions. For xeVa we define \xea\ to be a{x] for each new a. We now define ||a = 6|| by (2). If a, beVa this duplicates a known result. If aeVa, beVa+1, beVa, this is an acceptable definition, since Q)a c Va 3)b = Va, hence ||#ea|| is determined by (1) and ||#e&|| by b{x}, so ||G(a;ea)(aje6)|| and ||C(a;e6)(icea)|| are both determined. Similarly in the other cases. So (2) holds for Va+V Now if ae@b, beVa+1, beVa we define: ||ae6|| for U \\K(xeb)(a = x)\\ xe9b
this is an acceptable definition since for each xe3)b \K(xeb) {a = x)\\ is determined by (9) and (2). 7-2
196
Ch. 3 Predicate calculi
We have to show that (1) holds in F a + 1 . The only case to consider is beVa+1, beVa, aeSb. In this case S)b = Va, so that aeVa, then we have \\aeb\\ = \\K(aeb)(a = a)\\ ^ U \\K{xeb){a = x)\\.
(11)
xe9b
Now take xeQ)b = Va. Then by (9) \\K(xeb)(a = x)\\ < b{a] = \\aeb\\, thus
U \\K(xeb)(a = x)\\ ^ \\aeb\\, xe9b
then by (11) we infer (1) for F a + 1 . It remains to check that (3)-(7) inclusive hold in Va+1. By (2) we conclude (3), since \\C(xea) (xea)\\ = 1. Also (2) gives (4). LEMMA
We have thus whence
(i). IfxeSb then \\xeb\\ n ||6 = c\\ ^ \\xec\\. ^ ^ ^ ^C(pceh) ( ^ c ) | | = ^ n n ^xec^ ? ||ae&|| n \\C(xeb) (xec)\\ ^ \\xec\\, ||a;e6||n 0 ||O(#e&)(#ec)|| ^ ||a;ec|| if xeQ)b. xe2b
By (2) the lemma follows. (ii). If a, beVa and ceVa+1 then (7) holds. If ceVa then by assumption (7) holds. Thus suppose that ceVa. Then LEMMA
2c = Va. By (8) and Va c Va+1 we have Sib c Sc. Then by lemma (i) \\K{xeb) {a = x)\\ n \\b = c\\ < \\K{xec) {a = x)\\9
whence using (1) ||ae6|| n ||6 = c\\ ^ U K\\(xec) (a = x)\\ xe@c
= \\aec\\ by (1), which is (7). We now verify (5) for oc + 1. Let xeQ)a. Then, by lemma (i) we have ||a?ea|| n ||a = 6|| ^ \xeb\. So \\K(a = 6) (6 = o)H n ||ajea|| ^ ||aje6|| n ||6 = c\\. Case (i) beVa, by lemma (i) ||ge&||n||& = c|| ^ ||^ec||.
(12) (13)
3.32 Truth-definitions for set theory
197
Case (ii) beVa. Then 3)b = Va9 so that xe3)b. Then by lemma (i) we again have (13). Therefore (13) holds in either case. By (12), (13) we obtain = c)\\n\\xea\\ < \\xec\\. So
\\K(a = 6) (6 = c)\\ U \\xea\\ ^ ||a;ea|| -> \\xec\\
and then
\\K(a = 6) (6 = c)\\ ^ \\G(xea) (xec)\\
whence
\\K(a = b)(b = c)\\ < fl ||C(sea) (n;ec)||.
(14)
One can start with xe@c and go through a similar argument to obtain \\K(c = 6) (6 = a)|| ^ n ||0(sec) (sea)||.
(15)
xeQic
By (4), (2), (14) and (15) \\K(a = 6) (6 = c)|| ^ ||a = c||, which gives (5). We next verify (6) for a + 1. Let xeQJc, By (5) we have \\K(a = b)(b = x)\\ <\\a = x\\. Therefore \\a = b\\ n \\K(xec) (b = x)\\ < \\K(xec) (a = «)||, summing both sides over xe3)c and using (1) we get (6). Finally we verify (7) for oc + 1. Let xeQsb. By lemma (i) ||a;e6||n ||6 = c\\ ^ \\xec\\. Therefore Then by (6)
\\K(xeb) (a = x)\\ n ||6 = c\\ ^ \\K(a = x) (xec)\\. \\K{xeb) {a = x)\\ n ||6 = c\\ ^ \\aec\\.
Sum on the left over xeS)b and we obtain (7) by (1). We have now obtained a universe V and a Boolean value for each atomic statement (aeb), where a,beV. So we have given a generalized truth-definition for set theory. Now write ^ ^ _ 1; for h (j> for
fi is an 5^-theorem.
It is possible to ^-prove that if h^ then t=^, i.e. all ^-theorems take the value 1. Thus C becomes a model for SP. The proof must take place in some system which can deal with ordinals because they are essential to the construction of V. The system £P is such a system.
198
Ch. 3 Predicate calculi
An interesting application of this method for denning truth for set theory is that by suitable choice of the Boolean Algebra 0&, we can show that A.C., G.C.H., etc. take values different from 1 and hence are non-theorems of SP. Thus it is impossible to ^-prove them. On the other hand we can find another type of model, namely an 'inner model9, constructed as follows: we start with the null set and by a process of transfinite induction we define all the sets which can be obtained from it by repeatedly performing the operations allowed for the construction of new sets from old ones, e.g. union, complementation, etc., In this way it seems clear that all sets are given an ordinal and so V is well-ordered, so that A.C. holds. It can also be shown that G.C.H. holds in this model. All this can be done in the system SP. Thus A.C. and G.C.H. are consistent with set theory if this is itself consistent. Altogether A.C. and G.C.H. are independent of the other axioms of set theory. The full details are lengthy. 3.33 Predicative and impredicative properties In a second order predicate calculus we can have bound predicate variables and hence we can form properties Ax
property variables of the first order,
X2, X'2,...
property variables of the second order, etc.
The order of a property is the greatest of (i) the order of a free property variable in it, (ii) the successor of the order of a bound property variable in it. Then \x
r>/A/A\—
*' ^ e
revers
ibility of
3.33 Predicative and impredicative properties
199
lie', here A is to be a property of order the same or less than that of the variable X. We obtain this result from l i e ' by everywhere replacing related occurrences of X in the part of the tree above the upper formula of l i e ' by the property A which must be of the same or less order as X, A property is called predicative if it fails to be defined in terms of itself, otherwise it is called impredicative. Thus in ^^ we only have predicative properties. If we try to give a definition of validity to ^l?\ which contains impredicative properties, then we get into trouble because we should require (AX) (/>{X} to be valid if and only if ^{A} is valid for all properties A, but one of these is 3£
D91 Topx for (Au,v) (u, vex.-^.un vex) & (Ay) (y <= x-^Hyex); read lx is a topology'. D 92 xTopy for Topx & y — 2#; read 'x is a topology for y\ D93 xOpy for Topx &y ex; read ' y is an open set in the topology x\
D94 xCly for
Topx&Op(x-y);
read 'y is a closed set in the topology x\ D 95 TopSpx for (Ey) (Topy & x = Sy); read 6x is a topological space'. D 96 IndisTopx for (Ey) (x = {y, 0}); read 'a; is an indiscrete topology'. D 97 DisTopx for (Ey) (x = Py)\ read (x is a discrete topology'.
200
Ch. 3 Predicate calculi
D98 uTopxNeighy for (Ev) (v ^ u&yev &xOpv) &u c x\ read 'u is a neighbourhood of?/ in the topology x\ D 99 yTopxLimz for 2 c= a; & Topx & (^4w) (uTopxNeighy -> (i£#) (# e w n z & # 4= ^/)); read '?/ is a limit point of the subset z of the topological space x\ D100 Z*TOPX for $(yTopxLimz) [) z; read 'the closure of the subset z of the topological space x\ D101 IntzTopx for y(zTopxNeighy); read 'the interior of the subset 2 of the topological space x\ D102 BdyzTopx for z* T o ^ n ( # - z ) * T o ^ ; read 'the boundary of the subset z of the topological space x\ D 103 zBaseTopx for z ^ x & (^y, u)(yex& uTopxNeighy ->(Ev)(yev&vez&y c ^)); read '2 is a base for the topology a?'. D 104 Se^a; for Topx & (#z, ti;) (zBaseTopx & w"z = w); read ' x is separable', x is separable if a; has a countable base. D105 zDenseTopx for z* To ^ = X; read'2 is dense in the topological space x\ D106 uGovw for w ^ Su; read '^ covers w\ D 107 Sep [y, z] Topx for y^opx n s = A & z*T°vx n y = A; read '^/ and z are separated in the topology x\ D 108 GonnTopx for .4 (y, z) (a? = y u z &fifep[y,z] T o ^ . -».y = A v z = A); read 'a; is a connected topological space'. D 109 HausSpx for Topo; & (^4^, v) (^, v e 2# & ^ 4= v. -> (.E/^, ^') (^n wr = A & wTopxNeighu & w'TopxNeighv)); read '# is a Hausdorff space'. (,4y) (fe) (?/(7ora & z(7ora &z^y& (Ew) (w"z c
Compx for
and so on. If we are dealing with a single topology for a topological space
3.34 Topology
201
then we can omit Topx in the above definitions. The whole of general topology can now be formalized without undue difficulty. H I S T O R I C A L REMARKS TO C H A P T E R 3
Predicate calculi differ from propositional calculi by the adjunction of quantifiers, whose intended meaning always has something to do with the cardinal number of things which satisfy a certain statement. Quantifiers were first introduced by Frege (1879). Somewhat later and independently quantifiers were used by Pierce who introduced the term 'quantifier'. Thereafter their use becomes general, though the notation for them varies. The various orders of predicate calculi is due to Russells' theory of types and perhaps to Frege's Stufen and Schroder's Mannigfaltigkeiten. Lowenheim and Skolem in effect gave a treatment of the first order predicate calculus with equality. But the first explicit formulation of the classical predicate calculus of the first order as a formal system in its own right is in the first edition of the book by Hilbert and Ackermann (1928). Thereafter it was much studied as a formal system. Many-sorted predicate calculi were discussed by Schmidt (1938) and Wang (1952). Models by Kemeny (1949) among others. Predicative and impredicative predicate calculi arises from the unqualified use of the concept of' all'. Russell's (1906) P.M. vol. 1, Ch. II, vicious-circle principle, designed to avoid paradoxes, was 'no totality can contain members defined in terms of that totality'. The term 'impredicative is due to Poincare (1905) who condemned impredicative definitions, as did Weyl (1918). In developing the classical predicate calculus of the first order we again use the direct formulation due to Gentzen (1934) and further studied by Schiitte, (1950-1,1960). Prop. 4, the elimination of M.P., is Gentzen's Hauptsatz, the demonstration we give is due to Lorenzen (1951). The prenex normal form is due to Skolem who also found other normal forms. Props. 9,10 and 11 are due to Herbrand (1930) as is the discussion on -ff-disjunctions, hence their name. Much has been contributed to the concepts of validity and satisfaction by Tarski (1933). Prop. 12, the completeness theorem of the classical predicate calculus of the first order is due to Herbrand (1930), Godel (1930), Lowenheim (1915) and Skolem (1920). Cor (ii) is due to Lowenheim (1915). Prop. 13, the independence of the axioms and rules of the classical predicate calculus of the first order was first considered by Godel
202
Ch. 3 Predicate calculi
(1930) and the consistency by Hilbert-Ackermann (1928). The discussion of theories is due to Tarski (1935-6). The classical predicate calculus with equality is implicit in the work of Pierce and Schroder, but its first treatment as a system in its own right is in Hilbert and Ackermann (1928) and again in Hilbert and Bernays (1934-6). Prop. 19, the elimination of axiom schemes, is due to Skolem (1959) who used a formulation of set theory due to Godel (1940). Early attempts at finding a decision procedure for the classical predicate calculus of the first order were unsuccessful (because as we shall see in a later chapter there is none), so research turned to finding decision procedures for special classes of statements. One of the earliest of these was by Lowenheim (1915). He gave a decision procedure for the monadic predicate calculus of the first order. This was followed by work by Skolem (1919, 1920) and Behmann (1922) on the monadic predicate calculus of the second order and of the first order with equality. Prop. 22 is due to Bernays and Schonfinkel (1928) and Prop. 23 is due to Godel (1933), Kalmar (1933) and Schiitte (1934) A detailed account of all known decision procedures for special classes of statements has been given by Ackermann (1954), see also Church (1956). A related type of problem is the reduction problem. Here we try to find special classes of statements such that any statement of the predicate calculus of the first order is equivalent as regards validity to one in the special class, there is a corresponding problem for satisfiability. A simple case is the class of statements in prenex normal form. These special classes are called reduction classes. Prop. 24 is due to Lowenheim (1915) and Godel (1933). The Skolem V- and ^-normal forms are of course due to Skolem (1925) and lemma (iv), the restriction to exactly three universal quantifiers is due to Godel (1933). Prop 25, where we have only one predicate and that one binary is due to Suranyi (1943) and Kalmar (1947) whom we follow closely. Many reduction types with a variety of prefixes have been found by Kalmar and Suranyi, and account of them is given by Suranyi (1959), see also Church (1956). The method of semantic tableau is due to Beth (1955, 1959) and Hintikka (1953, 1955) and Prop. 26 is due to Craig (1953) and Kleene (1952, 1967). The idea of a resolved predicate calculus is due to Hilbert [H-B, II], but the 'T/' symbol (written i) and its use is due to Peano (1897), Frege (1893, 1962) and (1905, 1956). The V symbol is due to Hilbert [H-B].
Historical remarks to Chapter 3
203
He demonstrated two theorems about the elimination of the e-symbol, the first is virtually Prop. 35. An historical account of the predicate calculus has been given by Hermes & Scholz (1932). Many examples are to be found in Church (1956). The system 88!FC is suggested by Kalmar & Suranyi's reduction type whose only predicate is a single binary one. Definitions D26 and 27 go back to Leibniz He took two classes to be the same if they contained exactly the same members. This is the extensional approach, and is used in classical mathematics. But one might consider two classes to be distinct, even if they contained exactly the same members on the ground that the rules for membership were different. This is called the intensional approach, we shall have to speak about it again in later chapters. The concept of set and class as defined in D 33 and 34 is due to v. Neumann (1925) who used it to avoid the syntactic paradoxes which had crept into set theory since Frege and Cantor. The definitions D 37-57 are largely from P.M. But the definition of an ordered pair has received simplification at the hands of Wiener (1912) and Kuratowski (1921). The rules R2'-9" are modifications of the axioms used by Godel (1940) in his account of set theory. Prop. 31 on normal classes is due to Godel (1940). When P.M. first appeared it was considered to have a few blemishes. One was the complicated type theory. Several writers have tried various ways of simplifying this, notably Leon Chwistek (1921, 1927) and F.P.Ramsey (1926), with his simple theory of types. Another thing thought by some to be a blemish is the axiom of infinity. Perhaps the reason for this may be compared to the reasons for considering Euclid's axiom of parallels to be a blemish on his work in geometry (first formalized by Hilbert (1922)), namely, one might think that it should follow from the other axioms. But apparently this is not the case. In fact in Godel's (1940) formulation of set theory the axioms previous to the axiom of infinity are consistent, this is seen by taking ae/3 to be always false. But with the axiom of infinity this is no longer the case, because this axiom postulates the existence of a set. There are many ways of formulating an axiom of infinity, all one requires is the existence of some set with an infinity of members. The last thing thought by some to be a blemish in P.M. is the axiom of reduction. Here this is avoided by the distinction between classes and sets. Set theory is very useful for making models of various mathematical conceptions. Thus we easily get a model for ordinal numbers. So we give
204
Ch. 3 Predicate calculi
a short account of them. Fuller accounts are given in Cantor (1895, 7), Godel (1940), Bernays-Fraenkel (1958), Sierspinski (1928, 1958), Bachmann (1955). We have indicated in Ex. 41-4 inclusive how to develop the algebra of ordinals. Ordinals and cardinals were first invented by Cantor, but finite ordinals and cardinals were known to the Greeks. Ex. 43 (xi) is known as Cantors normal form for ordinals. When we introduce ordinals we immediately require a new axiom, namely the axiom of infinity, its object is to ensure that the ordinals we define are sets, so the process of ordinal construction can proceed. We largely follow Godel's (1940) account of ordinals and cardinals. The alephs are defined as certain ordinals, but there may be other cardinals that are incompatible with the alephs, if we define cardinals as classes of similar classes. It is at this point that we come across A.C., first introduced by Zermelo (1904). Prop. 32 is due to Cantor and Prop. 33 to Cantor and Bernstein (1905). Cantor's theorem immediately gives rise to C.H. and G.C.H. For many years the logical position of these and A.C. was unknown. Then Godel (1940), by constructing the constructible universe, was able to find an 'inner model' in which A.C, G.H. and G.C.H. were all satisfied provided that set theory itself is consistent. Thus A.C, C H . and G.C.H. are consistent with set theory provided set theory is consistent. The proof of this takes place in set theory. Godel's method of inner models, as shown by Shepherdson (1951,2,3), is incapable of showing that the negations of A.C, CH. or G.C.H. are consistent with set theory. Many years were to pass before Cohen (1963, 4, 5) developed an entirely new method for dealing with independence proofs, this was based on the denumerable model found by Skolem. This was a remarkable breakthrough, the original paper was couched in such strange terms that only the most resolute of professional logicians could understand it. Now however, monographs are appearing and the method is explained so that it is available to the general mathematician. There is a monograph by Cohen (1965), Another method of doing the same things was discovered by Solovay and Scott (1970). This proceeds by forming Boolean valued models and gives a generalization of twovalued truth. This method arose because it was noticed that the main feature of Cohen's 'forcing' method was the semi-order it gave rise to. Accounts of this method are given by Rosser (1969), Scott (1966a, 6), Jensen (1967). By suitable choice of Boolean Algebra models can be found in which A.C or C.H. or G.C.H. fail. The complete Boolean Alge-
Historical remarks to Chapter 3
205
bras required came in after Boole, they are discussed by Halmos (1963) and Sikorski (1960). We just show how to set up the model. The full proof that it is a model for set theory together with the independence proofs is given by Rosser (1969). Thus with Godel's result that A.C., C.H. and G.C.H. are consistent with set theory the final result is that A.C., C.H. and G.C.H. are independent of the other axioms of set theory. The proof takes place in set theory so that the result just stated only holds if set theory is itself consistent. This is parallelled in mathematical history by Caley's proof in Euclidean geometry that the axiom of parallels is independent of the other axioms of Euclidean geometry provided these themselves are consistent. The elimination of the e-symbol is due to Hilbert and two theorems about it are given in H-B (1934-6), we give one of these because it is tantamount to the consistency of A.C. with the other axioms of set theory. That is we give an effective method of converting a contradiction in set theory plus A.C. into a contradiction in set theory itself. The chapter closes with some topological definitions, many of which occur in the latter parts of P.M., they show again how useful the system S? is for talking about all sorts of mathematical concepts. Other works on set theory are: Suppes (1960), Halmos (1960), Skolem (1962), Sierpinski (1951), Fraenkel and Bar-Hillel (1958), Fraenkel (1953) with a complete bibliography to 1953 and Fraenkel (1946). EXAMPLES 3
1. Complete the demonstration of Prop. 6. 2. Complete the demonstration of Prop. 8. 3. Put into prenex normal form (Ax) CpxC(Ax')p'x'x(Ax")p"x% B(Ex) (Ex')pxx'(Ex')
(Ex)pxxf.
4. Put the «^>-proof of the prenex normal forms of C(Ax)
CpxpfxC(Ex)px(Ex)p'x,
C(Ex)
NNpxNN(Ex)px,
BK(Ax) Cpxpfx(Ax) Cp"xp'x(Ax) CDpxp"xp'x, into normal form.
206
Ch. 3 Predicate calculi
5. Obtain H-disj unctions for (Ex) (Ax') (Ex") DDpxx'xpxx'x"Npx"xx', (Ax) (Ex') (Ex") (Ax")DDpxx'x"Npxxx"px"x'x, (Ex) (Ax') (Ex") (Ex'") DDpx"x'xNpx"'x'x"px"'x'x. 6. Obtain semantic tableau for the statements of Ex. 5. 7. Find the equivalent of (Ax, x', x") (Ey) KDpxx'p'x'yCKp"x"xp'x'yDpyx"p"xx" according to Prop. 25. 8. Find which of the following are ^-theorems: (Ax, x', x") (Ex®\ x®>, x®)
DCpxx^x^px'x"x^Cpxx^x^ px"x'x®\ (Ax, x') (Ex", x'") CCpxx"px'x"'Cpx"xpx'"x'.
9. Find which of the following are J^-theorems: (Ex) (Ax', x") (Ex"1) GKpxx'px'x"Gpx'"xpx"x', (Ex) (Ax', x") (Ex1") CCpxx'px'x"Cpx'"x"px'x. 10. Reduce (Ex) (Ax', x") (Ex"1) (Axir) CCpxx'xCpx"x"'x'Gpx"xiYxf"pxiYx'x to the form given in Prop. 24. 11. Put in resolved form, using the e-terms (Ax)(Ex')(Ax")
Show that:
for
$ft'$"(x = x'),
&e
for
SAr&"(x = x").
9t?X = MX, 2«X= VxX, S {x, x'} = x U x', X{x} = x, £"{x} = x, C(X c X') (Q)X c CK(X c X') (X" c Xm)CX«X* c X'«XXm,
Examples 3
207
13. Show that (X x X')n {X" x Z i v ) = ( I n X") x (X'n X iv ), U X') = S Z U S I ' , X = SPX, CN£mX{X' c SS(Z x X')), X c P2X,
2 2 / = V,
14. Construct an applied predicate calculus of the second order which has exactly one constant predicate R which is binary. State axioms for order and express 'every non-empty class has a least member in the ordering R\ 15. Set up axioms for a group using one ternary predicate. 16. Find the Skolem $-normal forms and the Skolem F-normal forms for the following: C(Ax)px(Ex)px9 C(Ax) Cpxp'xC(Ex)px{Ex)p'x, C(Ex) (Ex')pxx'(Ex')
(Ex)pxx'.
17. Give the demonstration of Prop. 24, lemma (iii). 18. Show that the conditions (A), (B), (C) and (D) below are necessary in order that (Ag, £', £") (Er/) ^{£, £', g", TJ} be satisfiable. There is a nonempty set S of 4 x 4 tables which satisfy <j) such that: (A) If To is a table of S, v, v\ v" = 1, 2, 3, 4 then there is a table T of S such that: [T/1] = [TO/1] and [T/123] = [T0/v', v', v"]. (B) If To is a table of S then there is a table T of S such that: [T/1] = [TO1] and
T = [T/1114].
(C) If T 1; T 2 are tables of 2 then there are tables T, T", T" of S such that:
[T/l] = [T^l],
[T/2] = [T 2 /l],
T = [T/1224],
[T'/2] = p y i ] ,
[T'/l] = p y i ] ,
T' = [T/1214],
[T"/2] = [Tj/1],
[T'/l] = [T 2 /l],
T" = [T"/1134].
208
Ch. 3 Predicate calculi
(D) If Tl9 T2, T 3 are tables of S then there is a table T of such that: [T/l] = p y i ] ,
[T/2] = [T 2 /l],
[T/3] = [T 3 /l].
19. Apply Prop. 20 to decide which of the following are J^-theorems. (i) (Ex) (Ax') GBp'xpBp'x'p, (ii) B(Ex)Gpxp'x(Ex,x')Cpxp'x, (iii) (Ax') G(Ax) CpxCpx'p'xCpC(Ax)pxp'xf. 20. Apply Prop. 22 to decide (Ax, x', x") (Ey,yf)
Cpxx'Cpyx"DKpyx'py'x'Kpxx"py'x".
21. Apply Prop. 22 to decide (Ax, x') (Ex")
KCpx"xKCpxx'pxx"Cpxx"GNpxx'Kpx"xpx'x".
22. Find the binary SFC statement equivalent to (Ex) (Ax') (Ex") Cpxx'x"px'x"x as regards satisfiability according to Prop. 24. 23. Continue Ex. 22 to find a binary J^-statement with exactly one binary predicate which is equivalent to (22) as regards satisfiability according to Prop. 25. 24. Ditto for (Ex) (Ax')(Ex")\(Ax'") Cpxx'x"px'x"x'", (Ex) (Ax')(Ex", x"1) Cpxx'x"x'"x"px"xx'x"xm. 25. Demonstrate Prop. 12 for the system I^Q. 26. Demonstrate Prop. 14 for the system I^c. 27. Show that the singularly I!FC is decidable. The only predicates in the singulary I!FC other than / are singulary. 28. Show that B(X= Y)(Au)B(ueX)(ueY) and B(X = Y) (A U) (XeU) (YeU) are independent. [See Robinsohn, J.S.L. 4, 69.] 29. Show by formula induction X= Y B where F is free for X, Y in
Examples 3
30. Show and
209
B<j>{ Y}(AX) C(X = Y) 4>{X} B
31. Show
B(XeY) (Eu) (u = X) K(u = X) (weY).
32. Show
X = y(yeX).
33. Show
(A£,',...,^)B
where S is like y except for containing ijr at some places where y contains ^ and £', ...,£,(v) exhaust the variables with respect to which those occurrences of 0, i/r are bound in y, S.
where \]r is like 0, except for containing 8 at some places where $ contains y and £',..., £^p) exhaust the variables with respect to which there occurrences of y, S are bound in
jy 7 / V
T7"\
Rel (X, Y)
/ A
\ ID V
Tr
(Au, v) BuXvu Yv X=Y
36. Show
{u, v} = {x, y) DK(u = x)(v = y) K(u = y) (v = x)
37. Show
(u,v) = (x}y) K(u = x)(v = yY
38. Show
(Ay) (y = Fy).
39. Show
(Av) B(v = y) vXx y = X'x
40. Show (Ax)BKUnX{xeSiX){Eu)(Av)B{u = v {Ax)BKVnX{xe9X) (Ay)B(y = X'x)yXx, Y&nX (Aw)B(weY) (Eu) K{ueX) (w = Y&nX
Z&nX
(Y'u,u)y
(Au)C(ueX)(Y'u = Z'u) Y =Z
210
Ch. 3 Predicate calculi
41. Defining the sum a + b of two ordinals a, b as the ordinal isomorphic as regards order to the order type obtained by sticking the order type b at the end of the order type a, show that: (i) If 0 < b then a < a + b. (ii) b ^ a + b. (iii) If b < a then there is a unique ordinal c such that b + c = a. (iv) If b ^ a and d < c, then b + d ^ a + c. (v) a = b1 + c1 = 62 + c2> c i < c 2J then6 2 < 6X. (vi) If a = 6 + c, c is called a remainder of a and 6 is called a segment of a. Show that the number of remainders of an ordinal is finite. (vii) An ordinal a is called decomposable if a = b + c, 0 < b, c < a. Otherwise indecomposable. Show that the least positive remainder of a positive ordinal is indecomposable. (viii) If cx < c2 are both remainders of an ordinal a, then cx is a remainder of c2. (ix) The least positive remainder of an ordinal a is a remainder of every other remainder of a. (x) The only positive remainder of an indecomposable ordinal is itself. (xi) The only positive indecomposable remainder of an ordinal is its least positive remainder. (xii) If c is indecomposable and b < c then b + c = c. (xiii) If c > 0 and b + c = c whenever b < c then c is indecomposable. (xiv) Any ordinal a is the sum of a finite decreasing sequence of decreasing indecomposable ordinals. (xv) If
a = c1 + c2+...+cn,c1
> c2 > ... > cn
and
cl9c29...,cn
in-
decomposable then cx is the greatest indecomposable ordinal < a. (xvi) If A is a set of indecomposable ordinals then HA is indecomposable. (xvii) Every ordinal can be uniquely represented as a finite sum of non-decreasing indecomposable ordinals. 42. Defining the product ab of two ordinals a, b as the ordinal order isomorphic to the set of ordered pairs (c9d} c < a,d < b ordered by last differences, show that (i) If b± < b2 and 0 < a then abx < ab%. (ii) If ax < a2, b± ^ b2 then axbx ^ a262. (iii) We can have ax < a2, b > 0 and axb = a2b. (iv) If a = be, 0 < 6,1 < c, then b < a and c ^ a.
Examples 3 (v) If abx < ab2 then bx < b2. (vi) If axb < a2b then ax < a2, (vii) If ab± = ab2, 0 < a then bx = b2. (viii) lie < ab t h e n u n i q u e l y c = ab1 + d,b1
211
< a.
(ix) If 0 < a then b = ac + d, d < a, uniquely. (x) If c is indecomposable, 1 < c then ac is indecomposable, (xi) If 0 < a then the least indecomposable ordinal greater than a is aco.
(xii) If c is indecomposable and positive then the next greater indecomposable ordinal is ca). (xiii) Every indecomposable ordinal is divisible on the left by every lesser positive ordinal and the quotient is indecomposable. (xiv) An ordinal is prime if it is greater than unity and is different from the product of any two lesser ordinals. Show that every ordinal > 1 is the product of a finite number of primes. (xv) Show that the number of right divisors of an ordinal is finite. 43. Defining ba for ordinals a, b as the ordinal order isomorphic to the order type of functions over a with a finite number of non-zero values in b ordered by last differences, show that: (i) This order type is that of an ordinal. (ii) If 0 < a < 6, 1 < c then ca < cb. (iii) If 0 < a < 6, 0 < c then ac < bc. (iv) ca+b = ca.cb, 0 < a, 6, c. (v) If b is a limit ordinal, 1 < a then ab is the limit of all ordinals 0 a for c < b. (vi) (aa)c = abc, 0 < a, 6, c. (vii) o)a is indecomposable for a > 0. (viii) If 0 < a, 1 < c then a < ca. (ix) If 0 < d, 1 < c then there is exactly one ordinal a such that ca ^ d < ca+1.
(x) Every indecomposable ordinal ^ co is of the form a)a for some a > 0. (xi) Every ordinal a > 0 can be uniquely expressed in the following normal form: „ d = fc)ci. 71^ + (i) 2 . 7l2 + . . . + (if™. 7lm
w h e r e cx> c2> ... > cm a n d nx,n2, ...,nm a r e i n t e g e r s . 44. A n e-ordinal is a n o r d i n a l e w h i c h satisfies e = co6. S h o w t h a t : (i) I f e0 = co + a)*0 + w(6>CU) + ... t h e n e 0 = 6>e«.
212
Ch. 3 Predicate calculi
(ii) e0 is the least e-ordinal. (iii) If 1 < c < e0 then e0 = ce«. (iv) If 0 < c, c0 = c, cn+1 = G)cn then limcn is the least e-ordinal not less than c. 45. Show that there exist ordinals which satisfy a = o)a. [Consider c0 = ^> cn+i = Mcn>
c
= limcn.]
46. Show that: (Ax, x') D(x = x') K
--->z{n\ x',...,x(0)}
can be expressed in the form
A
2 HK{z',..., z(7r)} FK{cj), ..., = , J, ?}, where 5 stands for 2',..., z(7r) and j for /c = l
x',...,xf®. Hence show that (E%)(A%)F{$,..., = , J , j } is equivalent to one of (Z?s) (^l j) HKFK for some /c, 1 ^ K ^ A. 48. Show that DN(Ax) (Ey) <j>{x, y] (Ex, x', x", x'") (DKKKKK
x")
is generally valid. Ackermann J.8.L. ai (1956), p. 197. 49. Show that (i) (Ex) DN4>{x, x) {Ay) NK
are generally valid. Oglesby (1962).
Chapter 4 A complete, decidable arithmetic. The system Ao
4.1 The system Aoo In this chapter we construct the formal system Aoo. It is a very simple arithmetic with familiar fundamental concepts. These are: the natural number zero, the successor function, the operation of repeatedly applying a function, the operation of forming functions by abstraction, equality and inequality between numerical expressions, and the logical connectives, conjunction and disjunction. The atomic statements are equations and inequations between numerical terms, compound statements are built up from atomic statements by conjunction and disjunction. Negation, material implication and material equivalence are definable, but existential quantification and universal quantification are unrepresentable. We give definitions of AQ0-truth and of A00-falsity for closed AOostatements, and show that they are exclusive properties. We also show that a closed A00-statement is A00-true if and only if it is an Aoo-theorem. Thus the system Aoo is consistent in the sense that Aoo-theorems are Aoo-true; and is complete in the sense that A00-true AOo-statements are Aoo-theorems. We give a procedure which applied to a closed A00-statement will terminate and tell us whether it is A00-true or is A00-false. Thus the system Aoo is decidable. 4.2 The Aoo-rules of formation To construct the system Aoo we first list the A00-signs and attach a type to each proper A00-symbol and give each Aoo-symbol a name which will assist the reader in understanding how the system was first conceived. Parentheses round type symbols are usually omitted by association to the left as explained in Ch. 1. (See table overleaf.) The required unending sequence of variables is obtained by repeatedly attaching primes, thus: x, x', x", .... The table order of the proper and improper symbols is called their lexicographic order. In the system Aoo [ 213 ]
214
Ch. 4 A complete, decidable arithmetic. The system Aoo symbol 0 8 = =1= & V
name
type I
11 Oil
oil 000 000
J
ui(iu)
X
i
zero successor function equality inequality conjunction disjunction iterator operator variable
The improper isymbols are, as usual: X ( )
abstraction operator left parenthesis right parenthesis generating sign
the natural numbers are represented by certain formulae of type i called numerals. These formulae are defined by the following rules: (i) 0 is a numeral, (ii) if v is a numeral, then so is (Sv), (iii) these are the only numerals. Thus 0, (SO), (8(80)), (8(8(80))), ... are the numerals. Note that a numeral is a formula, distinct numerals are distinct formulae. An Aoostatement is an A00-formula of type o, according to the universal rules as given in Ch. 1. The only abstracts allowed are of types u, ui, uu, etc., i.e. (kx
to denote undetermined numerals, to denote undetermined formulae of type i, called numerical terms, to denote undetermined variables of type i, to denote undetermined statements, to denote undetermined functors of type a, ui, etc.
4.2 The Aoo-rules of formation
215
In the last case the type can be introduced as a subscript if desired. From now on we usually omit the concatenation sign. Thus = otfi stands for an undetermined equation between numerical terms, &
(<* = /?) for
D113
(&>ft)for
D114
(?W^) for
= a/?,
(a 4= fi) for
+ a/?,
&$ft,
v#\
We shall frequently use the parenthesis convention, thus: ^ Vftv x stands for ((<}) vft)vx)
an
d hence for ((V ((V (j>) ft))x)>
The AoO-rules of consequence
The system Aoo has the following axiom schemes: Ax 00 .1
(oc = a),
Axoo.2.1
Offa#0)f
216
Ch. 4 A complete, decidable arithmetic. The system Aoo
Ax 00 . 2.2
(0 4= Sot),
Axoo.3.1 Axoo. 3.2
where a, ft are closed numerical terms and p is a closed function of type in. i... i is the result of striking out each parenthesis and the zero in the Sn- times
formula (Sn) and then replacing each occurrence of S by a corresponding occurrence of i and replacing the parentheses by association to the left. Ax 00 .4.1 Ax oo .4.2
(X£. p{$) PP'... P* = p{p}pf... /?<*>, (7
where /){£} is of type i... i and both sides are closed and £, £' fail to occur Sn-timea
free in p{TL] and I \ is free for £, £' in p{Tt}. When written in full the parentheses are put back by association to the left, viz.: This axiom allows us to apply an argument to a function. We really only need functions of at most two arguments but it is sometimes useful to have functions of any number of arguments. This makes these two axioms more complicated. In them p{£,} is of type i... i so it must be of the form: where a{£, £',£", ...,£(7r)} is a numerical term, or it could be; (Xg'(Xg*(...(Xg<'-»5)...)) or or
(H'(H"
with appropriate modifications if n = 0,1,2. By repeatedly applying Axoo- 4.1, 4.2 we can change any Ef® to a new variable. The first axiom scheme states a familiar property of equality and the two parts of the second axiom scheme state a familiar property of the successor function. We require both parts because if we worked with only one part then some properties of inequality which we require would fail. The third axiom scheme shows how the iterator operator acts. This may become clear if Ax00. 3.2 is written in ordinary mathematical notation: suppose that / is a function of two arguments, then {{Jf)m{n+l)=f{n,{Jf)mn)).
4.3 The A00-rules of consequence
217
If we write g[n, m] for ^ffmn we obtain: g[n + l,m]
=f[n,g[n,m]].
g[3,m]=f[2,g[2,mj\
Thus
since g[0,m] = m by Axoo. 3.1. In general g[n,m] = / [ » , / [ » - 1 , [ f [ » - 2 , ...,/[2,/[l,/[0,m]]]... ]]]]. The system Ao0 has the following rules of procedure: ^{a} v
where a and J3 are closed numerical terms and co is a closed A00-statement and is subsidiary as in R 1, (a #= /?) and (Sot 4= #/?) are the main formulae. Note that since a and /? are closed in R 1 then a variable whether free or bound is unaffected by applications of R 1. The remaining rules are labelled in a different manner because they are some of the rules of &c. In listing them, except in one case, we omit the condition that the A00-statements in them be closed because they can only be used in A00-proofs when this is so. In the exceptional case the condition must be stated otherwise free variables could be introduced into an Aoo-proof. I.
Remodelling rules 0)' V
w'vfv^vw' II.
permutation
Building rules (a) -X— dilution
(b'\ ^yC°
^V0)
composition
218
Ch. 4 A complete, decidable arithmetic. The system Aoo
In II (a) (j) is closed. In II (&') the order of the premisses is immaterial. The Aoo-statements a>, a)' are subsidiary and may be omitted, the other Aoo-statements are the main formulae. The A00-statement x is secondary and must be present. We have omitted parentheses by association to the left and the outer pair is usually omitted. The rules are known by the names beneath them. 4.4 Definition of A00-truth We say that a closed numerical term 7 determines a numeral v under the following conditions: (i) 7 is v. (ii) v can be obtained from 7 by replacements of the following kinds: (a) replace an occurrence of ^paO by a corresponding occurrence of a; (b) replace an occurrence of Jrpoc(Sj3) by a corresponding occurrence of (c) replace an occurrence of X£.p{£}/?/?'.../?(7r) by a corresponding occurrence of p{/?}/?'... /?(7r); (d) replace an occurrence of X£./?{£}/?/?'... /?(7r) by a corresponding occurrence of X£' ./ where £ and £' fail to occur free in p{TL} and TL is free for £ and £' in p{Tt}. If 7 determines the numeral v then 7 = v is an Aoo-theorem and the above replacements give a special type of Aoo-proof of 7 = v. We say that an A00-statement is A00-true if and only if it satisfies the following conditions ^ 0 . (i)
then N["$®K)"] v f can be A r deduced from hypotheses $', (j)'\ ...,
'") is $', since (j)' is A00-false then one or both of
330
Ch. 7 Ao-Definable functions. Recursive function theory
We denote the partial recursive function of 6 arguments whose table has g.n. K by:
= v iff
(#£) (DW»[*; i/',..., jX*>, g, y] = 0).
We let 3CK denote the range of ®[AC; V], we write this as
7.12 Productive sets A set ^ is productive if there is a partial recursive function p such that whenever 2£K c y then /o/c is defined and pKeSf — 2E. p is called a ^roduction function for ^ Thus the set ^fr is creative if it is recursively enumerable and its complement is productive. P R O P . 15. / / ^r is a creative set then we can find a recursive (1-1) production function for it. L e t / b e a (partial) recursive productive function for the creative set Wr. We define a function/' as follows: «^0 is the null set, hence / 0 is defined, (&0 c ¥r), we set/'O = / 0 . Suppose that/'O < / ' l < / ' 2 <... < / V have been defined, we give the following procedure for calculating f'(SK): Let table A calculate the recursively enumerable set 3?JSK0 ^r and let table B do the following calculations: Start calculating f(SK), if this is found, find a g.n. for the recursively enumerable set {/(/S/c)} U XSK say JI^SK) SO that
Then start calculating /[^i(#/c)], if this is found, find a g.n. for the recursively enumerable set {flh^Sfc)]} U ^hx(sKh s a y M ^ ) * then start calculating f[h2(SK)], and so on until we have found /(<S*),/[fti(<S*)], • • • # O T ( S K ) ] .
(15)
Table J. is set to stop when it produces a numeral, e.g. let n be a gr.n. for a table which recursively enumerates ^ K n ftr, then generate the natural numbers and when the natural number v has been generated find vl9 v2 so that v — {vv v2}, then set going table n on argument vx for v2 moves, if it produces a natural number /i then table A stops, if not proceed to Sv. Table B is set to stop when it has calculated the series (15). Table A will
7.12 Productive sets
331
eventually stop unless 3£Slc <=
exists and f(SK)G^SKf]¥r,
(16)
^1(^) = {/(^)}U^
(17)
thus arhl(Sld c ~¥r. Again f\hx(SK)] exists and KK{SK)\e£hi{BK)(\¥r. Now
Srh2(SK) = {RhiSK)]}
(18)
U &hl(SKh
and so on. Thus the calculation performed by table B will eventually stop. Hence either table A or table B will eventually stop. We set )
> * ^ b l e Estops first, \ Max [/(Sic^flhjlSK)],... ,f[hsirK)]] if table B stops first,
v
;
t h u s / ' is a recursive function. Further if 3CSK c f t w e see from (16), (17), (18) that f(SK),f[h1(SK)l...,f[hsif.K)(SK)] are all distinct and are in ^"^fl ffr, thus the maximum of these SS(f'K) numbers is greater than or equal to S(/K). T h u s / ' is strictly increasing. Also the maximum is in 3£$K 0 ftr. Hence / ' is a recursive production function for tfr, it is (1-1) because it is increasing strictly. The choice of value oif'(SK) given in (19) is settled for us if we let the two tables do one move alternately beginning with table A. This completes the demonstration of the proposition. We say 's/ is (l-l)-reducible to 88' for 'there is a (1-1) recursive function /such that v e stf if and only iffve&\f may omit some values of &. If si is (l-l)-reducible to ^ we write si < x 3$. LEMMA.
For any numeral K we can find a numeral 0 such that
O[/c; v, S[[TT, TT]] is a partial recursive function of v, n effectively given by K, hence we can find a g.n. for a table for it, say gK, where g is a recursive function. Then .r .,-, o,r
332
Ch. 7 Ao-Definable functions. Recursive function theory
whence
O[/c; v, S[[gK, gic]] = O[g/c; v, gtc],
b u t from P r o p . 4
<3>[gK; v, gK] = ^>[S[[gK, gic]; v],
put 6 = S[[gK, gK] and the result follows. 7.13 PROP.
Isomorphism of creative sets 16. Any two creative sets are isomorphic.
The demonstration falls into two parts. (i) any recursively enumerable set is (l-l)-reducible to any creative set. (ii) If s/ ^ ! 8ft and 8ft < x stf then si and 8ft are isomorphic. Ad(i). Consider the creative set <€r with (1-1) recursive productive function/, we show that SCe is (l-l)-reducible to #V. We define a set v
10
otherwise.
Then given /i and v the set <S^ can be effectively enumerated. (Define a function y as follows: generate the numerals and when the numeral n has been generated find 7rx and TT2 SO that n = {T^, TT2}, then set going table 6 for argument n1 for 7r2 moves if this has produced v then set yn = f/i, otherwise yn undefined.) Let g-^fi, v] be a gr.w. for a table for <3^. Now Q>\gi[x',v\\x\ is an undetermined value of a function of x,x' which can be effectively found given v, thus for some recursive function gr2. By the lemma O[gr2v;A,7r] =
O[TT;A],
f
where n = S 1[g[g2v],g[g2v]], g being as in the lemma. Hence ^ is 2£n> with this value of n. If ve&0 then fne&g, by definition of ^ , hence / T T G ^ ; and so &n <£ ¥r, since S£w = { » then fne Vr&nd so fnetfr. (If Xw c: ^ r then since / is a productive function for ^V we should have fne¥r{\3tn whence fire&n.) If i>e#* then # ; and ^ are bothjnull, by the definition of <&% whence fireVr by property of/, (SCn c ^r). Thus pe^ i.e. where
iff
/Tre^r,
V G ^ iff
hve&r,
Az^ = fS[[g[g2 v], g[g2 v]] == /TT.
7.13 Isomorphism of creative sets
333
It remains to show that the recursive function h can be made (1-1). Given a g.n. of a table we can effectively enumerate an unending sequence of gr.Ti's of equivalent tables, for instance we can easily enumerate the g.n. 's for the sequence of tables for the functions: /, PSf, PSPSf,.... Now the values of the functions g, g2 are defined as g.n.'8 of certain functions so using the just mentioned fact we can define new functions *$i, *gr, *g2 which are strictly increasing and thus (1-1), and they will play the same roles as S[, g, g%. Thus the composition of these functions with the function/ (which by the lemma is (1-1)) as in the construction of the function h will be a (1-1) function *h satisfying ve3Te iff
This completes the demonstration of (i). Ad (ii). Let s/^ X0S and ^ ^1s/, then there are (1-1) recursive functions Jiif f, q such that „ .~ ves/
iff
veSS iff
gves/.
We wish to construct a recursive permutation function h such that vest iff It will suffice to show that we can effectively enumerate a set F, of pairs {A,/i} with the following properties: (i) if {A, /i} is in F then Aej/if and only if fie&,
(ii) {A,/i'},{A}/i"}erthen/i'=/i", (iii) {A',/4,{A'>}eF then A' = A", (iv) every numeral occurs as first member of a pair of F, (v) every numeral occurs as second member of a pair of F. Suppose that the first 2.n pairs have been chosen and that (i), (ii), (iii) are satisfied by them. We choose the (2. n + l)st and the (2. n + 2)nd pairs as foliows: (2.7T+ 1). We let the first member be the least numeral A' which has failed to appear as yet as first member of a pair (this ensures (iii) and (iv)). Then if /A' has also failed to appear as second member we take the second member to be/A'. If, however,/A' has been a second member let the corresponding first member be A"; in this way we form a sequence until we come to an / A w which is either the same as / A ^ for K < 6 or
334
Ch. 7 A0-Definable functions. Recursive function theory
has as yet failed to appear as second member. (Since only 2. n second members have been chosen at this stage then one of these possibilities materializes for 6 < 2n.) Only the second case can occur, because by (iii) if fMS6) = / A ( ^ for K < 6 then X{se) = A{SK) since / is (1-1). By (ii) this makes fX(d) = /A(Af), this is impossible by our assumption that (S6) was the first numeral such that /A(Sf<9) = /A(;Sr/c) for some K < 6. Thus only the second case can occur, we take / A w as the second member of the (2.7T+ l)st pair. We see by this construction that property (ii) has been preserved because the first member of the (2. n + l)st pair has previously been absent. Also X' estfii and only if /A' e3S (by the property of/) and this is so if and only if A" G j / ( b y the property of F) and this is so if and only if/A" e 38 (by the property of/) and so on until if and only if/A ( ^ e f . Thus A' G j / i f and only if/A(flf^e^? and the property (i) is preserved. (2.7T + 2). The construction is the same with the following obvious changes: second member
9
for first member\
»
f and vice versa.
(iv)
„
(v)
The construction of the recursive permutation h is now obvious. This completes the demonstration of (ii). Now let ^r' and #V" be creative sets. Since ^V' is recursively enumerable and <£>" is creative then by (i) %rf < J€r\ similarly #V" ^ tfr'. By (ii) <£>' is isomorphic to ^r". This completes the demonstration of the proposition.
7.14 Fixed point proposition P R O P . 17. Iffis a recursive function then a numeral 6 can be found such that O[/0; v] = O[0; v] for all v. (I.e. both sides are undefined, or both sides are defined and have the same value.) For fixed n, given v, find
7.14 Fixed point proposition
335
is the table for the constant function n, Tn is the table whose g.n. is n, U is the universal table. Thus The function/o 9, (^#-/[(7#])> is a recursive function, let A be a g.n. of a table for it. We have ^ r . -. A r / x n . n n O[grA;*>] = O[d>[A; A];*>]
but
*[A;A]=/foA],
so that
Now take # = gA and we have finished. This proposition is known as the fixed point proposition. COR. (i). There exists a recursive function h such that for any numeral /i iffi is the g.n. of a table for a recursive function /, then for all v.
(I.e. both sides undefined or both sides defined and have the same value.) From the proof of the fixed point Prop, we have: where A is a g.n. of a table for the recursive function / o (7- Clearly this g.n. is a recursive function k of the g.n. of a table for the recursive function /. Now take g Q k for h. 7.15 Completely productive sets A set J / is said to be completely productive if there exists a recursive function / such that fn e (s/[) SC^) (J (<stf D 3£n) for each numeral IT. A completely productive set fails to be recursively enumerable. If sf= SCn forsome^then (
U <*> * , ) - 0,
and so fn fails to be a member. A completely productive set fails to be recursively enumerable in a strongly constructive sense in that the counter example, fn, can be produced to witness that j / i s distinct from 3£n. A completely productive set is productive. I f ^ <= j / t h e n ^ n <st = 0,whence/7TG^'7rn s#, and so sf is productive. P R O P . 18. If stf is a productive set, then <$/is a completely productive set. Let / be a productive function for the productive set s/. Let n be fixed. Take any numeral v. Enumerate 3En and stop when we find fv (e.g.
336
Ch. 7 A0-Definable functions. Recursive function theory
generate the numerals and when the numeral 6 has been generated then evaluate Unf[n\ 6V #2, #3], where 6 = {dv d2, #3}, if it is zero then test if #3 = fv if this is so then stop, otherwise proceed to Sd). The g.n. of this table is a recursive function of v, say gv. By the fixed point Prop, the g.n, of a table for the same calculation will be v itself for some numeral v. From the corollary to the fixed point Prop, we see that v is a recursive function of the g.n. of g, then we have v = hn for some recursive function h. Thus % {} hn \0 otherwise. We now show that j / i s completely productive with productive func-
hence f[h7T]es/(] 2£hlJ since/is a productive function for stf. Thus f[hn]es/9 If if
altogether f[h7r]es/()S£n.
f[hn]earn f[hn]es/
then then
SChn = {f[hn]}, ^hn^^,
but then /[^Trjej/n ^ w , since / is a productive function for s/9 which is absurd, hence «r, n -7 , -7 f[hn]ej/ and so In either case /[/&7r]e(^n ^ w ) U 7.16 Oracles At the beginning of this chapter we defined a calculable function of natural numbers as a set of instructions which applied to a natural number produced another natural number. Similarly for application to sets of natural numbers. We analysed this vague intuitive idea and gave a precise definition of calculable function, namely a function calculable by a Turing machine from a complete set of atomic instructions. We also considered the case when the calculation failed to terminate or terminated in something other than a natural number, such functions we called partially calculable. We now extend the Turing machine definition of calculable function of natural numbers to include the case where at some point in the calculation we require to know the value of a certain function / for a certain argument v.
7.16 Oracles
337
If / is a calculable function then we can merely add the extra piece of calculation to find/V and we have the case we have been considering. But if/fails to be a calculable function then we have a new situation. We appeal to an Oracle to tell us correctly the value offv. We could suppose that we are supplied with a second tape on which are written the successive values:/0,/l,/2,..., so that we need only copy down the value required when wanted. A function g which can be calculated given the correct values of a function / as required will be called an j^-calculable function, or a partially J^-calculable function if the calculation fails to terminate or terminates in something other than a natural number. Similarly for/',... J^-calculable functions if we require to know the values of several functions. We have defined primitive, general and partial recursive functions and we showed that partial recursive functions were partially calculable and conversely. Now add to the initial functions the functions/', ...,/(7r), the resulting functions will be called primitive, general or partial f, ...,/(7r)recursive functions respectively. If the functions /',...,/ (7r) are general recursive and if g is general/',... ,/(7r)-recursive then g is general recursive and similarly in the other cases. We have shown that a partial recursive function is A0-representable. Suppose that we add to Ao new axioms giving the values of the functions / ' , . . . ,f(n) for all arguments, this list will be unbounded and will fail to be given by an axiom scheme unless/', ...,/(7r) are recursive, but this case is without interest. This system will then fail to be formal, we call it Act/',...,/<*>]. In Prop. 3 of this chapter we showed that given a partial recursive function of K arguments we could find a numeral 6 such that the function g was Ao-represented by the A0-statement: where the value, of g[v',..., *A)] is v and where Un(K) is a primitive recursive function. We now obtain a similar result for/', ...,/(7r)-recursive functions. 19. / / the function g with 6 arguments is partially f,... ,/(7r) recursive and if the functions have 6', ..., #(7r) arguments respectively then g can be A0[f',...,j^^-represented by PROP.
. . ^ ^ [ / c , ^ , . . . , ^ ) , ^ , . . . , ^ ^ , ^ ] = 0)
(20)
338
Ch. 7 Ao-Definable functions. Recursive function theory
for some numeral K, conversely for each numeral K (20) represents a partial / ' , ...J^-recursive function of 6 arguments whose value for the arguments vr,..., v(KHsv.
Here D217
fg for
this is a register of the initial values of the function/, where / i s a #-place function and TJn%_t^n) is a primitive recursive (d + n + 3)-place function. K is called the index of the function gr. We proceed as in Prop. 2 of this chapter except that the complete table now has extra lines which correspond to compound acts. We have extra instructions #', ...,#(7r) such that when under instruction g(A), 1 ^ A < 7T, and observing a cipher we do nothing and pass to the passive instruction. But if observing a tally we test whether this is the last tally of the representation of a #(A)-tuplet {ju,',..., /^(A))} and if the test is affirma-^ tive we write down two ciphers and then S{JIL', ...,/^(A))} tallies and then, after a one cipher gap, Sf[/i',..., /^(A))] tallies and stop observing the last of these tallies and pass to instruction 8A. But if writing down these tallies we encounter a square already containing a tally then we pass to the passive instruction. If under instruction g(A) and the test just mentioned fails then we pass to the passive instruction. In other words if under instruction g(A) and observing a #(A)-tuple {/i\ ...,/^(A))} in standard position with sufficient ciphers to the right so that we fail to foul a tally then we write down the representation of {fi\ ...,/^(A))},/[/£', ...,/^(A))] leaving a two cipher gap otherwise the machine stops. The machine just described is called a Turing/',...,/^-machine and is said to/',...,/^-calculate. The extract from the 'table' for the function g which involves g(A) (the function g fails to have a table in the sense used previously) is
(($A) or 0), according as we foul a tally, where i£ is the table for the test described above and F^ directs us to write: ,,{/£', ...,/^(A))},/(A)[/£', ...,/^(A))] where commas denote ciphers and the test affirmed that we were observing the tuplet {/£', ...,/^(A))} in standard position. The instruction we are referred to after doing F^ must be unique (we finish J^(A) observing a tally) we call it instruction ($A), JP(A) fails to be a table unless/(A) is partial recursive, it is the oracle, telling us something we are unable to find out for ourselves. A complete table for (A)
7.16 Oracles
339
the calculation of a partially / ' , . . . ,/(7r)-recursive function g is a table of entries: N N AO RTT[X) Al RniX)
L I
(1 < A < TT'),
L I 0 ^ n[A\ n{2X) ^ T T ' + TT.
where
The passive instruction is omitted as usual and so are entries corresponding to the instructions q',..., q(n) here numbered Sn', ..., TT' + TT. The order of the table is TT'. The g.n. of a complete table for the calculation of a partially / ' , . . . ,/(7r)-recursive function is the numeral K determined by ) , where
K^ =
N R
according as there is T in the cor0 = 4.
TT(2U)
+ 9 Mod [4. S(n + TT'
responding entry.
Every numeral is the g.n. of some such table. Note that the g.n. is independent of the functions/', ...,/(7r), and is primitive recursive in TT. We define the function J exactly as before in Prop 2 except that we require additional cases corresponding to the compound intructions q',..., q(n\ here numbered 8n',..., TT' + n. If the g.n. of the situation is {/iniynln...
n
i y > - 1, l,#7r', A}
(if/^ is zero then the first component begins /i'). Then this changes to (modification as above if JLL is zero)
provided A = 0 or A = <0>nA', where m otherwise {/i, ju,', Sn', /i"} changes to {/i,/i',O,ju,"}. There are similar cases when the third component of the quartet is TT' + 2, ...,TT' + TT. The previous eight cases receive the qualification that the third component is to
340
Ch. 7 Ao-Definable functions. Recursive function theory
be less than or equal to n\ The g.n. K of the table will be (K\ ..., /c(7r)) where K\ ..., K^ are as just described. Then J(7r)[*r, v] is the g.n. of the situation which arises from the situation whose g.n. is v after one act (atomic or one of the new compound acts). In the new clauses write Pt{\'a, {/,...,p??)}]
instead of / > ' , . . . ,
where a is sufficiently large (greater than any {/i\ ...9/^ef)} that occur). This leaves J^\K, V] unaltered in value, it is a primitive recursive function of K, V, and fa,...,f (7r) a. D218
^,...,^)[^^fa,...,f ( 7 r ) a]
for the function J(7r)[/c, i>] just described. Note that J(7r)[/c, v] depends on 6', ...,6^n\ The term a has to be sufficiently large so that the required components of the ordered sets are in the register. Thus as the calculation proceeds a may have to change, it may have to depend on the number of moves already done or on the g.n.'s of past situations. Note that the functions /', ...,/(7r) enter into the expression D 107 in such a way that they fail to occur in JV,...,0(")[^> £> £'> •••> £(7r)L tlm is a primitive recursive function of the (n + 2) variables 7/, £, £',..., £(7r). We now define D219 0 will depend on d\ ..., #(7r), they should have been appended as suffixes. In this replace v' by fa,..., y(7r) by f(7r)a, where a is sufficiently large, then 0[/c, *>, fa,..., f(7r)a, A] is the <7.w. of the situation after A applications of the table whose g.n. is K to the initial situation whose g.n. is v Suppose the calculation terminates after A moves having calculated v from the argument set v\ ...,v(0). We then have ©[*, QF>[v',..., i#>, 1], Ya,..., p>a, A] =
fi(>',...,
i**>, P, 0],
provided a is sufficiently large. It remains to choose a. If in the course of the calculation we require the value of /'(X,...,..., /^0f)] then we have to write down at some place on the tape, and finish this compound act by observing the last tally written. Now the calculation terminates in the calculation of v
7.16 Oracles
341
so that the second of the two ciphers written in the compound act must ultimately be reached because the final result is: v',..., v{jI\ v and this is without consecutive ciphers. Thus either we shall have to erase the representation of {ju,', . . . j / ^ } and/'[/£', . . . , / ^ J or we shall have to replace the aforementioned cipher by a tally. If we erase {/i\ ...,fi{ef)} we shall have to reach the aforementioned second cipher in order to ascertain that we have reached the end of the erasion. In order to reach this position after having done the compound act we must do a further {/i'9...9/i^} + 3 atomic acts at least, namely atomic acts of moving one place to the left, whatever other acts the calculation calls for. Thus if we take oc equal to the total number of moves then a will certainly be sufficiently large. Thus if the calculation terminates in the calculation of v then
for sufficiently large A. Un$> *,,[*, r,',..., if>, v',..., «/»>, £, £]
D 220 for
AJL@[K,
Q<%\ ..., VW, 1],«/',..., •/">, g], W»» ly',.... vm, C, 0]].
Then Un$\..._ e^)[y, i}'> •••> Vm> v'> • • • > ^e)> £>C] is * primitive recursive function of the (d + n + 3) variables i),tj',..., rj(fft, v',..., iP\ £,, £. The value of
the function g can be expressed as ptHfii) [Unp
*,)[*, v',..., i/« f 'g,..., f')g, glt gJ = 0], 2,2],
if we use the unlimited least number symbol, but this expression is outside the system Ao[f',...,fM]. We can represent the function g in the system A o [/',...,/W] by {Ei)(Unf
^[K,v',...,vM,n,...,?%>£,*>]
= 0).
The machines described in the demonstration of Prop. 19 are called f-oracle machines, or f-oracle Turing machines. COR. (i). An enumeration with repetitions of partial / ' , functions is given by D220 for K = 0,1,2,....
...J^-recursive
COR. (ii). / / the function g of (A + A') arguments is partial / ' , . . . J'^-recursive where the functions have d\ ...,#(7r) arguments respectively and if the
342
Ch. 7 A0-Definable functions. Recursive function theory
last A' arguments of g are held constant so that g gives rise to a function g' of A arguments then g' can be represented in the system A o [/' ,/(7T)] by {Eg) ( Uf$\ ..., rf^',...,
= v
and where 8^t..., ^<w>[/c, y(SfA),..., y(A+A/)] is a primitive recursive function of K, v(SA),..., j/A+A') awd the function g is represented in the system A o [ / ' , . . . ,/(7r)]
by D220 ^i^A (A + A') instead of d. The demonstration is similar to that of Prop. 4 of this chapter, and so is the definition of the function 8. for the primitive recursive function of Cor (ii). The numeral K is the g.n. of a table which calculates the function g of (A + A') arguments. The numeral determined by is the g.n. of a table which calculates the function gf of A arguments where whatever the numerals v',..., vw. Thus the g.n. of some table for the function g' is a primitive recursive function of the g.n. of a table for g and of the values of the fixed arguments. We have defined primitive, general, and partial f-recursive functions, where f stands for/',... ,/(7r), if n is zero then omit f. Similarly an ^-recursive set of natural numbers is a set of natural numbers with an f-recursive characteristic function, similarly for an f-recursive statement. Similarly an ^-recursive enumerable set of natural numbers is the range of values of an f-recursive function. Similarly for f-recursively enumerable set of lattice points in 31Sn. We can now repeat a lot of the work we have done on recursive and recursively enumerable sets of natural numbers, using instead f-recursive and f-recursively enumerable sets of natural numbers. Thus the f-complete set of natural numbers is the set {/c, 6} where the f-machine with g.n. K produces 6. It is of the highest degree of unsolvability for f-recursively enumerable sets, that is to say, if some oracle would tell us if {/c, 6} was in the f-complete set then we could decide of any natural
7.16 Oracles
343
number whether it is in a given f-recursively enumerable set of natural numbers. The f-complete set is f-recursively enumerable but fails to be f-recursive, the demonstration is as for the complete set. In Prop. 19 we have reproduced a lot of Props. 1-4 incl. for f-recursive sets. In a similar manner we can produce Props. 8-18 for f-recursive sets and f-recursively enumerable sets, here we use \-decidable instead of decidable. For instance ^-recursive sets contain the field of sets generated by recursively enumerable sets, e.g. to decide if K is in SCd U 2EQ> we only have to decide whether {K, 6} is in ^? or whether {K, 6} is in ^f, if ^ is decided for us by an oracle then so is ^ and so we can ^-decide 2£e U 3£Q>. We omit the details of the extensions for f-recursive functions of Props. 8-18 incl. If we need them we shall refer to them as Prop. X for f-recursive functions. Thus we have \-simple, \-~hypersimple, ^-creative sets, etc. Note that by Prop. 16 ^-recursive and ^V-recursive sets are the same, where ffr is a creative set. Church's f-thesis is: an f-calculable function is calculable by an foracle machine. &{., 3?f denote the set of natural numbers calculated by /-oracle or e£/-oracle machines. ®{, Of denote the functions calculated by /-oracle or ^/-oracle machines.
if a < b and b ^ c then a ^ c, a ^ b and b ^ a iff a = b.
344
Ch. 7 A0-Definable functions. Recursive function theory
We define
a
for
for
a ^ b
b ^ a,
and
a >b
a 4= b. for
b < a,
we write a/b when both a ^ b and b ^ a fail, we then say that degree a is incomparable with degree b, a $£ b for a fails to be recursive in b, i.e. any function of degree a fails to be recursive in any function of degree b. Then for any degrees a and b we have the quadricotomy a
or a = b
or a > b
or
a/b.
Recursive functions form degree O. For any degree a, O ^ a. Thus O is the lowest degree of unsolvability, it is the degree of solvability. Now suppose that (i) / and g are both ^-recursive and (ii) that h is /, gr-recursive. For example: h[2. v] = fv, h[2. v + 1] = gv. Further if fv 9i> ^i a r e respectively of the same degrees as /, g, h then (i), (ii) hold for/ l5 gv hv Thus the functions which satisfy (i), (ii) for given/, g belong to the same degree. In fact they constitute that degree, since any function hx of the same degree as h also satisfies (i), (ii) for the same /, g. Further this degree is determined by the degree of/, g, for we still have (i), (ii) if we replace/, g by/ 1? g1 where fx is of the same degree a s / a n d g1 is of the same degree as g. We denote the degree of h by (a U b) where a, b are the degrees of/, g respectively. We call it the l.u.b (least upper bound)
ofa. and b. We have if
a^aub,
b^aub,
a ^ c and b ^ c then a u b ^ c.
Thus degrees form an upper semi-lattice. Further we have ^ - .-. , a^b
-
iff
a u b = b,
a u a = a,
O u a = a,
a u b = b U a, a u (b u c) = (a u b) u c Thus parentheses may be omitted when writing l.u.b.'s, in fact
a!ua 2 u ... ua, is exactly the degree of a function h such that if g$ is of degree a^ then g{ is A-recursive and h is gl9 ...^-recursive.
7.17 Degrees of unsolvability
345
If v is zero we agree that ax U ... U ay is O the solvable set. If/ is of degree b and gi is of degree a^l < i ^ v, then / is gl9 ...^-recursive if and only if b ^ ax u ... U a,,. We say t h a t gl9 ...,gv are recursively independent if none of them is re-
cursive in the others; in degrees, if a^ < ax U ...**... U a^fails for 1 < k < v, where *K denotes the omission of 2LK. If a 1? ..., av are recursively independent then so is any subset of them. For v = 1 independence is nonrecursiveness, for v — 2 independence is incomparability, for v > 2 independence implies pairwise incomparability but the converse fails. What we have said about degrees of functions carries over to degrees of statements or of sets, for these are determined by their characteristic functions. If g is of the same degree as / then g is /-recursive and a gr-recursively enumerable set (the range of a gr-recursive function) is an /-recursively enumerable set, hence, the gr-complete set, which is ^-recursively enumerable is also/-recursively enumerable and so is recursive in the/-complete set. Similarly the /-complete set is recursive in the g-complete set. Thus the degree of the /-complete set depends only on the degree of /. If a is the degree of/the degree of the/-complete set is denoted by a' and is called a complete degree, a' is called the jump of a. / / a < b then a' ^ b'. If g is of degree a and / is of degree b and if g is /-recursive then the gcomplete set being gr-recursively enumerable is also/-recursively enumerable and hence is recursive in the/-complete set, thus a' < b \ a < a' Because the /-complete set fails to be /-recursive, this follows from the analogue of the complete set §7.8. a ' u b ' ^ ( a u b)' We have a < a u b hence a' ^ (a u b)', similarly b ' ^ (a U b)' and so a' U b ' ^ (a u b)\ Thus altogether degrees form an upper semi-lattice with a jump operation superimposed on it. A degree is called recursively enumerable if it is the degree of a recursively enumerable set. If Rf[x,x\ ...,x{se)] is an/-recursive statement then (Ex)Rf[x,x\ ...,x(S0)] is/-recursively enumerable. For generate the
346
Ch. 7 Ao-Definable functions. Recursive function theory
($$#)-tuplets {v, v', ,,.,v(-S0)} and when the above tuplet has been generated/-decide whether Rf[v, v', ...,i£S0)], if this holds then write down in a list, this enumerates the ($#)-tuplets {V, ...,v(S0)} for which (Ex)Rf[x,v,f..., v ^ ] . Hence this set is recursive in the/-complete set. If the degree of / is a then the degree of (Ex)Rf[x, v',..., v{Sv)] is ^ a'. Note the/-analogue of Cor. (v), Prop. 8. If a set £f and its complement are both f-recursively enumerable then £f is f-recursive.
Thus O' besides containing the field generated by recursively enumerable sets also contains all sets £f such that Sf and SP are both O'-recursively enumerable and the field generated by them and recursively enumerable sets. Again a complete degree a' contains all a-recursively enumerable sets and all sets £P such that £f and £P are both a'-recursively enumerable and the field generated by these sets. The projection of a set Sf of lattice points in &SS7T is the set of lattice points {V, ..., jASn)} in 01Sn, where {v9v',...9 v^Sn)} is in Sf.
A set of lattice points is called arithmetic if it is obtained from a recursive set of lattice points by repeatedly taking complements and projections. A degree is called arithmetic if it is the degree of an arithmetic set. These degrees will contain functions representable in the system A, to be introduced in the next chapter, this system amounts to the system Ao plus negation, which failed to be A0-representable. Recursively enumerable degrees form a sub-semi-lattice of the semilattice of all degrees, because if/ and g are recursively enumerable then so is h defined by h(2. v) = fv, h(2. v + 1) = gv. Again arithmetic degrees form a sub-semi-lattice of the semi-lattice of all degrees, for a similar reason. The above function h is of degree (a U b) where a and b are the degrees of / and g respectively, so that if a and b are recursively enumerable degrees then so is (a u b). 7.18
Structure of the upper semi-lattice of degrees of unsolvability
The next proposition contains a construction which can be used in a wide range of cases to construct degrees standing in such and such relation to each other. We shall give several such propositions and also some extensions of the method of construction.
7.18 Structure of the upper semi-lattice of degrees of unsolvability
347
P R O P . 20. Given a non-recursive set stf of natural numbers, sets 3$, ^ of natural numbers can be found such that if a, b, c are the degrees of stf, &, *% respectively, where a 4= O, then b U c ^ a' and
(0) b ^ a,
b ^ c;
(1) c^a, (2)a0 3
c0; a ^ c.
Each of these conditions gives rise to an unending progression of conditions [0, K], [1, K], [2, K], where [v, K] says that the recursivity in question fails for the machine with g.n. K. Let a, /?, y be the characteristic functions of the sets s#, SS, *£ respectively. We shall define /? and y by stages so that at stage 6 exactly the first gO values of /?, y will have been chosen where gO < g(86).
(21)
Instead of saying that the first gd values of /? have been chosen we can say equivalently that the number
has been chosen. Then fiv — Pt[bd, Sv] for v < gd, whence fiv = Pt\b(8v), Sv],
since
v < g{Sv).
At stage 0 gO = 0 and bO = 0 (the null set). In the step from stage 6 to stage Sd the g(S6) — gd additional values of /?, y will be chosen so that however the definitions of/?, y are subsequently completed the (Sd)th of the conditions [0, 0], [1, 0], [2,0], [0,1], [1,1], [2,1], [0, 2],..., [0, K], [1, K], [2, * ] , . . . will be satisfied. Thus when the definitions of /?, y are completed all of these conditions will be satisfied, and so (0), (1), (2), will be satisfied. We shall pay careful attention to the form of the operations employed so that when we are done we shall be able to say that /?, y and g are of degree ^ a' and so b U c ^ a'. We now describe, by cases, how the g(S6) — gd additional values of /?, y are to be chosen. In proceeding from stage 6 to stage Sd, we can think of the process as 'extending' hd (which incorporates the first gd values of J3) to h(S6) (which incorporates the first g(S6) values of/?), and similarly for c.
348
Ch. 7 Ao-Definable functions. Recursive function theory
At any stage 0 (when only gd values to be given finally to /?, y have been chosen), it will be convenient to call any function J31 which is a characteristic function of a set and which possesses the first gd values of /? an ' extension' of /?, similarly for y. Case 0. Bern [3,0] = 0, then 0 = 3. [0/3], let [0/3] = K. We must extend W to h(Sd) and C0 to c(#0) so that condition [0, K] will hold for all extensions /?x of fi and yx of y. That is we must render it impossible, for every such pair of extensions, that k,i[K,v>Og,eg, gx, g j = 0]]2,
(22)
where £ = {£l5 £2}, hold for each numeral v, i.e. the r.h.s. of (22) is defined, for which the condition is
and equal in value to the l.h.s. of (22). In fact we shall render this impossible for v = gd. Subcase 0.1. For some extensions y2 of y and some A Un\tl[K, g6, aA, c2 A, A1? A2] = 0. This is equivalent to saying that there is a pair A, n of numerals such that n 4= 0 and c6nn is an extension of C0, in which case (A£)x\Pt[7T, £] < 2], so that IT = (ev e 2 ,..., e^>, for some numeral JLL, where el9..., e8/i are 0 or 1, so that the extension represents a characteristic function of a set. Thus C7<1[/c,9f0,aA,c0n7r,A1,A2] = O. Thus the subcase hypothesis becomes (Eg) (£a 4= 0 & tug2 = & -=- ^ & g1? 2 < 2 & (^,) f l [«[g a , V] < 2] & *7 Og, C0n£2, glsa, g1)2] = 0). This is of the form
(Eg) R%{6, gd, c0, g},
where i?0{A,/^, v, K} is a primitive recursive A00-statement. Let us write Xo
for
(v.g)[B%{6,g09c0,$\,
(23)
then Xol, X02 are the A, n with the above properties. Now we first extend (if necessary, i.e. if XOfl > gd) the choice of values _ of y so that x
7.18 Structure of the upper semi-lattice of degrees of unsolvability
349
and secondly we choose forfi[gd]a value ^ 1 different from that given by the r.h.s. of (22) when v = gd. Then as desired (22) will fail when v = gd for any /?, y having the values already chosen (including those chosen at stage 6). What this gives as value for fi[gd] is
After taking
g(80) = Max [8(gd), XOt J,
(24)
which ensures (21), thirdly we choose all values (if any) of fiv, yv for v < g(S6) not already chosen (either at stage 6 or in the further choices just described) to be 0. Then f>(88) = (|*£) K * 0&(AV)a$[Pt[g,V] = Pt[W,V]] &Pt[£,gd] = 1 ^ X 0>lj2 &S£ = g(S6)l c(Sd) = (pig)[| 4= 0 & (AV)ge[Pt[£,V] = Pt[cd,7)]] = Pt[X0_ „ £]] These give with (23) us equations of the form (25) where ^o,i{^'/^» p» ^} a n d A!o,2{A,/^3^, AC} are partial a-recursive being defined exactly when: (Eg) RQ{\,/I, K, £,}. Subcase 0.2. Otherwise. Then the values already chosen at stage 6 render (22) impossible for v = gd no matter how ft, y are completed as extensions of bd, c6. However to ensure (21) we take g(8d) = 8(gd),
(26)
and extend ft, y by choosing sothat
filgff] = r\s0] = o. b(86) = (Wfl, c(80) = (cd)"l.
Combining the two subcases using (24), (25) with our formulation of the subcase 0.1 hypothesis
otherwise,
}
(2?)
350
Ch. 7 A0-Definable functions. Recursive function theory
and similarly for c(8d). Thus (
^'AOgdwce}}
'
where #o,'i{A,/£, V, K} and Xo,2{^,M<, v, K} are partial a'-recursive, but since it is completely defined then it is a'-recursive. Similarly, from (23), (24), (26):
xio{e,gd,td}>
(29)
where #O,O{^J/^ ^} is a'-recursive. Case 1. Rem [3,6] = 1. Similar to case 0. Case 2. Rem [3,6] = 2. Then 0 = 3. [0/3] + 2, let /c = [6>/3]. We must extend W to 6(A8'(9) and C(9 to c(S0) so that {2, /c} will hold for all extensions /?x of y5 and yx of y, i.e. we must render it impossible, for every such pair of extensions, that: *v = [({*£) [Vn\tl[K, vM, c£, it U\ = 0]]a
(30)
hold for all v. Subcase 2.1. For some v and some extensions j3l9yl9 of /?,y respectively the r.h.s. of (30) is defined with value opposite to av. This is equivalent to there being numerals A, n', n", v such that n\ TT" * 0, (ADX [Pt[nf, a < 2], (.4£)A[P*[>r", g] < 2], and
Un\fl[K, v, (hdfn', {tOfrr", A1? A2] = 0 and A2 # av.
This amounts to ]&(^^[P^[g3,9/] < 2] 2,
(cd)n g 8 , g 1}1 , g l j 2 ] = 0 & £ 1>2
where ^1 This is of the form
1 < /* < 4, so that
g=
(Eg) R%{6, gd, W, cd, Q,
where J?2 is a-recursive. Put:
(31)
X2 for ([L£)Bi{O,g0,bO,c69Q. We first extend (if necessary) the choice of values of /?, y so that
7.18 Structure of the upper semi-lattice of degrees of unsolvability
351
hX21 = (hd)nX2>2, cX2}1 = (cd)nX2}3. Then (30) will fail for v = X2A
for all /?, y having the values thus far chosen. After taking g(Sd) = rnax[S(gd),X2>1l
(32)
secondly we choose the further values (if any) of fiv, yv for v < g(S6) to be 0. Then: f>(80) = (,ig)[g * 0&(AV)gd[Pt[£,V]
= = Pt[X2)1,V]]],
i(S0) == #f.i{0, gd, W, cd},
whence
(33)
(34)
and similarly for c, where ^2.i{A,/^, v, K} is partial a-recursive being defined exactly when (i?£) i?|{#, A, /i, v, ^}. Subcase 2.2. Otherwise. We shall show that in this subcase, for some v, the r.h.s. of (30) is undefined for every pair of extensions /S1,y1 of/?,y respectively, after showing which the subcase can be treated similarly to subcase 0.2. Accordingly, suppose that (for reductio ad absurdam), for every v, the r.h.s. of (30) is defined for some extensions fil9 yx of/?, y respectively. That is, for each v, there is A, n', TT", such that
(A£)A[Pt[n",£] <2], mn' = wn"9mn' +wed = A and
Un\t ±[K9 V, Wnn\ cdnn", A1? A2] = 0.
This can be expressed in the form
where R is primitive recursive. Put X
for
(\L£)B{e9gO9bd9c09v9®.
Then X12 is the value of the r.h.s. of (30) for some extensions fiv y± of J3,y respectively for which that r.h.s. is defined, But since subcase 2.1 is excluded, the r.h.s. of (30) when defined has the value av. Thus writing out the X in X± 2 in full, for each v
for the fixed 6 under consideration, which makes a recursive, contrary to hypothesis.
352
Ch. 7 A0-Definable functions. Recursive function theory-
Combining the two subcases
where Xt'^i^'/1'
v
> K}> n = 0,1,2 are a'-recursive.
Combining the three cases
These equations and gO = /?0 = yO = 0 define g, ft, y simultaneously by recursion on 6 from functions Xo\ Xi> X% • By ^r^ setting up a recursion for the triplet {gd9 W, cd} and then taking components it follows that g, b, c are recursive in the functions Xo > Xi> X%'• Since the latter are recursive in a! so are g, b, c and so are /?, y. This completes the demonstration of the proposition. The above type of demonstration can be used when the conditions required to be fulfilled can be put into a progression of conditions each stating that one of the conditions fails for table 6, 6 = 0, 1, 2,.... But the demonstration of the next proposition is of a different type. Here the required functions are obtained as the 'limit' of a decreasing sequence of functions. In the system !FC we shall write <j)->\Ir for
Example of the priority method. Solution of Post's problem
P R O P . 21. Two recursively enumerable sets of natural numbers neither of which is recursive in the other can be A0-defined.
It is clear that both such sets are neither recursive nor complete. Thus they will be of incomparable degrees lying between the degrees O and O', also they are recursively enumerable degrees. We wish to define two recursively enumerable sets of natural numbers j / a n d 38 such that stf^T38 and &^T<$/, i.e. and
38 # Xf
for
6 = 0,1,2,....
7.19 Example of the priority method. Solution of Post's problem
353
These are respectively equivalent to (Ex) (xea?*-> xlftf)
and
(Ex) (x e@ <-> xeSCf) for
0 = 0,1,2,....
It suffices then to show that there are two functions/and g such that fOes/ ^fdeSCf
and
gde$8 <-• gOe&f for
(9 = 0,1,2,....
(i)
The method we use is known as the priority method because in the course of the construction certain things are given priority over other things in order that the latter will be unable to upset what has been achieved by the former. It is more complicated than the diagonal method which we have used many times. The construction proceeds by stages. We start with two lists of numerals the ja/-list and the ^-list. From time to time we shall place a ( + ) or a (—) against selected members of either list. Once a (+) is placed against a member of either list then it will remain unchanged thereafter, but a (—) may later on be changed to a (+). The members of the j^-list which receive a ( + ) shall constitute the set s/ and the members of the ^-list which recive a (+) shall constitute the set 3$. It will be apparent from the construction and the manner in which we place the (+) signs that the sets si and 0ft are recursively enumerable. We shall also have two lists of markers the ^/-markers and the 88markers. These are respectively denoted by O',l',2',...,
and
0", 1",2",...
These are, as their name implies, used to keep the place. In the course of the construction we shall put markers against members on the like list of numerals and at other times we shall move some markers down the list in which they are. An essential feature of the construction is that each marker ultimately comes to rest. The numeral fd against which the marker dr comes to rest will be thefd in (i), and the numeral g(6) against which the marker d" comes to rest will be the gO in (i). The numerals which receive a (—) in the course of the construction are candidates for J& or 3$, according to which list they are in. A list like 0' 0" T l"
9'
is called a priority list because in the course of the construction if the position occupied by any marker gets a ( + ) against it then all other markers of the other kind which have so far been used and are further
354
Ch. 7 A0-Definable functions. Recursive function theory
down the priority list are moved still further down the list of numerals they are associated with in order that what is achieved by the (—) signs used at that stage will never thereafter be fouled. The construction takes place by stages. We define s£v = the set of natural numbers in the j/-list which have received (4- ) at the end of stage v, 88V is similarly defined. At any stage in the construction we shall call a member of either list of numerals free if neither it nor any numeral further down its list has any mark or marker against it. At any stage in the construction we shall say that a member of either list of numerals is vacant if it is without a (+). We now give the construction. Stage 1. Place marker 0' against 0 in the j/-list. Stage 2. Place marker 0" against 0 in the ^-list. Stage 2v+ 1. Place marker v' against the first free numeral in the ja/-list. Let a^\ a{{\ a2v\ ..., a{vv) be the current positions of the markers in the s£list. Let condition C^d, v) be: (i) a^v) is vacant, (ii) Unl[d,a$\@2v,v,l] = 0. Place (+) against the least a^\ 0 < /i ^ v, say af$, which satisfies condition C^/i, v}, place (—) against each vacant numeral in the ^-list whose membership of 882v is used in the evaluation of Un\[d, a{0\&2v, v, 1], then move all markers pt" in the ^-list for which JLC0 < JJL" < v (this is all the markers /i" as yet in the ^-list for which JLC0 ^ JJL) in order down to the first available free places in the ^-list. If each a%\ 0 < /JL < v, fails to satisfy condition C{/i, v} then pass to stage 2v + 2. Stage 2v + 2. Place marker v" against the first free numeral in the ,^-list. Let b(ov\ b^\ b2v),..., b^ be the current positions of all the markers in the ^-list. Let condition C2{6, v} be: (i) 6(/) is vacant, (ii) Unl[d,b%\j/2v+liv,l] = 0. Place (+) against the least 6^ 0 ^ /i ^ v, say 6^, which satisfies condition C2{/i, v}, place (—) against each vacant numeral in the j^-list whose membership of £#2v+1 is used in the evaluation of Unl[jLt0,6^, ^v+u v> ^]»
7.19 Example of the priority method. Solution of Post's problem
355
then move all markers [if in the j/-list for which fiQ < ft ^ v (this is all the markers jaf as yet in the s/ list for which /i0 < /i) in order down to the first available free places in the j^-list. If each b^ fails to satisfy condition C2{/i, v) then pass to stage 2v + 3. The success of the construction depends on the following: Each marker ultimately comes to rest. We proceed by induction. Marker 0' never moves. Marker 0" can move at most once. Marker v' can move at most once for each position of markers 0", 1",..., (v — 1)" whence if these only move a bounded number of times then marker v' can move only a bounded number of times. Similarly marker v" can move at most once for each position of markers 0', 1', 2',..., (v — 1)'; whence if these markers can only move a bounded number of times then marker v" can only move a bounded number of times. Let/# be the final position of marker 6', and let gd be the final position of marker d". We show that LEMMA.
fdesi'++fde&f
and
g6e88
Suppose th&tfdes/ then f6 gets (+) at stage 2^+1 for some v. The signs introduced at this stage remain unchanged thereafter. For if 6 <{i marker fi" is moved below all these (—) signs and thereafter can only move further down and new markers are introduced after all markers present in the appropriate list, thus markers fi" for which 6 < fi never foul these (—) signs. Markers for which ju, < 6 are already at rest, for if such a marker moved then by our construction marker 6' would have to move contrary to hypothesis that it had already reached its final position fd. Thus the evaluation of Un\[dJ6,88%v, v, 1] is the same as that of Un\[0, fd, &,v,l] because the answers to ' is A in 88^ ?' will be the same as the answers to 'is A in 88%'. Thus fdlSCf, in order to give /# a (+) condition C^d, v) must be satisfied, and this means that/#eiTf*». Now suppose th&tfde&f*", then there is a ju, such that for all v >ii
Unlldjd, 88, vy 1] = Un\[dJ6,8§2v, v91],
because if v is sufficiently large then the resister of 882v will contain all the information that is required of 88 for the calculation of
356
Ch. 7 Ao-Definable functions. Recursive function theory
Thus if fdeSCf then we shall have Unl[djd,@2v, v, 1] = 0 for all v > /i whence condition Cx{d, v) part (ii) will be satisfied, hence at a certain stage fd will be the least such marker position and so will satisfy the whole of condition Cx{d, v}, thus/0 will get a (+) and so will belong to si. Thus (i) is satisfied and so si 4 T ^- Similarly 88 ^ T ^- This completes the demonstration of the proposition. It is known as Posts' problem. Note that / and g fail to be recursive functions. This is seen as follows: Let 0 be in si if and only if ad = 0. Then if ^ is a recursively enumerable set there is a numeral 0 such that v is in <% if and only if
where b is the characteristic function of the set 8ft, because ^ being recursively enumerable there is a partial recursive function, t, (and a fortiori one partial in 6) such that (1 if v is in # , tv= \ [undefined otherwise. The function t is given by the rule: for argument v generate ^ and if v appears in the generation let tv = 1, otherwise t is undefined, we can find a table which does this, let its g.n. be 0. But fd is in s/ if and only if
hence fd is in <$/ if and only if fd is in ^ , by definition of ^ . But if ^ <= s/ then fd is in ^ n s/. If / is recursive then s/ is creative, by definition of creative, this is impossible because then si would be of degree O' and hence 8% would be A -recursive.
7.20 Complete degrees 22. A degree a is complete if and only if a > O'. We first show:
PROP.
LEMMA. TO
any degree a we can find a degree b such that b' = b u O' = a u O'.
7.20 Complete degrees
357
Let h be a function of degree a. Define a function k of degree < O' as follows: k[v, v'] = ^(px) [Un\[v, v, x, (mx)l9 (wx)2] = 0 & Cora# [x, v% where D 222
[0 if undefined. Comp [v, v'] for (Ax)MinivVtmvl
[Pt[v9x] = Pt[v',x]].
The function k is of degree ^ O', because to find the value of k[v, v'] we first have to decide whether (Ex) (Un\[v, v\ x, (wx)v (wx)2] = 0 & Comp [x, v'])
(35)
and this is of degree ^ O', if the decision is favourable then it is a recursive process to find the least numeral which satisfies, if the decision is unfavourable then the value of k[v, vf] is zero. Define a function g as follows:
f go = o, \g(8v) = Then
^0 = 0, g\ = (h0)ny0, g2 = g(8v)= <^0> n y0 n ... n <^>V
(h0)ny0\hiyyl,...,
for some y0,yl,y2, ...,yv.
Now let/x = Pt[g(Sx), x], let b be the degree of/. Then for each v
l^v^fx'),^,^]
= 0->k[v,gvn(hv)] = (fx') + 0
for some nf ^ n (we have Comp [(/#'), ^^n<^^)] and
liv, v, (jx'},nl9nA = 0, n
l '
where n = mk[v,gv hv]). (Note that if (35) holds then k\y, v'] =(= 0, because if (\LX) \Urt\\y, v,x, (wx)v (TDX)2]
= 0] and x — (wx)1 = (wx)2 = 0
and table v with input v after zero moves fails to produce zero (it produces the null set)) the gr.n.'s of the initial and final situation are different if input is v and output is zero. Therefore
k[v,gv\hv)] # 0^(Ex)Un\[v,v,(fxf)fx1,x2]
= 0.
(36)
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Ch. 7 A0-Definable functions. Recursive function theory
Now g is h, ^-recursive, therefore the product on the r.h.s. of (36) is M-recursiveor (37) b ' < a U O', On the other hand, since hv = f[S(m[gv])] for each v, we can substitute f[(Sw[gv])] for h in the definition of g to obtain a formula according to which g is /, ^-recursive. Then using hv = f[(Sw[gv])] again, we see that AisaJso/.i-reoursiveor (38) a ^ b y O'. From (38) it immediately follows that and it is elementary that
auO'OuO', h 11 O' < h'
(39) (4.ft\
(O ^ b hence O' ^ b' hence b U O' ^ b'). From (37), (39), (40) we obtain: b' = b u O ' = a u O '
(41)
and the lemma is demonstrated. We now turn to the demonstration of the proposition. Since O < a then O' ^ a' so that a complete degree is ^ O', if O' ^ a then a u O' = a and (41) becomes b' = b u O ' = a and so a is complete. PROP.
23. Given a degree d there are degrees a and b such that a' u b' = a u b = a' = b ' = d'
and a/b and d < a < d', d < b < d\ We give a demonstration for the case d = O and then show how it can be modified for d > O. We construct functions oc and 0 of degree a and b respectively such that
a'ub'^aub^O'
(42)
and then show that the other properties follow. Let yv = {ocv,flv},a, /? are characteristic functions so y will only take the values 0, 1, 2, 4. We proceed by stages as in Prop. 21 at stage 6 the first gd values of a, /? have been chosen. gO = 0 so gO = 0. a, b, c, g are the register functions for oc, /?, y, g respectively.
7.20 Complete degrees
359
For all 6, K we shall have OLV = /3V iff v^gd-l,
(43)
(Ex) UU\[K, K, ax, xv x2] = 0 iff ot[g(2. K + 1) -L- 1] = 0,
(44)
(Ex) Un\[K, K, hx, xv x2] = 0 iff y%(2. K + 2) ^ 1] = 0,
(45)
g0 = 0 g(Sd) = S([LX) [ax + j3x&x> gd\ From (44)
(46)
a' ^ a u g, where g is the degree of g.
From (45)
b ' < b U g,
hence
a' u b ' ^ a u b u g,
but from (46) g ^ a u b and so a' u b ' ^ a u b whence a U b = a' U b ' . Let C2K+1 be the condition that y satisfies (44) and (43) for gd^v<
g(Sd).
Suppose that g(2. K) and y[g(2. K)] have been defined. Case 1. g(2. K+ 1) and a[g(2. K+ 1)] can be defined so that UU\[K, K,an, TT-L,n2]
= 0 for some n = {TT^^} < g(2./c+l).
This is equivalent to = 0&(AxXXi[Pt[x±,xf]
(Ex)(Un\[K,K,a[g(2.K)Tx1,x2tl,x2t2]
< 2]
&
This is of the form
^ , i = g(2.K)+wx1). (Ex)x{K,a[g(2.K)],g(2.K),x},
/T7 x
r
r /rt
...
/rt
x
^
where X {x, x', x", x"1} is recursive. Write X Setg(2.K+l)
for
(iix)x{K,a[g(2.K)],g(2.K),x}.
= SY, where Y for ilfaa; [gr(2 . K), X2] and c+l)] = c[g(2.K)T({Pt[X1,x'],Pt[X1,x']}y(0,l)
i.e.
av = fiv = Pt[Xly Sy [g(2. AC)]] for g(2.K) ^v ^ X2,
(unnecessary if g(2.tc) ^ X2) and aY = 0, j3Y = 1. This ensures condition C2 K+1. Thus being defined exactly when (ifo) ^{AC, h(2. K), g(2. A:), X).
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Ch. 7 Ao-Definable functions. Recursive function theory
Also g(2. K + 1) = ^ { ^ J H%. K), g(2. K)} being defined under the same conditions. Case 2. Otherwise. Then y[g(2.K+l)] = y[g(2.K)]n<0,1>, g(2.#c+l) = Sjr(2.#c), so fc(2.*+l) = JI(2.K)\0,
1>,
then condition C2K+1. Cases 3, 4. A(2. K + 2) and gr(2. K -f 2) are defined similarly but reversing the roles of a and /?, then C2#/c+2 7 satisfies (45) and (43) etc. Altogether K{8K) = ^ ' {/c, TIK, gK}, g(8/c) = xl R hie, g/c},
whence ^ and g are of degree < O', hence so are a and /?. Now O ^ a ' ^ a' u b', O ^ b ^ a' u b ' which with (42) yields a ' u b ' = a u b = a' = b ' = O'.
(47)
If any of a/b, O < a < O', O < b < O' fail then we shall have a = a' or b = b' which is absurd. If a/b fails then say a < b, whence a' < b ' and so b = b', which is absurd. If O' ^ a then O' < O" < a' which is absurd. If a/O' then a u O' > O' whence O' = a ' u O ' > a u O ' > O', which is absurd. If a = O then b = b ' which is absurd. For arbitrary degree d. Let tkx.oc{2.x}, \x/3{2.x} be characteristic functions of degree d. The values of a,/3 for odd arguments are determined as in the case when d = O, except for obvious changes to take account of the fact that a(2.y) and fi(2.v) are determined prior to the construction. Letting a, b be the degrees of a, jS respectively we have d < a ^ d',
d ^ b ^ d'
and the demonstration of the proposition is complete. COR. (i). / / a ' = b ' we can have any of a < b, a = b, b < a, a/b. Prop. 23 gives a case of a/b and a' = b', also of d < a and d' = a'. COR. (ii). / / a/b we can have a' u b ' < (a u b)'. Prop. 23 gives a case of a' u b ' = a u b < (a u b)' and a/b. C O R . (iii). Each complete degree is the l.u.b. of the set of lesser degrees.
By Prop. 23 we have degrees a,b such that d < a < d',d < b < d' and a u b = d'.
7.20 Complete degrees
361
COR. (iv). / / a < b we can have either a' = b ' or a' < b ' only. If a/b we canhavea' = b'. Prop. 23 gives a case of a/b and a' = b ' and of d < a and d' = a'. COR. (V). If a is a degree and a > O' then there fails to be a degree c, c > a, such that ifb
A->oo
enumerable. We require gM to fail to be recursive in gt1 for each /i. The witness that this is so will be L h[A,/i, 6], where h[X,/i, 6] increases with A—>oo
A but is bounded so that it is ultimately constant. We write k\ji, 6] for L h[A,/i, d], the final value of h[X,/i, ff]. Then we shall show that A-*oo
gjttfi,
Un\[6, k]ji,ff],fl*[A+ AJ[8X ^ fi], A], A, 1] = 0,
9
i.e. <& £\k[ji, &]] = 1, so that lc\ji, 6] is a witness to the fact that g^ fails to be recursive in g? by table 6. g is the register for g^. We define these functions by stages A = 0, 1, 2,.... The values of A for which Pt[A, 1] = 6 and Pt[A, 2] = /i will be devoted to showing that g^ fails to be recursive in gp by table 0. And we shall have for some n. We start with , 2] = / if and
f[X,/i, K] = 1, K = h[\,/t, ff],6 = Pt[\, 1] Un\[Pt[X, 1], K, f [A, A + AX[SA - /£], A], A, 1] = 0
362
Ch. 7 A0-Definable functions. Recursive function theory-
call this condition C^A; f[SA, /i, K\ = /[A, /i, K] otherwise,
if p! * /i and C^A and
= h [A, pi', 6] otherwise 6 > Pt[A, 1],/*' > Pt[SA, 2] = p.
Forfixedp, 0, h(A, p, 6] is bounded as A increases. For fixed /i, 6, h[A,/i, d] will be unbounded as A increases if and only if CyA' occurs unendingly often with ju, > [ir = P^/SA', 2] and 6 > Pt\X\ 1], say for pairs {/il9 6^, {{i2,d2},.... Then h[SA',{i,d] =fc[A',/*',0]n.If this is so then C^X' occurs unendingly often with Pt\A', 1] = 6r < 6 and Pt[SA', 2] = pi' < /i for some fixed ju,', 0r. There will then be an unending sequence of changes in /[A',/^', K] with K = h[A', ji', d']. But for fixed pi, K,f[A',/i, K] can only change once, hence there must be an unending sequence of changes in h[A',/i', d'] as A' increases and 6' < 6,/i' < /i. But this is absurd if we had taken {/i, 6} to be the pair earliest in the ordering of all ordered pairs for which this occurs. Now we want to show LEMMA.
]] = O
iff
Suppose r.h.s. then for A' sufficiently large and we can take then we shall have Un\[Pt[Af, 1],fc|>,6], f[A', A' + AJL8X - /i\, A'], A', 1] = 0. But this is C^A' so that if f[A',/i, K] = 1 then f[SA',fi, K] = 0 and so ^ = 0, K = k[/i,6], as desired. Iff[A',/i, K] = 0 then again g^K = 0 as desired. Suppose l.h.s. then condition G^A must have arisen for some A for whlch
ft[A,
/i9 0] = k[/i, d] (its final value), Pt[A, 1] = 0, Pt[8A, 2]=/I,K
=
k\ji9 d]
7.21 Sequence of degrees and
363
Unl[d, k\ji, 0], f [A, A + A1[SA - /i], A], A, 1] = 0.
Now for A' > A this last condition still holds, for it can only fail as A increases by a change in the value of f[A',/i', v] for /JL' + JU,JLL',V ^ A because these are the only values of/[A, /i, v] which are used in the calculation of
= 0,
as desired. This completes the demonstration of the proposition. Let Jf be the set of natural numbers and let < R be a recursive partial ordering of df9 i.e. v < R/c is a recursive predicate which gives a partial ordering of JV. C O R . / " with the recursive partial ordering < R is embedable in the upper semi-lattice of recursively enumerable degrees.
Let s/Q9 s/l9 <stf2... be a sequence of recursively enumerable independent simultaneously recursively enumerable sets. That is A[/i9v] = 0,1 is such that the set whose characteristic function is Ax.A[/i9x] is recursively enumerable and fails to be recursive in Axx'. A [x + A-^Sx — /i]9 x']. From these we can form a sequence of recursively independent, disjoint, simultaneously enumerable sets, for instance let 0$^ be the set of ordered pairs {v,fi} such that vestf^ having characteristic function AXB\JL9X~\. Let ^ be the union of the sets 3SK with /c < R /£, then ^ is recursively enumerable because the ^? 's are simultaneously enumerable and ^ R is recursive (in fact the #'s are simultaneously recursively enumerable
We will now show that <6V is recursive in ^ if and only if J>
364
Ch. 7 Ao-Definable functions. Recursive function theory-
it will then be clear that JT with the partial ordering ^ R is embedable in the upper semi-lattice of recursively enumerable degrees. Suppose ^ R / i , then ^ , 0 ^ = 0, and ^ is recursive in \xx'. B[x + Ax[8x JL V], x'] since
geC, «-> (JE£, T,) (g = {V,Q & £
It follows from the recursive independence of 33^33^3$^... that 9SV fails to be recursive in ^ . But then c€v fails to be recursive in ^ , since 3SV is recursive in ^,. Suppose v < R/^. Then ^ c= ^ and ^ is recursive in ^ since
, V) (I = {?, ^} & £ < B the last line follows since the J5's are disjoint.
7.22 Non-recursively separable recursively enumerable sets 25. We can find two recursively enumerable sets of natural numbers which fail to be separable by a recursive set. We shall show: recursively enumerable sets s/ and 8$ can be A0-defined such that if si £ ^ and 3$ c Qf where ^ and £& are recursively enumerable then we can find a numeral 6 such that 6 fails to be in the union of ^ and 2. Hence ^ U 3i fails to be the universal set of natural numbers and so ^ and 3) fail to be recursive. Let si be &(Ex') U[x, x'] and a be &{Ex') V[x, x'], where U[x, x'] for PROP.
[a;, 2], x'v x'2i x] = 0 & (.4a;") (a;" < «' -> C7^' [Pt[x91], < , »J,»] # 0) and a;' = {x[, x'2} and F[o;, a;'] for Un'[Pt[x, 1], a£, a?i, x] = 0 & (^a;/r) (a?" < a' -* Un'[Pt[x, 2], a?, a£, a?] 4= 0). [7 and F are primitive recursive, hence si and ^ are recursively enumerable sets. From U[x, x'] & V[x, x"] we get x' < x" & x" < x\ hence ((Ex') U[x} x'] & (Ex') V[x, x']) fails, thus si and 38 are disjoint. Now consider two disjoint recursively enumerable sets ^ , Of with j / c ^ & ^ c 0 . Let them be ^ a ; ' ) R{x9 x% $(Ex') 8{x, x'} respectively,
7.22 Non-recursively separable recursively enumerable sets
365
where R and 8 are primitive recursive. We now find a numeral which is in f H^. We have (Ex')R{x,x'}^(Ex')(Un'[A,x'vx'2,x] = 0), (Ex')8{x9x'}++(Ex')(Uri[ji,x'l9x'29z]
= 0),
for some A and ju,. Let 6 = {A,/*}, assume that d is in ^ , i.e. (Ex') R{6, x'}9 since # and ^ are disjoint then 0 is in Sd9 thus (2?#) (DV[A, x[, x'2,6] = 0), but (JEto') £{#, a;'} fails, whence (Ax') [Un[/i9 x'l9 x'29 d] + 0], hence (Ex') (Un[A, xl x'i 6] = 0)& (Ax")x, [(Un\ji, x'[, xl 6] * 0)], i.e. (Ex') V[6, x'], i.e. 6 is in ^whence 6 is in Sf which is absurd, hence 6 is in ^ . Similarly 6 is in Si. 7.23 Cohesive sets A set of natural numbers is cohesive if it is unbounded and if either its intersection with a recursively enumerable set is bounded or its intersection with the complement of that recursively enumerable set is bounded. Thus a set si is cohesive if it is unbounded and if for each recursively enumerable set 88 either s/(] 38 or s/f] £$ is bounded. A set of natural numbers is maximal if it is unbounded and recursively enumerable and its complement is cohesive. 26. Every unbounded set of natural numbers possesses a cohesive subset. • Let s/ be an unbounded set of natural numbers. Define a sequence of subsets of si as follows
PROP.
fl $EV if this is unbounded, fl 3Cv otherwise. Then s/0 ^ s/x ^ jtf2 3 .... Assume that s/ is without a cohesive subset. Then for each v there is a [i^ v such that ^ divides J ^ into two unbounded subsets, otherwise s/v would be a cohesive subset of s/. Hence the sequence s/0, s/v... contains a strictly decreasing subsequence 08Q,3Sl9... otherwise s/ would
366
Ch. 7 Ao-Definable functions. Recursive function theory
be constant from v onwards contrary to being divided into two unbounded subsets by ^ . Thus we can define a sequence of numerals as follows
Denote this sequence by ^ . By our construction, s/v+1 c; 2Cvors/v+1 £ 2£v. Hence dSv+1 c 3CV or dSv+1 c SCv. Since all but a bounded number of members of <€ must lie in &v+1, either ^ n #*„ or *2f n #"„ is bounded. This holds for all v, and thus ^ is a cohesive subset of stf which is absurd.
7.24 Maximal sets 27. A maximal set can be A0-defined. We describe a procedure for enumerating a set of natural numbers s/ by stages, and then we show that j / is maximal. SC* is the set of natural numbers produced by table v after /u, steps of the following: Do one step in the calculation by table v with input 0, then do one step in the calculation by table v with input 1 then do the second step in the calculation by table v with input 0, etc., every time an output appears write it down in a list. For any n, any stage Sv, and any numeral 6, we define the 7r-state of 6 at stage Sv to be e^e^... en9 where PROP.
fl
if OeXlv and v
A ^ v,
jo if de&i or v < A, forO ^ A ^TT. Clearly there are only 2 ^ Tr-states. We order them lexicographically (0 < 1). Note the following two properties of this ordering: (i) for fixed n and 6, and for v < fi, the 7r-state of 6 at stage fi must be at least as high as the Tr-state of 6 at stage v. e^ ^ e^v); (ii) for fixed stage v, for given 6 and #', and for n < n' if the 7r-stage of df is higher than the 7r-stage of 6 then the Tr'-state of dr is higher than the TT' -state of 6. To enumerate the set s/ we proceed as follows: We begin with a list of numerals each associated with a marker, v is associated with the marker v*. We then move these markers down the list of numerals according to
7.24 Maximal sets
367
certain rules. If at any time a numeral loses a marker then thereafter it remains markerless. The set si will consist of the numerals which are ultimately markerless. Stage Sv. For each fi, let julv) be the position of marker ju,* at the end of stage v. Compute #"f" for all A ^ v. Find the least numeral /i0, if any, such that for some ju, > ji0, julv) is in a higher fi0- state than /i^ at stage Sv. If this search is unsuccessful then pass on to stage (v + 2). The search terminates because if 6 > v then eA = 0 since if 6 occurs as output it will require at least 6 steps. If the search is successful, let /i± be the least fi such that fi^ is in a higher /£0-state than /i^ at stage Sv. Move marker /£* down to /4"\ and for each JLL > JLL0, move marker /i* down to (/i + /i± — /iofv). Place in si all the numerals fi such that fi^ ^ fi < fi^ that are not yet in si. (These are the only numerals to lose markers in stage Sv.) Then go on to stage {v + 2). We now show that si is maximal. Clearly si is recursively enumerable. (i). A is unbounded. It suffices to show that each marker ultimately comes to rest, then there will be an unbounded set of numerals which ultimately retain markers and so belong to s/. Assume otherwise. Let /** be the least marker which moves unendingly often. By construction and fundamental property (i) of 7r-states, after {/i0— 1)* reaches its final position /i* must move to positions of higher and higher /£0-states. But there is only a bounded number of different /^-states. LEMMA
(ii). For every n, either 3£nfts/ is bounded or SC\ ()s/ is bounded. Fix 7T. For each numeral 6 in J / , 6 must reach a final 7r-state /? as stage v increases. We say that 6 terminates in fi. Since s/ is unbounded, at least one 7r-state J3 must be associated with an unbounded number of members of s/ which terminate in/?. We show that exactly one 7r-state is associated with an unbounded number of members of si. Assume otherwise, let J3 be the lowest 7r-state associated with an unbounded number of members of si, and let/?'be another Tr-state associated with an unbounded number of members of si. It follows that there must be numerals ju,, v, 6, 6' such that n < /i < v, 6 is the final position of marker /i* and 6' is the final position of marker v*9 6 terminates in the Tr-state fi and 6' terminates in the 7r-state fi'. By fundamental property (ii) of Tr-states d' reaches and LEMMA
368
Ch. 7 A0-Definable functions. Recursive function theory
terminates in a higher /estate than 0. But this means, by construction, that marker /£* must be moved, which is absurd. Thus all but a bounded set of members of si (final positions of markers) have 7r-states different from a certain /r-state e0exe2 ...en. If desi and en = 1 then 0earw, if 6e^ and en = 0 then fle^. Thus all but a bounded number of members of si are in #*w, or all but a bounded number of members of si are in 2Fn. This completes the demonstration of the proposition. In this construction marker/£* moves to positions of higher and higher /^-states. When it comes to rest all markers further down the list have states at most as high as that of marker /i*. This gives the crucial property of there being only one 7r-state which is associated with an unbounded number of members oi si. At each stage we move the least marker /i for which there is a later marker of higher /estate. Ultimately each marker comes to rest, so, as the construction proceeds we treat all markers in turn, some possibly several times. 7.25 Minimal degrees 28. There is a non-recursive degree such that any degree of lower unsolvability is the solvable degree. It suffices to show that there is a non-recursive set s/ such that if a set 88 is recursive in si then either 88 is recursive or si is recursive in ^ . We construct the characteristic function g of a set si by stages, each stage will ensure that one of the requirements is met by one table. In toto each table will satisfy all the requirements. For each natural number v, we shall have, at the end of stage v, three functions hp,f{,f£, such that the following conditions are satisfied: For each natural number v, hv,f\,f^ are recursive, and fl,fl are characteristic functions. For each natural number v, hv is a strictly monotonic increasing function, flpi = flfi for fi < hp0, and for every A, f\fi #= flfi for some pi, hvX ^ /i < hv(SA). We call these intervals, the intervals determined by Kp. Thus for all natural numbers v, f{ and /£ are identical on the first interval determined by hv, and differ on every other interval determined by*'. For all /i < v the range of h? contains the range of hv, so that the intervals determined by hv are unions of bounded sets of consecutive intervals determined by h^. PROP.
7.25 Minimal degrees
369
1
In each interval determined by hf , f[ is identical with either/f or with /^, and similarly for /£. (In particular all four functions f\j£,f{,fi are identical on [0, h^O), i.e. the interval including 0 but excluding h^O. Finally for all ji < v, h^O < hv0. These are all the conditions. Suppose that we have defined hv, /{, /£ for all v. We define gv to be the initial segment of length hv0 such that gvji =/i/£ = /£/£, for 0 < /i < hv0. From the conditions we see that g0, gv ... is a chain of initial segments whose union is everywhere defined. Let g = U gv. V
Note that for any given v and any interval determined by hv, either g is identical with/j or g is identical with /£ on that interval, g is the characteristic function of the desired set $0\ It now remains to define hp,fi,f2 and to show that g has the required properties. Note that the functions hvj[jl are recursive for each v (but non-recursive in v). Stage 0. Let h° = 'kx.xjl = Xx.0,/20 = Xx. 1. Note that the conditions are trivially satisfied. The intervals determined by h° are all unit intervals. Stage 2J>+ 1. Let fi = 2v. By induction the conditions are satisfied for hf, f{,fg. Then/f and/£ differ on the interval [M), ^ 1 ) . See whether, on this interval O, #/f. If so define h8* = Xx.h^Sx), /f/* =/f/» = / f on [0, A^O), {h8P0 = hn), and let/f^,/^ be otherwise unchanged. Otherwise, then <3>v + f£ in the interval [h^O, hn) since ff,f£ differ in that interval, we define h8?, ff, /^ as before except t h a t / f ^ = /f/1 = /^ on It follows that h8?, /f, /^ satisfy the conditions. This stage ensures that g #= O,,, hence gr will be non-recursive. Stage 2y + 2. Let /^ = 2^+1. By induction the conditions are satisfied for h^J^, /^. Call g an available segment at ju, if g is defined on the segment [0, h^d) for some 6, and if, on each interval determined by h^ in this segment g is identical either with /f or with /^ in that interval. If g is an available segment at /i, we call g* an available extension of g if g* is an available segment at /£ and is an extension of g. In the interval [0, h^O) g can have only one value because//* and/^ are identical on that interval. Substage a. See whether there exists a natural number K and an available extension at fi, g such that for all available extensions at ju,, gr* of g, O[?*]/c
370
Ch. 7 Ao-Definable functions. Recursive function theory
is undefined. Recall that O[/]/c is defined if and only if the oracle is only required to consult arguments for which/is defined and gives an output. So that if/* is an extension of/then
on
[0,7^0), hs^0 = hn.
In this case ®[?]K is undefined, otherwise, O^ij/c and Oj *]K would be the same. This part of the definition ensures that hs^0 > h^O. (We could have taken/^ instead of/f.) Assume that hs^A has been defined for all A ^ 0, and that / f/*, have been defined on [0, h8^0). h8fld determines SO intervals on [0, hence there will be 2e different segments of length hs^6 that will be available at S/i. (Note that on [0, WO) the functions/f ^ , / ^ are identical.) Denote these segments by gl9 g2,...,g2ein some order, say lexicographic. Note that however g is eventually defined one of these segments will be an initial segment of g. We proceed through 2° subsets in order to reach our definitions of hs^(S6), fffi and/f^ on [hs^6,hs^(S6)). Then repeat with SO for 6 and so on. Substep 1. By remark (ii) above there exist available at JLC extensions sx and t± of segment g± and there exists a KX such that <S>[^]K and Oj*l]ic
7.25 Minimal degrees
371
are defined and unequal; furthermore 5X and tx can be taken to be of equal length. sv t± and K± can be found effectively. We can recursively enumerate the triplets (sl9tl9 /q); so run through these triplets until we come to one that satisfies, we know that there is such a one if substage c is reached. Define Substep SX. By remark (ii), there exist available at /i extensions s' and s" °f 3sA. U ^A a n ( i there exists a KSX such that Oj*']#CgA and O[,s"]/c^A are defined and unequal. By remark (i), there exists an available extension t of 9s\ U vx such that Oj^/c^ is defined. Hence we can get %A and tsx so that ssx is an available extension of gsx U ux and tsx is an available extension of 9s\ U vx a n ( i ®[V8SX]KSX a n ( l ®[VSX]KS\ a r e defined and unequal; furthermore ssx and tsx can be taken to be of equal length. As at substep 1, KSX, SSX, tsx can be found effectively. Define glsx^sx = ss\> 9SA.VS\ = ^sx- Observe that usx is an extension of ux and similarly for vsx. Let K be the least natural number greater than all members of the domain of u^e, (it is easily seen that the domain of u2e is non-null). We define h8fi(86) = K, = u2e
on
= v28 on
This completes the definition of A^/f^,/f>. It is easily seen that the conditions hold. Furthermore if substage c is reached, then g is recursive in ®jj. For we can decide g if we can decide, for each interval [h8Pd,hsr(Sd))9 6 = 0,1,... whether g agrees with/f^ or with/|^ on that interval (see note a). Assume that g has been computed on [0, hSfl6) then we can find A ^ 6 such that gx c: g and we can compute ] ®[,SA1/CA and ®^ /cA. One and only one of these must agree with <J>?A:A, because by construction of/f^ and/f^ either sx c: g or £A <= g and in the computation of $[,SA]/CA only arguments for which sx is defined are used. If the former agrees then g must agree with /f/* on the interval; otherwise g must agree with /f^ on the interval in question. This completes stage 2*> + 2. Let g be the characteristic function of £0 It remains to show that s/ has the desired properties. First s/ is non-recursive; otherwise g = O,
372
Ch. 7 Aa-Definable functions. Recursive function theory
for some v, contrary to the construction at stage 2v+ 1. Assume that 38 is recursive in srf\ that is cm = $>9V for some v. If stage 2j> + 2 terminates in substage b, then ®{J is recursive, hence 38 is recursive. If substage c is used then g is recursive in <&gv and so si is recursive in 38. This completes the demonstration of the proposition. 7.26 Degrees of theories P R o P. 2 9. For every set si there exists a theory 2T^ of the same degree as s/; furthermore, if J / is recursively enumerable, then ^^ is axiomatizable. By the degree of a theory we mean the degree of the set of the g.n.'s of the theorems in the theory. We shall find the required theories in the pure calculus of identity J*d\ this is an applied predicate calculus with identity as sole predicate. Clearly the truth or falsity of a statement in this calculus depends solely on the cardinal number of members in a model. We consider only non-empty models. We denote by £fpv the class of cardinal numbers for which the statement $ whose g.n. is v is true in a model of that cardinality. We call this class the spectrum of <j>. For instance the spectrum of (Ax) (Ey) (x #= y) is £{2 ^ x}. We list some properties of spectra. (i) Every spectrum is either bounded or its complement is bounded. (ii) There is a uniform effective procedure for going from a statement to its spectrum, i.e. recursive functions/and g can be found such that for a statement whose g.n. is v, if £fp v is bounded then g[v] = 0 and f[v] gives the spectrum in the ordering of all tuplets, and if the complement of the spectrum is bounded then g[v] = 1 and/[V] gives the spectrum of the complement. (iii) For every statement
7.26 Degrees of theories
373
quantifiers by conjunctions over 1, ...,J> and existential quantifiers by disjunctions, thus
isreplacedby fl 0 1
isreplacedby
£
Now use truth tables. This gives a uniform effective method of testing whether v is in the spectrum. If there are v quantifiers and if v < pi then the table for pi will give the same result as the table for v. We demonstrate this by induction on the construction of (j). If the matrix of <j) is atomic then it is x = x, for the case of one quantifier or is x = y for the case of two quantifiers, the result follows at once. To deal with negation it is simplest to take =f= as atomic as we did in the system Aoo, etc. The result follows at once. Let cj) be <j)' V <j)" and suppose the result holds for $' and for <j)". Then if
374
Ch. 7 Ao-Definable functions. Recursive function theory
and let Gv stand for Fv and ~ F8v. Then has spectrum the bounded set (y*', v",..., v^), its negation will have the complement of this bounded set in its spectrum. By J ^ we see that if we take a set 88 of statements of
_
and only if spectrum Qv' =^ J / . Using the recursive functions / and g from (ii), we have Ve^
iff £[>] = 0 and f[v] => tf,
or gr[i^] = 1 and f[v] <= j / . By f[v] c j / w e mean that the tuplet whose number is f[v] is a subset of J / . From this we see that for all natural numbers v, ves/ if and only if ~ GVE^^; hence s/ is reducible to « ^ . Again ve^ if and only if a certain sequence effectively determined by v is a subset of s/ or contains jrf. Thus e ^ is reducible to si. Finally 3~^ is recursively enumerable if 38 is. Hence 3~^ is recursively enumerable if si is. Thus by Prop. 15, Ch. 8 3~^ is axiomatizable if si is recursively enumerable. (Strictly this remark should be listed as a corollary to Prop. 15, Ch. 8.) 7.27 Chains of degrees 30. There exist non-denumerable chains of degrees. By Zorn's lemma there exists a maximal chain of degrees. This chain is without a greatest element, otherwise the jump of that element could be used to form a longer chain. Assume that this chain is countable, then by Mult. Ax. it contains a sequence of degrees a0, a 1? .... Again by Mult.
PROP.
7.27 Chains of degrees
375
Ax. choose a set from each of these degrees, say J/ 0 , J/ 1 ? ... where s/v is in av. Now form the set £8 of ordered pairs {V,JH} where v is in s/^ This set is above all the sets $£vy hence the degree of ^?, say b, satisfies av < b, for all natural numbers v, contrary to the maximality of the chain. The demonstration of this proposition is highly non-constructive. Bear in mind that there are at most a denumerable set of degrees beneath any given degree, because there are only a denumerable set of tables. The ordinal numbers of the second class have the property of being non-denumerable yet there are only a denumerable number of ordinals less than a given ordinal in this class. 7.28
Recursive real numbers
We finish this chapter with a brief account of recursive real numbers and recursive analysis. Rational numbers are easily introduced into the system Aoo as ordered triplets of natural numbers. Thus {A,[i, v} will play the role of
^ (in ordinary mathematical notation).
From this idea definitions of + r x r — r = r
{A.8v' + K.8v,/i.8v'+/i'.8v,8v.8i/ ± 1} for {A,/^} + , { A » ' } .
We leave the rest of the definitions to the reader, also the demonstration that they are satisfactory in that the laws of algebra are obeyed. Having got the rational numbers the recursive real numbers can be defined in any one of four different ways as follows: A recursively enumerable sequence of rational numbers given by the recursive function/is called a recursive fundamental sequence if and only if W-rfAr
t(
>T gv < /I, A,
where g is a recursive function, called the convergence function of the sequence. (i) A recursive fundamental sequence of rational numbers is called a recursive real number. From this definition it is easily shown, as in analysis, that if the recursive functions / and / ' both define recursive fundamental sequences
376
Ch. 7 Ao-Definable functions. Recursive function theory
of rational numbers then so do / x r / ' , / + r / ' , /—r/'> e^c-> a n ^ define the product, sum and difference of the recursive real numbers defined by the recursively enumerable sequences/and/'. It is easily shown that these definitions are satisfactory, in that the laws of algebra are obeyed. (ii) If g is a recursive characteristic function then the real number whose binary expansion is v.glg2g3... is called a recursive real number. It is possible to define the product, sum etc., of two recursive real numbers on this definition, details are left to the reader. (iii) If a recursive set of rationals is a left Dedekind section then the corresponding real number is called a recursive real number. Definitions of product, sum, etc. follow as for analysis, they are denoted by x s, + s , etc. (iv) If/, g are recursive functions and iff/i, g/i are interpreted as rational numbers and if//* < r g/i and gfi — rffi
Definitions of product, sum, etc., are easily supplied and shown to be satisfactory. These definitions give us four classes of recursive real numbers and these four clases are all the same if we interpret recursive as general recursive, but they differ if we interpret recursive as primitive recursive. It is easy to show that recursive real numbers form a field. Call this
7.28 Recursive real numbers
377
definitions are satisfactory in that the laws of algebra hold. The recursive complex numbers are easily shown to form a field ${i). We get the pair
P B O P . 32. S'(t) is algebraically closed.
We indicate the demonstration. Start with a constructive proof of the fundamental theorem of algebra, and formalize it within recursive function theory. One point which requires note is in finding the maximum of a terminating sequence of recursive real numbers. Let pl9p2,...,pn be a terminating sequence of real numbers defined by the respective sequences /[I, x],f[2, K], ...,/[TT, K] of rational numbers with convergence functions g[l, K],g[2, K], ...,g[n, K] respectively. Then the recursively enumerable sequence Max[f[l, K], ...,f[n, K]] defines the recursive real number Maxs[pv
...,pn]
with convergence function Max[g[l, K], ...,g[n, K]].
Analoguously for Min. Another point that requires comment is the formation of a recursive function of recursive complex numbers. This is achieved, if the function is h, say, and/[A] is a recursive fundamental sequence defining the complex number p, we form ^[/[A]] and if h is continuous we see that A[f[A]] is a recursive fundamental sequence defining h[p]. As a basis for analysis $ and ${C) are a bit unsatisfactory, for one thing they are enumerable, this may be a liability, or it might have little effect. But the Bolzano-Weirstrass theorem: Every bounded set of members of $ has a limit point in $ fails. Consider a recursively enumerable set si which fails to be recursive. Its characteristic function will correspond to a non-recursive real number p. If g enumerates si the sequence
is a bounded sequence of rational numbers (hence members of S) with p as sole limit point. But the sequence fails to be a recursive fundamental sequence. For if it were with convergence function h, where h is recursive, then we should have
& + &Z +- + &&«<{ Hence
7 ~ o < ^ ^ [ 0 7 ^ •••] log 2
for
^<
f°r
^
378
Ch. 7 Ao-Definable functions. Recursive function theory
hence to decide whether QK = 6 we need only consider gl, ...,gh2d whence g defines a recursive set contrary to construction. But if we are allowing only recursive real numbers, then perhaps we should only allow recursive sets of real numbers. The theory of recursive real numbers may be extended to real numbers of degree a. This again would give a denumerable set of real numbers, this theory remains unexplored. H I S T O R I C A L R E M A R K S TO C H A P T E R 7
Turing machines, as the name implies, were invented by Turing (1936) and were described by him in detail. Post (1936) also had the same idea, but only sketched it, Turing wanted to analyse into atomic acts what one does when one computes. The term 'partial recursive function' is due to Kleene (1938). The account of Turing machines we give is due to Kleene (1952). About the same time as Turing invented his machine several other methods for defining calculable functions were invented independently and for different reasons. The system of X-conversion due to Church (1941) was one of these. Turing showed that the class of functions definable by his machines was identical with the class of X-definable functions. Curry (1958) produced a Combinatory Logic in order to invent a logic without variables and so avoid the difficulties of substitution. Post (1943) used a theory of productions, this was the result of analysing the concept of substitution. General recursive functions as defined by a system of equations were due to Herbrand, Godel (1933, 1934) and Kleene (1963). More recently Markow (1951) has given a theory of algorithms which is a significant modification of the productions of Post. More recently Smullyan (1961) has produced an equivalent theory of elementary formal systems. All these systems have led to the same class of calculable functions, hence Church's Thesis. Partial recursive functions are largely due to Kleene (1938); allowing them makes the theory of recursive functions more elegant, rather like homogeneous co-ordinates in analytic geometry as opposed to Cartesian co-ordinates. The idea of a tape, which we use from the beginning, is due to Turing (1936). 'Passive instruction' appears to be due to Kleene (1952), as is the bracket and dot notation in describing tables of machines. The demonstrations Props. 1 and 2 are due to Kleene (1952). The idea of a universal machine goes back to Leibniz, as already noted, but Turing (1936) was the first to give full details as to how to build one. But his universal
Historical remarks to Chapter 7
379
machine is not so powerful as that envisaged by Leibniz, but apparently it can do anything that can be done by man. The undecidability of the classical predicate calculus of the first order was first demonstrated by Church (1936); Turing (1936) gave another demonstration as an application of his machines. The concept' essentially undecidable' and lemmas (i), (ii), (iii) and the corollary are due to Tarski (1949, 1953), Prop. 7 is due to Turing (1936). Prop. 8 was first stated by Post (1944) and Cor. v is often called Post's theorem. Props. 9 and 10 are Post's (1944). The theory of complete, creative, simple and hypersimple sets was initiated by Post (1944) and the constructions in Props. 11, 12, 13 and Cor. (i), (ii) are due to him (1944). Prop. 14 and corollaries (ii), (iii) and (iv) are due to Myhill. The concept of degree of unsolvability is Post's (1944), who defined the various classes of sets just listed in order to try and find a set whose degree of unsolvability was strictly between the degree of solvability and the degree of a complete set. This problem which remained unsolved for several years, required a new method for its solution, it is known as Post's (1944) problem. The theory of degrees was further developed by Kleene & Post (1954) and an authoritative account was given by Sachs (1963). The term 'productive' is due to Dekker (1955). Props. 15 and 16 are due to Myhill (1955). Prop. 17 is sometimes called 'the recursion theorem' or the 'fixed point theorem' is due to Kleene (1952). Prop. 18 is again due to Myhill. The term 'oracle' is due to Turing (1936) and Prop. 19 is Kleene's (1952) formulation of the analogues of Prop. 2 for oracle machines. Prop. 20 is due to Kleene & Post (1954), the construction here is fairly straightforward, but that in Prop. 21, which solves Post's problem is quite a different matter. The method of proof is due to Friedberg (1957) and Mucknik (1956) independently, it is now called the priority method, a term due to Sacks (1963). It is so called because in the course of the construction certain things are given priority over others in order that part of the construction which satisfies the required conditions may not be upset as the construction proceeds. Prop. 22 is due to Friedberg (1957), Prop. 23 and its corollaries are due to Spector (1956). Prop. 24 is due to Shoenfeld, Kleene and Post, and Prop. 25 is due to Kleene. The notion of cohesiveness is due to Rose & Ullian (1963), Prop. 26 to Dekker & Myhill (1960), the demonstration of Prop. 27 comes
380
Ch. 7 Ao-Definable functions. Recursive function theory
from Yates (1965) who gave an elegant simplification to a result of Friedberg (1958). An account of computability has been written by Davis (1958) and another by Hermes. A very detailed account of recursive functions has been given by Hartley Rogers jun. (1967), and one on degrees of unsolvability by Sacks (1963). The theory of degrees of unsolvability is in a very untidy state at present. Most results state that the upper semi-lattice of degrees of unsolvability lacks such and such a nice property. The methods used are very ingenious, it seems that further progress requires some new method, more powerful than the diagonal method of Cantor and the priority method of Friedberg (1957). The motivation for the construction of simple and hypersimple sets is as follows. Post (1944) considered the problem of the reducibility of recursively enumerable sets of natural numbers. He invented several different kinds of reducibility. A set si is reducible to a set 88 if the problem of membership of si can be reduced to that of membership of 3$. The kinds of reducibility considered by Post are: (a) Many-one reducibility, (m-l)-reduciblity.
where / is a recursive function, if / is a (1-1)-function then this type of reducibility is called one-one reducibility. (b) Truth-table reducibility. (m-1)-reducibility can be put down in the form of a table thus: vest f
A generalization of this is:
vest
AQfrag, f2{v}e@,
...
fg[v]{v}e@
where there are 2g{p) rows, giving all the possible ways of placing (+) or (—) in the g{v} columns. The functions /,... are recursive, then if we can decide S3 we can, using the table, decide si. If g is a bounded function
Historical remarks to Chapter 7
381
then this method of reduction is called reduction by bounded truthtables, otherwise reduction by unbounded truth-tables. (c) Turing reducibility, (the only one we have considered). This is a Turing oracle machine depending on the characteristic function of a set 88. This machine with input v will produce output 0 if vestf and output 1 otherwise. Then membership of s/ has been reduced to membership of £8 in the most general way possible. We can develop a theory of degrees for each of these methods of reduction. Post (1944) wished to find if there was a degree of unsolvability between the degree of the complete set and the degree of solvability. He found that a N. and S.C. that a recursively enumerable set be (1-1)complete was that it be unbounded and that its complement contain an unbounded recursively enumerable set. Hence he set out to find if there were a simple set, such a set would then be of (1-1)-degree between the degree of the complete set and the degree of solvability by (1-1)reducibility. He was able to construct a simple set but found that it was of the same degree of unsolvability as the complete set with regard to reducibility by unbounded truth-tables. This particular simple set is obtained as follows: Consider the unbounded sequence of mutally exclusive bounded sequences: ^
( 3 j 4 ) ( 5 ? 6 > 7 > 8 ) #<# ( 2 , + i > 2 -
+ 2,..., 2-+i)....
Given a creative set ^ , generate the elements of the simple set Sf which we constructed, placing each in a set £fv Also generate the elements of ^f, and as the element v is generated, place all the natural numbers in the i>th sequence of cr in £fx. The resulting set S^ is a generated set, and hence is recursively enumerable. Since S^ contains £?, £? contains £fv As 9* is simple, «$^, and hence SP^ is without a recursively enumerable subset. Moreover &?1 is unbounded because ^ is unbounded. For each element of ^ , the corresponding sequence in or has only those of its members that are already in £f also in SPV and hence at least one element in SP^ Hence «S^ is simple. In like manner we see that a natural number v is in <& or ^ according as all of the integers in the yth sequence of cr are in SPly or at least one is in S?!. If then we make correspond to each positive integer v the sequence of 2V natural numbers (2" + 1 , 2" + 2,..., 2V+1) and the truth-table of order 2V in which the sign under v is (+) in that row in which the signs under the 2vf{fiy$ are all (+), and in every other row the sign under v is (—),
382
Ch. 7 A0-Definable functions. Recursive function theory
we have a reduction of *€ to ^ by unbounded truth-tables. Thus simple sets are useless as candidates for sets of lower degree of unsolvability than the degree of the complete set. The counter example just given suggested to Post to see if there was a hyper simple set. He was able to construct one, in fact the construction we have given, and showed that it was of lower degree of unsolvability than the complete set by reducibility using unbounded truth-tables. But he was dubious that they were of lower degree by Turing reducibility. Post ended his remarkable paper with the following remark. 'Indeed, if general recursive function is the formal equivalent of effective calculability, its formulation may play a role in the history of combinatory mathematics second only to that of the formulation of the concept of natural number.' Prop. 27 on the existence of a maximal set is due to Yates (1965). Prop. 28 on the existence of a minimal set is due to Spector (1956), we rely on the demonstration given in Rogers (1967). Recursive real numbers have been considered by Turing (1936), Rice (1954), Klaua (1961), Goodstein (1961) and Specker (1949). We have omitted to say much about recursive ordinals and recursive cardinals, for the latter see Dekker and Myhill (1960). A vast number of examples on the subject matter of this chapter can be found in Rogers (1967) and in Shoenfield (1967). Much of the matter discussed in this chapter and in chapter 5 has been worked out by Smullyan (1961) for general systems, he uses the methods of Post (1944).
EXAMPLES 7
1. Find diagrams for the following machines: (i) Started observing a tally replaces it by a cipher seeks the first cipher to the right replaces it by a tally and stops. (ii) Started observing a number in standard position will decide if this is the last member of the representation of a /c-tuplet and will then write down K after a one cipher gap. (iii) Started observing a number in standard position will find the first cipher to the left will move the number two places to the right and replace the cipher by , I, i.e. it replaces, V, by , 0, v9.
Examples 7
383
(iv) Started observing a number in standard position will decide if it is even, if so prints cipher followed by a tally, otherwise prints cipher followed by two tallies. 2. Write down the diagrams for the tables which calculate the following functions: + , x , exp, P, —, Av Bv A2, B2, 2r, Ilr. 3. Similarly for the function Un. 4. (i) Show that it is impossible to find a uniform method for deciding if two primitive recursive functions ever take the same value, (ii) Show that it is impossible to find a uniform method for deciding of two recursive functions whether they are interlocked, i.e. fv < gv < fSv for all numerals v. (iii) Show that it is impossible to find a uniform method for deciding of two recursive functions whether they are asymptotically equal, i.e.fv = gv for all sufficiently large v. 5. Show that the family of recursive permutations form a denumerable group under composition. 6. Show that isomorphism of sets is an equivalence relation, si and 38 are sets of natural numbers. Write: si ~ 38 if there is a partial function / such t h a t / j / = 38, si ~ 38 if there is a partial (1-1) function/such t h a t / s / = 38, si £ £8 stf is isomorphic to 88. Show that J / ^ ^ - > J ^ ~ & and si ~ @->si - 3S. stf ~ 88 fails to imply si ^ SS, stf ~S#
fails to imply si £ 38.
If si and 3% are bounded then si ^ 3$\ si' ^ 38 and si ~ 38 are equivalent. 7. If si, 3# are recursively enumerable then stf~38
& S3
~38.^>si^3S.
8. Let si ^ 38 then, si recursively enumerable <->«^ recursively enumerable, si immune <-> 38 immune, si productive <-> 38 productive. 9. Give the analogue of Prop. 17 for/-recursive functions. 10. Cor. (i), (ii), (iii), (iv), (v), (vi), (vii), Prop. 5, Cor. (i), Prop. 7,
384
Ch. 7 A0-Definable functions. Recursive function theory
Cor. (i), (ii), (iii), Prop. 8, each give rise to an oracle function, e.g. Cor. (i), Prop. 5 gives rise to / , where //c = { according as f\K\ v = 0 for some numeral v or otherwise. Find the relations of order between the degrees of these functions. 11. Show that it is impossible to find a uniform method for deciding whether a recursive function is never zero for sufficiently large argument. 12. Show that the set of gr.w.'s of tables for recursive functions fails to be recursively enumerable. 13. Show that an unbounded recursively enumerable set si is of the form J / = ^ r U ^ , where #V n & = 0, Wr is creative and 0* is productive. [Consider s/(] #, sin f5.] 14. Let 3P be productive and si be recursively enumerable, show that j / c ^ - > ( ^ _ j / i s productive), & <=• si-> (jtf U & is productive). 15. Let ^ be the /-complete set, show that ^ fails to be/-recursive. 16. Show that ^ is /-productive. 17. J / is the set of natural numbers v such that 9£v is bounded, show that si and si are productive. 18. Give details of an oracle machine in which the characteristic function of the ' oracle' set is written down on a second tape. 19. Give a demonstration of Prop. 21. on the lines of Prop. 24. 20. Give a demonstration of Prop. 24 on the lines of Prop. 21. 21. Find the table for the machine to calculate 2£$ as described in Prop. 27. 22. Show that it is impossible to decide if the intersection of two recursively enumerable sets of lattice points is void, or is bounded or is unbounded. 23. Show that it is impossible to decide of a recursive function p whether Spv = p(Sv) for some numeral v. 24. Find the table for the function of Prop. 18, whose g.n. is v. 25. Show that the following problems are recursively unsolvable: (a) To decide if $„ is a constant function. (b) To decide if v is in 3CK. (c) To decide if two tables give the same function. (d) To decide if 9£K = XV. (e) To decide if 2£K is unbounded. 26. The direct product of two sets si and 38 is the set of ordered pairs (v,/i), where vesi and pieSS, Show that the operation of direct product
Examples 7
385
is uniformly effective for recursive sets and for recursively enumerable sets. 27. Show that: (i) ^ j and ^ m are reflexive and transitive; (ii) \isi ^X3$
thenj/^m^;
x^
then
m J*
then
J/^JP;
(v) \i si ^m3ft and 3$ is recursively enumerable then si is recursively enumerable. 28. Show that there exist two non-recursive sets which are incomparable with respect to < m. 29. The m-reducibility ordering is an upper semi-lattice; i.e. any two degrees have a unique least upper bound. Furthermore the least upper bound of two recursively enumerable degrees is a recursively enumerable degree. 30. Show that si ^ mSS may hold while si ^ mS fails. 31. Show that the complete set is 1-complete (i.e. complete w.r.t. ^ x ). 32. Show that %(x is total) is productive. 33. Show that s/ is 1-complete if and only if s/ is m-complete. 34. Show that: (i) If si is simple then s/ is non-recursive; (ii) if s/ is simple then s/ is non-creative; (iii) if s/ is simple then s/ fails to be m-complete; (iv) if s/ is simple then si fails to be a cylinder; (a cylinder is a set of the form 3$ x JT, where Jf is the set of all natural numbers). 35. Show that the following classification is mutually exclusive and exhaustive: (i) The class of recursively enumerable sets, (ii) The class of immune sets. (iii) The class of non-recursively enumerable sets which are the union of an unbounded recursively enumerable set and an immune set. 13
SML
386
Ch. 7 A0-Definable functions. Recursive function theory
(iv) The class of sets si such that if £8 is a recursively enumerable subset of si then there is a recursively enumerable unbounded subset ^ of si U 38, and it is impossible to find a uniform effective method for finding an index for <% from an index for 38. (v) The class of productive sets. 36. Show that the join and the direct product of two simple sets is a simple set (the join of two sets si and 38 is the set = 2x&xesi v y = 2x + 37. Show t h a t ' s i recursive in ^ ' is transitive, but that 'si recursively enumerable in 38' fails to be transitive. 38. Show that the intersection of two hypersimple sets is hypersimple. 39. Show that the four definitions of recursive real numbers are the same. 40. Supply the definitions of sum, product, etc., for recursive real numbers defined by method (ii). 41. Supply the demonstration of Prop. 31. 42. Supply the demonstration of Prop. 32. 43. Show that there is a primitive recursive function g such that ]] = ®[g[x,y];z\.
Chapter 8 An incomplete undecidable arithmetic. The system A
8.1 The system A The system Ao is complete with respect to A 0 -truth hence it is impossible to extend it in order to get more A0-true closed A0-statements as theorems in the extended system, but it is possible to extend it so as to be able to express concepts which are unrepresentable in Ao. The system A is a primary extension of the system Ao obtained by adjoining the universal quantifier and a rule for its use. Thus A is richer in modes of expression than Ao. We accordingly introduce the symbol A of type 0(0i) which we call the universal quantifier.
D224
(Ag)m
for 4(Xgfl£}).
We also add the rule: <j){rj}VG)
,. generalization.
I\ is free for 9/, £ in 0{I\}, the variable y must fail to occur free in the lower formula and £ fail to occur free in 0{F J, co is subsidiary and may be omitted. We also allow free variables in A-axioms and rules, otherwise II &' would be useless. In II d oc can be a variable. The chief advantage of the system A over the system Ao is that some Ao-metatheorems become A-theorems. For instance (Ax) (x = x) is an A-theorem while in the system Ao we have instead: a = a is an Aotheorem for each closed numerical term a. Rule lid' has restrictions on variables, these are required for the following reasons. We require TL to be free for rj in
13-2
388
Ch. 8 An incomplete undecidable arithmetic. The system A
We require £ to fail to occur free in 0{F J in order to have (Ax)
(J.&) (a? = a?) V o)
x1 = # v o)
We require 7/ to fail to occur free in co to avoid x = Ovx 4= 0 (^#) (a = 0) v s =f= 0" We require 9/ to fail to occur free in 0{I\} to avoid #= x (Ax')(xr = x)' Note that the restrictions on variables are effective, we can always easily decide whether they are satisfied. Also note that once a quantifier is introduced into an A-proof then it remains in that A-proof till the end. Thus an A-proof of a closed A0-statement is an Ao-proof of that statement and an A-proof of a closed A00-statement is an A00-proof of that statement. Also note that the class of closed numerical terms is the same as in Aoo. 8.2 Definition of A-truth The gr.w.'s of the A symbols will be the same as before except that the g.n. of A will be 11 and that of of® will be (12 + 6). We give a truth-definition 3~ for closed A-statements in that we augment the definition ^ of A 0 -truth by the addition of: (vi) 0 is of the form (AE) i/r{^} and x]r{v) is A-true for each numeral v. Here £ fails to occur free in ijf{y}. An A-statement will be called A-valid if: (vii) Let £', £",...,£<"> be exactly all the free variables in #{£',£",...,gw} and let these variables fail to be free in the statement-form ^ { r ; , n , . . . , r ( ^ } , then ^{£',£",...,P*} is A-valid if and only if ^{/c',/c", ...,/c(|;)} is A-true whenever K'9K",...,/<&) are numerals. If v is zero the A-statement is already closed.
8.2 Definition of A-truth
389
We define Falsity for closed A-statements by the addition of the following to the definition !FQ of Ao A0-falsity: (vi) cj) is of the form (A£) ifr{£} and i]r{v} is A-false for some numeral v. Here £ fails to occur free in i]f{v}. We say that the A-statements ft{£',£",...,&*} and 0"{£',£", ...,£(A)} containing exactly the free variables shown are T-equivalent if for any numerals K', K", ..., i6v\ where v = Max [A, /i], $'{Kr, /c", ..., K^} is A-true if and only if (J)"{K\ /C", ..., K(A)} is A-true. Negation can be A-represented if we add to the definition of N["
D 225 N[ '^"] is the result of everywhere interchanging throughout
and and and
#=, V, A.
With this definition a closed A-statement
(v) suppose ^ is a closed universal statement (AE) i/r{Q and has been shown to be A-false, then, i/r{v} has been shown to be A-false for some numeral v. By our induction hypothesis it is impossible to show that i/r{v} is A-true, hence it is impossible to show that cf>{/i} is A-true for each numeral JLC, thus it is impossible to show that (AE) i/r{^} is A-true. 2. The system A is consistent with respect to A-validity. We have to show that each A-theorem is A-valid. The A-axioms are A-valid. We replace free variables by numerals and proceed as in Prop. 2, Ch. 4. The result then follows by (vii). Secondly we have to show that the rules preserve validity. In the case of the new rule II d' the conclusion is A-valid if and only if the premiss is A-valid so that 11 dr preserves A-validity. For the other rules we replace free variables by PROP.
390
Ch. 8 An incomplete undecidable arithmetic. The system A
numerals, same variable by same numeral, and proceed as before, except in the case of R 1:
Replace free variables by numerals throughout and this becomes oc'jd} = fi'{S\ v &>' {AS) 0'{q'{g}, g} V a>'
Some variables free in a{£} may be bound in ^{a{£}, £}. We show that R 1 preserves validity by induction on the construction of the main formula. If this is atomic then on replacing variables by numerals we are left with R 1 applied in Aoo and the result follows. The other cases are easily dealt with except the one just mentioned when the main formula is a universal statement. By our induction hypothesis a!'{6} = j3'{6} V a)' <j>'{*"{d}, 6} v a)'
for all numerals 6, but this says that if the upper premisses in (1) are valid then so is the lower premiss. The result now follows. 8.3 Incompleteness and undecidability of the system A PROP.
3. The system A is incomplete with respect to A-validity.
We have to find an A-valid A-statement which fails to be an A-theorem. D 226 Pr/A[£, w] for the primitive recursive A00-statement such that PrfA[K, v] is Aoo-true if and only if K is the g.n. of an A-proof of the A-statement whose g.n. is v. This is constructed as in the previous systems. But note that the g.n.'s of variables have been increased. D 227
s[£, y] Godel's substitution function for A.
This is constructed as before and is primitive recursive. Consider N["Prfh[x',s[x,x\~\"] let its g.n. be K. (We suppose that PrfA fails to contain x.) Let v be the g.n. of JV["Pr/A[y,$[7c, #c]]"]. Then v = S[K,K] is an Aoo-theorem. Suppose N["PrfA[x', v]"] is an A-theorem, then so is N[i(PrfA[xf, S[K, K]]"] by R 1. There will then be a numeral n such that n, v], here n is the g.n. of an A-proof of the A-statement whose g.n.
8.3 Incompleteness and undecidability of the system A
391
is v. But from the A-proof of the A-theorem N["PrfA[x', v]"] we can obtain an A-proof of N["PrfA[n, v]"] by everywhere replacing free occurrences of the variable x' by the numeral n. But it is absurd that PrfA\jT,v\ and N[(iPrfA[n, v]"] are both A-theorems, since Aoo is consistent, if they are A-theorems then they are Aoo-theorems. Thus N["PrfA[x'9 v]"] fails to be an A-theorem. We now show that N["PrfA[x', v]"] is A-valid, we have to show that N["PrfA[n,v]"] is A-true for each numeral n. Now N["PrfA[n,v]"] is a closed A00-statement so if it is A-true then it is A00-true hence AOoprovable. PrfA [£, TJ] is a primitive recursive A00-statement hence exactly one of PrfA[n9 v\ and N["PrfA[n,v]"] is Aoo-provable. Suppose that PrfA [TT, V] is Aoo-provable, then n is the g.n. of an A-proof of the A-statement whose g.n. is v. Thus N["PrfA[x', S[K, K]]"] is A-provable, whence so is N["PrfA[x',v]"]9 and hence so is N["PrfA[7T, v]"] for each numeral n. N["PrfA[7T, v]"] is closed and lacks quantifiers hence its A-proof is an Aoo-proof. Since Aoo is consistent we must have N["PrfA[n9v]99] for each numeral n. Thus N["Prfk[x',v]"] is A-valid but fails to be A-provable. Thus A is incomplete with respect to A-validity. 4. The system A is undecidable. A decision-procedure for A would give one for AOj because an A-proof of a closed A0-statement is an Ao-proof of that statement. Since Ao is undecidable then so is A. PROP.
COR. (i). A-theorems form a recursively enumerable set which fails to be a recursive set. A-theorems can be recursively enumerated similarly to the Ao-theorems. If they formed a recursive set than A would be decidable. 8.4
Various properties of the system A
P R o P. 5. If an A-disjunctand is an A-theorem then one of the disjunctands is an A-theorem and by examining the A-proof of the disjunction we can find which disjunctand is an A-theorem, and find an A-proof for it. The demonstration is similar to that of Prop. 4, Ch. 6, by theorem induction.
392
Ch. 8 An incomplete undecidable arithmetic. The system A
COR. (i).
Thus T.N.D. fails in A. COR. (ii). x = 0 v x 4= 0 fails to be an A-theorem. COR. (iii). Ibis a derived A-rule. P R O P . 6. An existential A-statement is an A-theorem if and only if a particular instance can be found and is A-provable. We have to show that if (E£) 0{£} is an A-theorem then so is ^{a} for some numerical term a. The demonstration is similar to that of Prop. 7, Ch. 6. If {£} contains a free variable TJ then so may a. Note that I & is still absent from the rules.
7. The scheme lid' is reversible. We have to show that if (A£) ^{£} v co is an A-theorem then so is ^{£} v o). It suffices to show that if (^4£)^{£} is an A-theorem then so is
COR. (i). / / {A£)(j){£) is an A-theorem then so is ^){y} for each numeral v and these proofs are obtained the one from the other merely by substituting the numeral v in an A-proof-form. Here £ fails to occur free in
<j){v}vw{v}
this becomes
JT-\ i~—L-f
which is still an application of II d.
8.4 Various properties of the system A
393
A similar result holds for (Ag) (j){g) v a). Hence the corollary. Each primitive recursive function / is explicitly A00-definable by an Aoo-formula p of type u, ui,... according to the argument set, such that pv = K is an A00-theorem if and only if fn = k where v and K are the numerals which A00-represent the natural numbers n and k. The same holds in the system Ao. Also each partial recursive function is A0-represented in the sense that if the value of g{n', n",..., # } is k then there is an Aoo-statement $%', 7/",...,rfe\£, g} such that (Eg)
Hence T
394
Ch. 8 An incomplete undecidable arithmetic. The system A
for some primitive recursive functions p. This says that Val(S, 1,8)7[/c]nftm[/c]n(9,9> is equivalent to a primitive recursive function p, but this we have found to be absurd. Another example of an A-true A-statement which fails to be an A-theorem is (Ax) (Ex', #") (JTV^I/c, X, X', X"] = 0), if the numeral A: is chosen as the g.n. of the table of a general recursive function which fails to be primitive recursive. P R O P . 9. / / (A£) (ETJ) ^{g, rj} is an A-theorem then so is
10. The system A contains an irresolvable statement. We have to find a closed A-statement $ such that both
COR. (i). (Ex)PrfA [x, v] V (Ax')N["PrfA[x, v]"] fails to be an A-theorem. For if it were then by Prop. 5 so would be one disjunctand, and this we have just seen is absurd. Thus T.N.D. fails in A. 11. A-falsity fails to be A-definable. We have to show that A lacks a statement Fals{x} with exactly one free variable, such that: (i) Fals{v) can be shown to be either A-true or A-false,
PROP.
8.4 Various properties of the system A
395
(ii) Fals{v) is A-true just in case v is the g.n. of an A-faise closed A-statement, (iii) Fals{v) is A-false just in case v is the g.n. of an A-true closed A-statement or fails to be the g.n. of a closed A-statement. Suppose that Fals{v} is A-definable and has the above properties. Consider Fals{s[x, x]}, let its g.n. be K, and let v be the g.n. oiFals{s[K, K]}, then v = S[K, K] is an A00-theorem. Thus Fals{v) states its own falsity. If Fals{v) is A-true then V is A-false, i.e. Fals{s[K, K]} is A-false whence Fals{v) is A-false, this is absurd by Prop. 1. If Fals{y] is A-false then V is A-true or fails to be a closed A-statement, but V is FCLIS{S[K, K]} which is a closed A-statement hence Fals{s[K, K]} is A-true whence Fals{v) is A-true, but this is absurd. Thus altogether Fals fails to be in the system A. ClstatAv for V is a closed A-statement. D 228 COR. (i). A-truth fails to be A-definable. Suppose that Tv is A-definable so that Tv is A-true just in case V is an A-true closed A-statement. Then ClstatAv &N["Tv"] would have the properties required of Fals, which is absurd. The set of A-theorems is recursively enumerable and so is A-definable, but the set of A-true closed A-statements fails to be A-definable and hence fails to be recursively enumerable. The theorems of any formal system are recursively enumerable, hence the A-true closed A-statements fail to be the theorems of any formal system. The argument of Prop. 11 can be applied to any formal system which contains the recursive function s[a;,a;], or in which this function can be represented, and which contains R 1 . Apply the argument to Aoo. Here A00-falsity fails to be Aoo- definable because A00-truth is general recursive and fails to be primitive recursive, negation is available so is Clstat00 which is primitive recursive so Fals is general recursive and hence unavailable in Aoo. Apply the argument to the system Ao. Here the A0-true statements form a recursively enumerable set which fails to be a recursive set. If Fals were A0-definable then A0-false closed statements would form a recursively enumerable set, we could then argue as in Prop. 8, Cor. (v) of Ch. 7 that the set of A-true closed A-statements would be recursive, which is absurd.
396
Ch. 8 An incomplete undecidable arithmetic. The system A
8.5 Modus Ponens P R O P . 12. Modus Ponens is a derived rule in the system A, A plus T.N.D. plus I(c). First we make a modification to the system A. We base it on ^c instead of OVL3F'G. This means that we take N as a primitive symbol and use the definitions of 4=, & and A in terms of = , v , N and E. Prop. 5 fails in A. We proceed as in Prop. 4, Ch. 3 by induction on the cut formula. We have to show: <^±
where x is present, but 0) can be absent. (a) $ is atomic. Then $ is an equation oc = /?. We proceed by theorem induction on N$ v X- If ^is is an axiom T.D.N. then % is 0 and we are finished. Use theorem induction on HN
o) v <]>
using the same rules as before but with 0) as the subsidiary formula instead of HN
where 2'JV0 is a part disjunction of 2JV0 and x is x' v X ">we have diluted with S"JV0 V #", where ZJV0 is S'iV^ v ISNtfr. If the result holds for the upper formula then we have \ v v' —* II a, « V (x' v ^") as desired.
8.5 Modus Ponens
397
The formula above 2iV0 v x niight be just %', where x *s x' v X"> this case we have ,
ll
as desired. Lastly the formula above EJV0 v # might be just S'JV0. In this case ^ is invalid. We now show that if o) v
(ii) if
a' = /?' is an occurrence of an invalid ancestor in the lower formula then there are two occurrences of an invalid ancestor of a = /? in the upper formula, etc. for more cancellations, (iv) if
j/-—r^—777
— is a case of rule I I 6 and if a' = B' is an
occurrence of an invalid ancestor of a = /? in the lower formula, then a' = /?' are two occurrences of an invalid ancestor of a = /? in the two upper formulae, (v) i f X v X V * ^
* = PVX
is
cage
f R x andif
, = ^/i
lower formula is an invalid ancestor of a = /?, then a/;/ = yffw, if invalid, is an invalid ancestor of oc = /?, but if aw = /?'" is valid then a" = fi" must be invalid and is an occurrence of an invalid ancestor of a = /?. We now omit all invalid ancestors of a = /?. Clearly we are left with a tree. At each node in this tree we have an application of the same rule that was used at that node before in case of rules I a, II a, b, c, d, e and I b provided that an invalid ancestor is in the subsidiary formula, otherwise
398
Ch. 8 An incomplete undecidable arithmetic. The system A
we get a repetition. Rule R 1 either remains a case of the same rule, or if an invalid ancestor is a main formula it can become a case of rule II a. A-Axioms, beingvalid are unaffected. T.N.D. reduces to an axiom SOL 4= 0. Altogether the new tree is an A-proof-tree of co, as desired. Note that the invalid ancestors of oc = /? must be in the subsidiary formulae of all rules except 16 and R 1, because otherwise a = /? would get governed by ND, N, E or NE and this is impossible. This completes the case when <}> is atomic. The cases when ^ is compound are dealt with exactly as in Prop. 4, Ch. 3. This completes the demonstration of Prop. 12. (See Ex. 8, 3.) 8.6 Consistency In Ch. 1 we denned a formal system jSf as an ordered quartet (£f, J^, J / , ^ ) , where Sf is a displayed list of distinct signs, !F is a list of rules of formation, si is a list of axioms and ^ is a list of rules of consequence. In each case it must be possible to decide by a fixed procedure in a limited amount of time whether an object belongs to one of these lists or is foreign to them. The list of JSf -symbols can be replaced by a list of natural numbers, and jSf-formulae by ordered sets of natural numbers. Then to be an J§?-axiom or an JSf-rule of consequence or to be a well-formed jSP-formula are, by Church's Thesis, recursive properties of natural numbers and hence expressible by recursive predicates. Thus a formal system 3? can be translated into arithmetic, the natural numbers playing the part of the names of J5f-formulae or of sequences of J^-formulae. We have already carried this out for the systems Aoo, Ao, A. We say that a, formal system J§? is definable in a formal system ££' if J§?' contains an jSf'-formula *df[
8.6 Consistency
399
such that Prfy>[K, v] is A0-true if and only if V is an J5?-proof of V, being Ao-true it is then an Ao-theorem. The result now follows. COR. (i). The system Ao is A0-definable. We have already given the required A0-statement Prfo[K, v]. We have just shown that any formal system is A0-definable and we have shown that Ao contains all recursive arithmetic. Thus we might attempt to reformulate all the work we have so far done in Ao. In particular we might attempt to formulate in Ao the results we have obtained about consistency. Anything we have said so far which fails to be Ao-formulable will also fail to be constructive, that is, if, as we do, we identify constructiveness with recursiveness, as in Church's Thesis. Consistency for a formal system J§?can be defined in several different ways. Con^ (i) Some particular closed J§?-statement fails to be an jSf-theorem. If j£? contains arithmetic one usually chooses the statement (0 = 1), or rather its counterpart in JSf. Cong (ii) For each closed JSf-statement
N[tcThm^[K]99].
For the system Aoo this fails to be A00-definable but can be A0-defined and A-defined. It fails to be A00-definable because Thmoo[x] and hence its negation N["ThmOo[x]"] are both general recursive and fail to be primitive recursive, they are then Ao- and A-definable. Now Ao is complete, hence N["Thmoo[KY9] is Ao-provable since it is A0-true.
400
Ch. 8 An incomplete undecidable arithmetic. The system A
(i) GonAoo(i) is Ao- and A-provable but fails to be Aoo-provable. For the system Ao, GonAo(i) fails to be A0-definable because N["Thm [x]"] fails to correspond toarecursively enumerable set. If it were a recursively enumerable set then Ao would be decidable. But ConAo (i) is A-definable, namely (Ax) N["Prfo[x, JC]"]. Now each of N["Prfo[v, #c]"] is an Ao-theorem because it is A0-true. If ConAo (i) were A-provable then N["Prfo[v, K]"] would be Aoo-provable in the same number of steps for each numeral v. This is absurd because to show that V fails to be an Ao-proof of V we shall have to find the components of v, in particular the first component and this operation is unbounded as v increases. On the other hand JV["Pr/0[V,/c]"] is an Aoo-theorem for each v, to see this we refer to Prop. 1, Ch. 4. There we gave a standard method for proving equations, in this we reduced the proof to the use of Ax. 1 and R 1 only so that only equations between identical terms arose, thus ( 0 = 1 ) fails to arise. The system A is too weak in rules of proof to be able to prove ConAo(i). (ii) ConAo(i) fails to be A0-definable, it is A-definable but fails to be A-provable. For the system A, ConA(i) is N["ThmA[K]99]. This is an A-statement. Now (0 = 1) is an A-theorem if and only if it is an A00-theorem and hence an Ao-theorem, hence if we could A-prove N["ThmA[K]99] then we could A-prove N[iiThmAo\K\"'\ which we have shown to be absurd. (iii) ConA(i) is definable only in A and fails to be an A-theorem. We note the following relations between the first three definitions of consistency, where we suppose that the J§f-truth definition is such that <j) and N["
8.6 Consistency
401
would be an A-theorem, which we have shown to be absurd. ThmA[Neg[K]] is an A-theorem so its negation fails to be an A-theorem by Prop. 2 hence by Prop. 5 N["ThmA[K]"] is an A-theorem if the disjunction is an A-theorem. But this is absurd. (iii) ConA(ii) is A-definable but fails to be an A-theorem. (ii) ConAo(ii) fails to arise because Ao is without negation. (i) ConAoo(ii) is A-definable but fails to be an A-theorem, for if it were an A-theorem then so would be one of N["Thmoo[x]"]9 N["Thmoo[Neg[x]Y'] by Prop. 5 and this is absurd. (iii) CW^(iii) is (Ax) (N["Thm
t-*T[v]~4>.
(i)
Consider N["T[s[x, #]"], where s is Godel's substitution function. Let its g.n. be K, and let v be the g.n. of N["T[S[K,K]]"], then v = S[K, K] is an Aoo-theorem. We have t- (f>, be a theorem of the system J§?' it is clear that the system J§?' must be somehow related to the system JSf. We have already shown that Ao contains a truth definition for itself and that A fails to contain a truth definition for itself. Let us now investi-
402
Ch. 8 An incomplete undecidable arithmetic. The system A
gate more closely to see if we can find how the system A might be extended so that in the extended system we could have a definition of A-truth. Consider the following primitive recursive properties and functions, all of which are explicitly A-definable. D230 prv: V is an A-statement and 'prv' is equal to that variant of a prenex normal form of V which is obtained from V by moving the quantifiers which occur in V to initial positions in the order in which they occur in V from left to right and relabelling the bound variables with different scopes as x',x", ...,#(/c) and relabelling the free variables from left to right as X^SK\ ..., x^K+7T\ (If K is zero the first set is absent, if TT is zero the second set is absent.) v fails to be the g.n. of an A-statement and prv is zero. D 231 st[v, v\ v"]: V is a variable, V" is a numerical term, 'st[v9 v'', *>"]' is that A-formula $ which arises from V when in V we everywhere replace the variable V by the numerical term V". v' and v" fail to be as stated above and st[v, v'', v"] is 0. It follows at once that a closed A-statement has the same truth value as any of its prenex normal forms. D232 If y = <*',..., *<*>> then mv = , ...,/c^2>>, if 6 < 8 then mv = 0. TQV\ V is an A00-true AOo-statement. This has already been defined. Thus V is without free variables, E or A. TKv: V is an A-true closed A-statement whose prenex normal form has exactly K initially placed quantifiers. We can define Tv T2,... successively by: T(sK)V: 'prv' is (A£)
(A)
8.7 Truth definitions
403
This scheme defines a sequence of A-statements for K = 0,1, 2,.... The prenex normal form of TKx contains 2./c initially placed quantifiers. Thus the number of quantifiers increases with K, and if we write down TKx in full primitive notation then the numeral K will fail to occur. We wish to define a statement T*v which says that V is an A-true A-statement, i.e. T*v is A-true if and only if V is A-true. To do this we require to have a statement T[x, x'] such that T[x, K] is TKx. Suppose we add such a statement to the system A, if necessary, then we can proceed as follows: T*x for (Ex') (x' <x&prx& T\$rx,x']). Then T7* is the required truth-predicate. We already know that such a statement is absent from the system A. The only place in the above set of definitions where we were uncertain about being in the system A is the replacement of the scheme (A) by a similar scheme with a free variable instead of the numeral K. Thus we are unable to A-define some statements P defined by the scheme:
P[x, Sx'] = B[x, x\ P[p[x, x', x"l x']]J where Q and R are previously defined A-statements, and the variable x" is bound in R and p is a previously A-defined function. The symbol = is to mean that if one component is an ^-theorem then so is the other, the system j£? being one in which P can be defined. In Ch. 11 we shall show that predicates defined by the scheme (B) can be explicitly defined in a primary extension of A, in which the equivalences (B) are theorems. The truth-definitions for the systems Aoo, Ao and A amount to these systems having a standard model. In fact the consistency of these systems with respect to their truth definitions simply says that they have a standard model. In Ch. 12 we shall say something about non-standard models.
A formal system is said to be of degree a if the g.n.'s of its theorems form a set of degree a. 8.8 Axiomatizable sets of statements A set £? of JSf-statements is said to be axiomatizable if a formal system «S?' can be found such that the set if is contained among the jSf'-theorems.
404
Ch. 8 An incomplete undecidable arithmetic. The system A
15.-4 set SP of S£l-statements is axiomatizable if and only if it is recursively enumerable. Let ^ be the closure of a recursively enumerable set 88 under some relation R. Thus ^= R £ ff & R ^ £ PROP.
where i?"^ 1 is the set of things which stand in the relation R to some member of 3C, and C0 (see D 43, 74) is the intersection of the classes 2E se which satisfy &. Suppose that there is a primitive recursive relation Q such that Q is a symmetric subrelation ofR, i.e. Q{v, /i} -» (${/£, y} & R{v, /i}) for all v, /i, and such that for each v e 88, Q{v, ju] for an unbounded set of /i (then R{v,ju] for an unbounded set of /i if ve88). Then there exists a primitive recursive set si such that ^ is the closure of si under R. For instance si = $(Ex')x[Q{fx', x}] (i.e. the set of things which satisfy the displayed condition) will do, where / enumerates &&, and / is primitive recursive. For each ve8S there is a fiesi such that Q{v,/i} and hence Q{fi,v} therefore the closure of si under Q and hence under R, includes &#, hence the closure of si under R includes ^ . Conversely since Q is a subrelation of R, si is included in %\ Finally si is primitive recursive. Thus the closure of si under R is ^ . Let SP be a formal system and let R be the relation of deducibility in SP so that R{/i, v) if and only if CJLL' is a proof of V. Suppose that SP contains conjunction and that ^ and 0 & ^ & ^ & . . . & 0 are interdeducible in £P. Let Q{/i, v) be that primitive recursive relation which holds if and only if '/i9 is
8.8 Axiomatizable sets of statements
405
sively axiomatized in SP, and hence formalized without the aid of additional symbols or rules of inference. For example, if &" is a system employing higher types which express an analytic theory of numbers, then there exists a system which expresses the corresponding elementary theory of numbers, its theorems being those theorems of 2T which are without higher types. Also, for example, if 0t is any recursive set of non-logical (individual, function or predicate) constants containing at least one constant predicate, then there exists a system whose theorems are exactly those theorems of S" in which no constants other than those of St occur. To take a final example, suppose that SP is completable and hence that there exists a complete and consistent system SF whose set of theorems is recursively enumerable and whose axioms and rules include those of SP. Then, provided that the set of formulae of SP is recursively enumerable, SP can be completed without use of additional symbols or rules. If y is the set of A-true A-statements and if R is the deducibility relation in A and if 86 is a recursively enumerable subset of &" and if 92 is its closure under R, then we can find a primitive recursive subset J / ' of £8 such that if we form a formal system with stf' as axiom set and R as deducibility relation then ^ is its set of theorems, hence ^ is recursively enumerable and contains 0$. Of course in this way we fail to obtain
16. The set of A-true A-statements is productive. We have to give a procedure for finding an A-true A-statement outside a given recursively enumerable subset of A-true A-statements. Let J be a recursively enumerable subset of 2T, the set of A-true A-statements, then we can find a primitive recursive subset s£ of 8$ such that we obtain a formal system SP with srf as axiom set, rules those of A and a recursively enumerable set *€ of ^-theorems. We add the A-axioms if necessary so that SP contains recursive number theory. Let Prfy be the proof predicate for SP and let s be Godel's substitution function for SP. We now proceed exactly as in Prop. 3 and Prop. 10, Cor. (i) and find that (Ax) N["Prf#>[x, v]"] is an A-true A-statement which fails to be in the set *jf and hence fails to be in the set 08. Thus given a recursively enumerable subset S8 of 3~ we have found a member of 3~ — 08. Thus 3" is productive. PROP.
406
Ch. 8 An incomplete undecidable arithmetic. The system A
COR. (i). An arithmetic based on the predicate calculus of the first order and containing recursive number theory is incomplete. It can be seen from the systems Aoo, Ao and A that the blame for incompleteness must be given to the universal quantifier, recursive function theory is comparatively blameless. In the system A we have an irresolvable statement G, G is A-true but fails to be an A-theorem. Now A is consistent with respect to negation by Prop. 1, thus we can add the statement G as an extra axiom, because N["G"] fails to be an A-theorem, this gives us a system A' which is a formal system and in which we can repeat what we did for A and obtain an irresolvable A-statement G\ we can then add G' as an extra axiom obtaining a formal system A" in which we again can find an irresolvable A-statement 6?", we can then add G" as an extra axiom and so on. In this way we generate a sequence of formal systems: A, A', A",..., from these we can form a formal system A(w) obtained from A by adding all of G, G\ G",... as extra axioms. But then we can begin all over again and find an irresolvable A-statement in A(w), say 6?(w), this we can add to A(w) as an extra axiom obtaining a formal system A(w+1), and so we can continue as long as we continue to produce formal systems. We shall have to stop somewhere in the constructive ordinals. Of course, in this way we fail to obtain a complete formal system. This argument applies to any consistent arithmetic. The consistency of A with respect to negation can be expressed in A as: (Ax)N["(ThmAx&ThmA[Negx]y'l this is ConA(ii). We have indicated that this is unprovable in A. The completeness of A can be expressed in A as: (Ax) (StatAx->(ThmAx v ThmANegx)), where StatA is the primitive recursive predicate such that 8tatAv is A-true if and only if V is an A-statement. Let G be an irresolvable A-statement, then both G and N["G"] fail to be A-theorems, hence we can add either of them as an extra axiom. Now G is A-valid so it seems natural to add G as an extra axiom, but we can add N["G"] and still have a consistent system according to Con (ii). Call A with N[' 'G''] as an extra axiom the system Ang. This A-statement is A-false so to make it Ang-true we should have to add some new elements to the numerals. We should have to add a new element a such that
8.8 Axiomatizable sets of statements
407
PrfA[<x, v] was A^-true then we should require as new elements pa for every primitive recursive function p, and so on. We shall see what these new elements are in a later chapter. This gives rise to what are called non-standard arithmetics. Here V is N["G"] Note that rule I b is only required when some axioms are disjunctions, for instance if T.N.D. is among the axioms.
H I S T O R I C A L REMARKS TO C H A P T E R 8
We introduce the universal quantifier in a manner similar to the way we introduced the existential quantifier. Instead of the universal quantifier we could have introduced negation. The rule II d' is again after Gentzen (1934), but it also occurs in many other formulations of various systems in which universal quantification is required. It means that we have to use free variables. With the advent of universal quantification, or negation, we immediately get incompleteness and other undesirable properties. Prop. 3, the incompleteness of the system A, is ultimately due to Godel (1933, 1934), and the method of demonstration follows his use of the diagonal method of Cantor. Props. 5 and 6 are again the sort of things the intuitionists like, it is the absence of T.N.D. which allows them. Prop. 10, the existence of an unsolvable statement, is again due to Godel (1933, 1934). It was this result which dashed the hopes of Hilbert (1922) of finding a proof for any true statement. As the theory of recursive functions progressed the structure of the set of theorems and of the set of true statements was more clearly disclosed, the first is recursively enumerable, the second productive and this fails to be recursively enumerable. Prop. 11, the impossibility of an A-definition of A-falsehood, and its corollary are due to Tarski (1933). The corollary depends on the system having a negation. Prop. 9 is related to some results of Kreisel (1951) and Prop. 8 is peculiar to the system A, it arises from the absence of any form of induction. Considering a formal system as an ordered quartet is due to Carnap (1937). In Prop. 13 the term ' basic' is due to Fitch (1942) who was the first to produce a basic system of logic. Ao is a much simpler system than Fitch's. Consistency was first studied by Hilbert (1904, 1934-6) who wished to show that mathematics is consistent, but according to a theorem of Godel (1933, 1934) the consistency of a system cannot be proved in that system itself, provided the system is consistent and contains a certain
408
Ch. 8 An incomplete undecidable arithmetic. The system A
amount of recursive number theory. The discussion that follows the definitions of various kinds of consistency is motivated by Godel's theorem (Godel's second incompleteness theorem). The concept of one formal system containing a truth-definition for another formal system is due to Tarski (1933). The investigation to find how the system A might be extended in order to contain the truth-definition for the system A is due to Bernays (H-B, 1936). We shall consider it again in Ch. 11. Prop. 14 is due to Tarski (1956). Prop. 15 is due to Craig (1953). Prop. 16 is substantially due to Dekker (1955). Non-standard arithmetic was first invented by Skolem (1934). We consider non-standard arithmetic in Ch. 12. EXAMPLES 8
1. Show that the scheme II b' is reversible. 2. Show that the Deduction Theorem fails for the system A. 3. Show that Modus Ponens is a derived A-rule. (Use Prop. 5.) 4. Show that ^ ^ * holds in the system A provided that variables J ^{}v o) ^{a}v * free in a are free in (f>{ot} and that £ fails to occur in 5. Investigate the system A plus T.N.D. and 16. 6. Investigate the system B, the anti-A system.
Chapter 9 A-definable sets of lattice points
9.1 The hierarchy of A-definable sets of lattice points Recursive sets of lattice points in 01K will be called 0>% or Jg sets. These sets are A0-definable by an A0-statement of the form (EE) 0{£, v,..., vM}, where 0{£, v,..., v(K)} is quantifier-free. We define a hierarchy of sets in A as follows: are the projections of J2f*-sets, are the complements of ^£-sets. The complement of a recursive set is another recursive set, thus i?0 is the same as ^ 0 , the projection of a set in 0t8lc is given by a statement of the form (EE) 0{£, v',..., p(/c)}, the complement of a set defined by
# £ ' , I',...,gw
v',..., 1*)}.
J£-sets are A-definable by A-statements of the form (Ag')iEg')...4>{g',g',...,&>,*,...,it*\. In each case there are n initially placed alternating quantifiers and <j> is quantifier-free, the initial set of quantifiers are alternately universal and existential, the matrix is primitive recursive. Any A-statement is equivalent to one of these forms. First, an A-statement is equivalent to any of its prenex normal forms. Secondly, a batch of consecutive quantifiers of the same kind can be replaced by a single quantifier of that kind leaving an equivalent A-statement, thus (
is A-equivalent to This process is called contraction of quantifiers. [
409]
410
Ch. 9 A-definable sets of lattice points 1. (i) If (^ is a ^-statement then N["
PROP.
(ii) Suppose that 0{T/',r\", ...,T/{K)} is equivalent to ^{r/^, ...,7)^{K))} where d',d", ...,# (/f) is a permutation of 1,2, ...,K then
...,9/W} is a
(i) Follows at once by definition and the fact that the negation of a recursive statement is another recursive statement, (ii) ^r{ri^f\ ...,7){diK)) is equivalent to
where Z7jf,..., U*(K) are identity functions and A',..., A(/c) is the inverse permutation to the given one so that <9(A/) = 1,..., #(A
PROP.
is equivalent to a 0i%lt-statement and to
(ii) A ^-statement is equivalent to a ^^-statement andtoa 2,%^-statement. Let i/r{^\ ...,£(7r)> y\ .--,y(K)} be the matrix of a ^-statement, this is equivalent to and to
(Eg**) (f{£'9...,&"\r,', . . . ^ v (gm # ^ ) ) .
Thus the matrix of a ^-statement can be changed to either of the above, we chose the first if n is odd and the second if n is even. This will convert the ^-statement into an equivalent &>%„-statement. A similar replacement will convert a J£-statement into an equivalent £%„-statement. By Prop. 1 ^ & (T/(SK) = 7}(SK)) is a ^"-statement if and only if 0 is a &%statement, also by Prop. 1 JV["0"] v (V{SK) * Vm) i s a ^ - s t a t e m e n t if
9.1 The hierarchy of A-definable sets of lattice points
411
and only if 0 is a ^-statement. Thus (ETJ^) (N[('0"] V (vm * Vm))is a ^%,-statement, it is equivalent to JV["5&"]. Thus ^ is equivalent to a J^-statement. A similar argument shows that a J^-statement is equivalent to a ^^-statement. This completes the demonstration of the proposition. We showed in Ch. 7 that a general recursive set of lattice points in 0£K is Ao-representable by the A0-statement for some numeral d, where Un^K) is a primitive recursive function. Thus, by contraction of quantifiers a ^-statement is equivalent to an A-statement of the form (Eg) (UnM[09pt[g, 2,1], v\ ..., it*\pt[g, 2,2], 0] = 0). Generally a ^-statement is equivalent to an A-statement of the form (Eg') (Ag*)... (Un«**M[d9 &\g',..., g^> »',..., iM\ &\ 0] = 0), similarly a JjJ-statement is equivalent to an A-statement of the form (Ag') (Eg")... (Un0*«+\0\ &\ g',..., g(^« v\ ..., zA>, ®\ 0] + 0). Conversely statements of the two above forms are 3P%- J^-statements respectively. Here 6 is a numeral. Thus we have an enumeration of SP%and of J£-statements. These sets will be called ^£[0], &K*\_6\ respectively. If we wish to exhibit the arguments we write &*[d; v\ ..., v^], etc. 3. We can find a ^-statement which fails to be equivalent to any ^-statement, we can find a ^-statement which fails to be equivalent to any 0*%-statement. Consider the A-statement
PROP.
This is (Ag')(Eg*)... ( P n ^ [ T , f f ) f , ...,p-i>,7 M ", ...,^>,^>,0] * 0), where there are n initially placed quantifiers. This A-statement is equivalent to 2t%\Q\q, v\'\ ...,y^] for some numeral 0, Suppose that (1) is equivalent to 0*%[O"',if9y"9 ...,^ ( / c ) ], replace 7/ by d" and we obtain an absurdity. Hence (1) is a J2£-statement which fails to be equivalent to any ^-statement. In a similar manner we can find a ^-statement which fails to be equivalent to a =2£-statement.
412
Ch. 9 A-definable sets of lattice points
COR. (i). We can find a ^^-statement which fails to be equivalent to any ^-statement, we can find a £KSjT-statement which fails to be equivalent to any ^-statement. The ^-statement £Kn[O;V>V"> •••,?/('c)] of Prop. 3 is equivalent to a &%nstatement. Hence we have found a ^^-statement which fails to be equivalent to any ^-statement. In a similar manner we can find a cS^-statement which fails to be equivalent to any ^-statement. Altogether we have: 4. There exists a hierarchy of A-definable sets of lattice points in MK defined by A-statements of the forms PROP.
nm
(J25P?]) &L\e\ &$0]... AC = o, i, 2,... &i[0] J2§[0]...0 = 0,1,2,....
Each type contains a set which fails to be equivalent to any of the earlier types and to the other type in the same column (if any). 5. SP^-sets form a ring, so do M^-sets. We have already shown this for ^o-sets for all K, Prop. 9 (i), Ch. 7. Suppose that the result holds for ^£-sets, and i?£-sets for all K. PROP.
is equivalent to (Eg) £s/[d";£',V', ...9rf*\ V (Eg") £fK[em;g\V'9
..
and this in turn is equivalent to {E£) ( J W J & Y , ...,VM1 V J2f[d'";&,y', -,*?*}) by induction hypothesis this is equivalent to
and this is a ^gff-statement. Again 0%.\O;ii', ...,vM] & is equivalent to & {Eg') Ms/[6'";g",V',
where
£=
9.1 The hierarchy of A-definable sets of lattice points
413
which is equivalent to (Eg, g') 2*8*\ff"\ g, g, 7)',..., v(K)] by induction hypothesis, but this is equivalent to a ^^-statement by contraction of quantifiers. In a similar manner we deal with i^-statements. This completes the demonstration of the proposition.
9.2 Assets An A-statement which is equivalent to a ^-statement and also is equivalent to a ^-statement will be called a ( ^ n J£)-statement, and denoted by Aj. COR. (i). (0>* n ^-statements form afield. K
P R O P . 6. A (0*sn H £
Sn)-statement
is &®K-recursive also Sfijf-recursive.
Suppose that
and also is equivalent to a i^-statement Then N["${i',..., | w }"] is equivalent to the ^ - s t a t e m e n t
Now
g*)}"]
is A-valid (it may fail to be an A-theorem) whence
(En) W{V, £,..., gw} v N["x{y, £',..., &>}"]) is A-valid. Hence (M) [f{V,£',....gfr)} v
N["x{y,£',....gw}]"]
may be used if we adjoin the unlimited least number symbol to the system A. Call this term/[£',..., £<">]. If ^ { / 0 ' , . . . , iW], v',..., iW} is A-true so is (Ey) ftty, v',..., iA>} whence so is ^ { / , . . . , vM}. Again (EV)(ijr{y, vf,...,iA)} v J ^ [ " ^ , "'> . . M ^ } " ] )
(2)
414
Ch. 9 A-definable sets of lattice points
is A-true, whence so is W\y'> • • • > ^ ] > "'>•••> ^ }
vN
V'x{f[v'>
• • •, ^ l *',..., ^>}"]
(3)
whence so is
whence so is whence, if 0{y', ...,JA>} is A-true then so is ^ { / [ J / , ..., J A ) ] , / , ...,*A)}. Thus 0{V,..., iA>} and ^{/[V,..., v(% *>',..., iA>} have the same A-truth value. In a similar manner we show that N[6C^>{vf9...9i^K)}99] has the same A-truth value as JV["x{/|>',..., ^(/c)], *>',..., ^ " J , whence ^{i^'5..., jA>} has the same A-truth value as x{flv'> •••^ (/f) ] J ^> •••> ^ } - Now ^{^, J^', ..., jA>} is a Jf^-statement and ^{v, ^', . . . , ^ } is a ^f^-statement. Thus a (^STT H i^J-statement is obtained from a ^f^-statement by substituting f[v',..., jA)] for v, it can be similarly obtained from a 2%^-statement. Now the function / is recursive in i?f^-statements, because its value can be found if we are given the truth-values of certain J f ^-statements, in our case the statement (3) with 0,1,2,... successively replacing/!)/,..., *A)]? the search terminates since (2) is A-true. Thus 0 is recursive in 2fKstatements, similarly it is recursive in 3P®^-statements. COR. (i). A {0*1 n ^-statement is recursive. This is Prop. 8, Cor. (v), Ch. 7. COR. (ii). A {&%„ n M%n)-statement has degree ^ O(7r).
^-statements and «2^-statements are of degree < O^n\ P R O P . 7. An A-statement ^-recursive a (&>KSn n £%n)-statement.
or Q^-recursive is equivalent to
Let the A-statement ^ be ^-recursive, where ^ is ae2£-statement. Then the A-truth-value of (j) is a recursive function of the A-truth-value of ^ . Thus for numerals v\ ..., *A) <j){v',..., v^} is equivalent to: for some 6, where
9.2 Assets
415
and f is the characteristic function of ijr{g'9...,£(/c)}. Thus
(Eg, v) ((Unf*[09 */,..., ift\ v, g, 0] = 0) ,K]})).
(4)
Now xjr is a J£-statement, hence its prefix begins with a universal quantifier, say it is AEAEA (omitting variables), so that ^ is a 2L\statement, then N["&"] has the prefix EAEAE. Now x'^X" is an abbreviation for (N["x'"] V x") &{N["x""] V #')• If x' is quantifier-free and if x" is ^ then ^' <-> ^ is of the form:
["f'"]),
(5)
where ^r' is quantifier-free. Ee is short for (Ev^). (5) is equivalent to rx'"] v ^ ) & (X' V ^ [ " ^ " ] ) ) . By contraction of quantifiers this is equivalent to an A-statement of the form: EAEAEAco, where o) is quantifier-free. The second clause in (4) is of the form This is equivalent to
and this is equivalent to EAEAEAGJ', where a)' is quantifier-free, by bringing quantifiers to the left and contracting. Thus (4), by contraction of quantifiers is equivalent to E A E A E Ao)'r where a)" is quantifier-free. Thus (4) is equivalent to a ^-statement. In general if ^ is a J£-statement then (4), and hence
416
Ch. 9 A-definable sets of lattice points
9.3 Sets undefinable in A P R O P . 8. There is a set of natural numbers which fails to be A-definable. Consider the set of numerals which satisfy N^'&l^v^ v]"]9 where v = {vv v2}, ^ it i s A-definable then it is equivalent to &*}[/<, v] for some numerals 6, K. We should then have for each numeral v N["£PlJy2, v]"] is A-true if and only if ^|[/c, v] is A-true, if we take v = {6, K} we obtain an absurdity. Thus we have given a description of a set which fails to be A-definable. The condition that a numeral v satisfies the conditions requires varying numbers of quantifiers according to vx and this number is unbounded. The system A is too poor in modes of expression to be able to define such a set. The complete set has degree O', it is the set {0,K}, where table 6 produces K. The set O" is the set {6, K} where table 6°', i.e. table with g.n. 6 recursive in the complete set of degree O', produces K, and so on. O(w) is the degree of the function/(w) such that/ (w V = /(Vl) v2, where/ (v ^ gives the complete set of degree vv
f'v = 0 <-> (EV) (Uni[vl9
Vl,
V* " J = 0),
then table vx produces v2. f'v = 0 <-> (EV) (Un\[vv Vv VV» ^ ^ = 0), then table vx recursive i n / ' produces vv <-> (EV) (Eg) (Unl[vl9 Vv & y2i v2] = 0 & (AQn{Pt\£, d = 0 - (E?) (Un^v fi, & f j = 0))) *-* (Etj) (AC,1) R{T}, t,'}, where E is recursive. Generally f(S/l)v = 0^{Er/) ( ^ , } #_# {Qrj{ll))m^^^^^^ where R^ is recursive and Q is E or ^4 according as ft is odd or even. We define/(w) by :/(6>V = /(j;iV2, so that/ (w V = 0 involves a varying number of quantifiers as v varies, and this number is unbounded. Having obtained/(6>) we can continue and get/ (w+1) ,/ (Gj+2) ,... and so on into the constructive ordinals. /(w) is called the transfinite jump of/. COR. (i). The set of g.n.'s of closed A-true A-statements fails to be arithmetically definable. For if it were then A-truth would be A-definable.
9.3 Sets undefinable in A
417
By using Props. 6 and 8 of Ch. 3 together with contraction of quantifiers and the equivalence of statements obtained the one from the other bychange of bound variable (without of course collision of bound variable) we can obtain an algorithm for finding the position of any A-statement in the arithmetic hierarchy. In general one is interested in finding the lowest place in the arithmetic hierarchy. In general the equivalences mentioned above fail to give a unique result, just as the prenex normal form fails to be unique. However the lowest place in the hierarchy can always be found because there are only a bounded number of ways of applying the equivalences. As an example consider the set of natural numbers v such that the set of values of the function given by table number v is recursive. The set of values calculated by table v is recursively enumerable. A set is recursive if and only if it and its complement are recursively enumerable. Thus the condition defining our set is where 9EV is the set of values produced by table v. More fully this becomes
(Ey) (Ax) This again is (Ey) (Ax) (N["(Eu, u') Un'[v, u, u', x]"] *-> (Ev, v') Un'[y, v, v', x]), where Unr is defined by D 122. Contracting quantifiers this becomes
EA(EE&AA), where we have left out the scopes of the quantifiers. This in turn is equivalent to a statement of the form:
EAAAEE and this in turn to one of the form EAE. Thus the set of table numbers of recursive sets is in 0%. 9.4 f-definable sets of lattice points
We can also develop the theory of/-definable sets of natural numbers. We give some results. The procedure is much the same as for the sets we have already considered. 0>%sf and 2L%t<s* are the classes in the hierarchy of ^/-definable sets of lattice points in 3%K. Similarly for functions we use 0*%** and St%*K The individual members of these classes are (&j) (Un*+"+\0,3, £',..., g^D, i/,..., i*>, % g , 0] = 0), 14
(i) SML
418
Ch. 9 A-definable sets of lattice points
where the sequence of alternating quantifiers begins with E for a ^£ j/ -set. A 2%'-set is just the negation of (i). These we denote by ^f [0; v', ..., j/*>] for the statement, and ^£ j/ [0] for the set. 0 is called the index of the set. We now collect some useful results. We often omit the dimension number K. Similarly for 21 and f. 2f is the ^/-complete set. 9. (i) 8ft is recursive in si if and only if 88 and 88 are recursively enumerable in si.
PROP.
(ii) si' is recursively enumerable in si. (iii) si' fails to be recursive in si. (iv) 88 is recursively enumerable in si iff 88 ^ x si'. (v) 8ft is recursive in si iff 8ft' ^ x si'. (vi) 88 e SP'gn if and only if 8ft is recursively enumerable in si^. (vii) 88 e 0%, n 24* if and only if 88 is recursive in si^\ (viii) If 3ft e SPfv then an index for 88 as a set recursively enumerable in si^ can be found from any ^^n index for 3ft, and conversely. (i) The demonstration is as for Prop. 8, Cor. v, Ch. 7. (ii), (iii) are dealt with as for the complete set c€. (iv) 88 is the range of a function recursive in s/, thus 8ft = 2£f for some 0, thus Xe88 iff <0, A> es/\ so 8ft ^ x stf'. Conversely if 88 ^ x s/\ so that Xe88 iff hAes/', where h is a 1-1 function. We have fies/' iS/i = (6, A) and table 0 in stf produces A. Thus generate the natural numbers, and when the natural number v has been generated, set table v\ going for v\ moves with input v\ if this produces the natural number A then write h~xX down in a list, this enumerates 88. (v) If 8ft is recursive in J / , by (ii) 88' is recursively enumerable in 88. By (iii) 88' =(= 0 . Hence 88' is the range of some function/recursive in 88. But if / is recursive in 88 then / is recursive in s/. Hence 8ft' is recursively enumerable in si, and, by (iv) 88' ^ ts/'. If 8ft' ^1s/\ Trivially 88 is recursively enumerable in 8ft and 88 is recursively enumerable in 88. Hence 88 ^18ft' and 88 <,X8S' by (iv). Hence 88 <,xs4' and 3 <^xsi'. Therefore by (iv), both 8ft and 3 are recursively enumerable in si. By (i) 8ft is recursive in si. If (vi) then (vii). (vi) By induction on TT; the result is trivial for n = 0, ^ - s e t s are recursively enumerable, similarly ^^-sets are recursively enumerable in si.
9.4 /-definable sets of lattice points
419
Suppose the result for n. Suppose that 88 = ^flSn), then we want to show that 38e0^8w. We have Xe2Cf87l) *-> (Ez) (Un\[6, A, fz, z, 0] = 0) <-> (Ez, w) (Un\[6, A, w, z, 0] = 0
& (4tO w (<* = 0 <-> uas/^>) & {Au)wz(wZz ^ 1)). Now uej*®*><->v$ear$n} <-> (2?w) (Un\[ul, u\, QW, w, 0] = 0 & ( , 4 ^ ( g ™ = 0 *->**&«/<»>) & (Au)ww(gu ^ 1)).
We next show that ue^n) is a ^-statement. Clearly ^ e j / ' is a ^ 1 statement. If ues/(n) is a ^-statement then by (ii) ue^S7l) is a 0>8nstatement. Substituting this in (ii) and reducing to prenex normal form we see that \e2£f(87l) is a ^^-statement. Now suppose that dSe0>s8n. Then 9S = &(Ey) 8{x, y}, where 8{x, y) is a cS^-statement, then 8 is a ^^-statement, hence by induction hypothesis 8 is recursively enumerable in j ^ ( 7 r ) . Hence 8 is recursive in by (iv), whence so is 8, hence by projection 88 is recursive in Hence (vi). (viii) Again by induction on n. For n = 0 the indices are the same. Assume the result for 8n, the result follows from detailed consideration of the construction in (vi). 9.5 Computing degrees of unsolvability 0 is the null set, 0' is the complete recursively enumerable set, 0 " is the complete set which is recursively enumerable in 0 ' , generally 0(£A) j s faQ complete set which is recursively enumerable in 0(A). The degress of these sets are O, O', O",..., O(5fA),..., respectively. Note that by the construction the g.n. of ve 0(A) is a primitive recursive function of v, A. These degrees form a useful measuring rod in the upper semi-lattice of degrees. Note that the degree of any recursively enumerable set is ^ O', and so is the degree of the complement of any recursively enumerable set. 14-2
420
Ch. 9 A-definable sets of lattice points
One frequently wants to compute the degree of a set. By the place of the defining statement of the set in the hierarchy of Prop. 4 one can find an upper bound to the degree of a set. Tofinda lower bound is more troublesome. We give one or two results and techniques. First, note that the sequence of degrees O(A) can be extended into the constructive transfinite. We first define Ow as the degree of the set of natural numbers v for which v\ e O(*#. 10. The set of g.n's of closed A-true statements is of degree CK The sequence of degrees O(A) is formed on a recursive plan. Hence the g.w.'s of yf eO^l* are the values of a primitive recursive function of v. For instance
PROP.
veO' *-> (Ew) (Un^vl w\, w\, v\] = 0) *-> (Ew) H'[w, v] say, veO" ++ (Ew) (Un\[vl w\, Vw, w\, v\] = 0), where V is the characteristic function of the complete set. Thus veO" <-> (Ew) (Ez) (Un\[vl w\, z, w% v$\ = 0 & (Au)ww(z™ = 0 ^> (Ev) (Un^uf, vl vlui] = 0) & (Au)ww(%T < 1)). ^(Eu)(Av)H"[u,v9vl
say,
where H is quantifier free. Generally if
> (Euf) (Au")... j ^ J then i;eO^ ^ (Ew, z) (Un\[vl w\, z,<
v\] = 0 & (Au)m(z™ = 0
<-(Ev,y) (Un\[ulv\ly,v%u\]
= 0 & (An)ww(zT <, 1))
& (Ax)wv(yr = 0 «(En1) (An")... j ^ j H9*[u' ...u,x\ reducing this to prenex normal form we see that
') {An")... j ^ j ^^B^^.u',
...,u<s\
Let g[v, A] be equal to the g.n. of veC^x\ the complete set of degree A then g is a primitive recursive function of v, A. We have > g[v,
9.5 Computing degrees of unsolvability
421
where &" is the set of g.n.'s of closed A-true A-statements. Thus O is reducible to 3~. Again, if v is the g.n. of an A-true closed A-statement ^,let (£l£) ^(j} be obtained from a prenex normal form of (j) after contraction of quantifiers. Consider (Ey) (Gj) (z = z&y = y& ^{j}), this is a ^ A -statement which defines a set 38. Then 38 is Jf if (j> is A-true and ^? is 0 if (j) is A-false. By (viii) Prop 9 given an index of 3$ as a set in ^ A we can find an index of 38 as set recursively enumerable in 0A. Thus ], A> eO, hence 3~ is reducible to 0w. Altogether ^" and O are of the same degree. H I S T O R I C A L REMARKS TO C H A P T E R 9
The discussion of the hierarchy of A-definable sets of lattice-points is due to Kleene (1943) and Mostowski (1946) independently. We use the 0
0
notation of Mostowski. Another notation is: 2 , II which can be further 7T
7T
extended when we allow property variables in the definition of sets of lattice points. We prefer the Mostowski notation in this chapter because the S-II notation leaves out the dimension number K. Thus most of this chapter is due to Kleene (1943) and Mostowski (1946), but Props. 6 and 7 are ultimately due to Post (1944). The algorithm for finding the place of an arithmetic set in the hierarchy of arithmetic sets is due to Kuratowski and Tarski. Addison has pointed out analogies in the Borel, Lusin and Kleene hierarchies (1955a, 19596, 1962a, 19626). EXAMPLES 9
1. Let G and P represent the operations of complementation and projection respectively. Let a sequence of such symbols denote a corresponding sequence of operations in reverse order of application. Thus PPC represents complementation followed by two projections. Show that CPPCP applied to a relation R can be expressed by a prefix AAE applied to R. Find prefixes corresponding to PCPCPC and GGCP. Find sequences of operations corresponding to EAEA and EAEEAA.
422
Ch. 9 A-definable sets of lattice points
2. Show that a bounded quantifier can be moved to the right past an adjacent universal quantifier, and that a bounded existential quantifier can be moved to the right past an adjacent existential quantifier. 3. Obtain a classification in the hierarchy for the following sets of natural numbers: (i) the numbers of the tables of simple sets, (ii) the numbers of tables of hypersimple sets, (iii) the numbers of tables of creative sets, (iv) the numbers of tables of maximal sets, (v) the numbers of tables of unbounded sets. 4. Carry out the details of defining the hierarchy of /-definable sets of natural numbers.
Chapter 10 Induction
10. i Limitations of the system A There are some universal A-statements which in classical mathematics are normally proved by the Principle of Mathematical Induction. Many of the so-called Laws of Algebra are statements of this kind. These laws can be stated in the system A and are A-true but many of them fail to be A-theorems. According to Cor. (i), Prop. 7, Ch. 8 if (AE) ${£} is an A-theorem then so is (j){y) for each numeral v and the A-proofs have the same number of steps, in fact they are obtained by substitution in an A-proof-form. Thus if ^>{v) for v = 0,1,2,... are all A-theorems but the lengths of their shortest A-proofs are unbounded then (Ag) $5{£} fails to be an A-theorem. This occurs in the case of the A-statement (Ax)(x + 0 = 0 + x),
(1)
written in fuller notation this is (Ax) (J^kx'x". Sx") xO = Ji^x'x". Sx") Ox).
(2)
If (2) were an A-theorem then each of 'x". Sx") vO = J{\x'x". Sx") Ov for v = 0,1,2,... would be A-provable in the same number of steps. By axiom 3.1 ^ / n , „ a ,n
J(kxx .Sx)*>0 = v. Hence by one more step we obtain by R 1 ".Sx")0v. (3) Thus if (1) is an A-theorem then (3) can be A-proved in the same number of steps for each numeral v. Now (3) is an A-true A00-statement, hence an A00-true A00-statement and so is an AOo-theorem. In fact an A-proof of 3isan Aoo-proof of (3). The Aoo-proof of (3) will use axioms 1, [ 423 ]
424
Ch. 10 Induction
3.1, 3.2, 4.1, 4.2, and rule R l only. The other axioms and rules are absent because (3) is without the symbols =(= & v and once these are introduced into an Aoo-proof without redundant parts they remain in that Aoo-proof from that place to the end and so occur in the A00-theorem proved. To Aoo-prove (3) we have to find the numeral determined by J^kx'x". Sx") Ov and to do this we have to use axioms 3.1, 3.2 to get rid of the iterator symbol and axioms 4.1, 4.2 to get rid of the abstraction symbol, axiom 1 might be used as a starting point, we also use rule R 1 . If v is zero then we can finish in one step by axiom 3.1. If v is Sv' then we can use axiom 1 or axiom 3.2 only; axiom 1 gets us nowhere, axiom 3.2 gives us v') = (Xx'x" .Sx")v'(J?(kx'x" .Sx")0v'). Write F for (kx'x" .Sx"), then if at any time we have arrived at (F*{F* ... (F*{JF0(Sv")))...)),
(3.1)
where F* is either FJJL for some numeral [i or is S, then we have a choice of two moves only; we could either replace one of the initially placed J^'s which is F/i by S using axiom 4.1 and R 1, or we could use axiom 3.2 and replace the part S{F0{8v")) by F*(JFW) using R 1. We finish up with an expression of the same form as (3.1) or possibly with Sv" replaced by 0. Let the order of (3.1) be the number of X's and
S(Xx'x".Sx'')0(Sv')
of order 2. We apply one of the two possible moves, the order either is decreased by one or is increased by one, such an increase can only happen v' times and a decrease can only follow a previous increase. Thus it is clear that we shall require at least 2. v' + 1 steps to obtain an expression of order zero which is a numeral. Extra steps can be added by applying axioms 4.1 and 4.2 alternately. The last step is an application of axiom 3.1 to get rid of
10.2 Possible ways of extending the system Ao
425
10.2 Possible ways of extending the system Ao We now wish to extend the system A so that more A-true A-statements become theorems in the extended system. We wish to do this in such a way that directness of proof is maintained, for then we can easily see, by arguments outside the system, that the resulting system is consistent in the sense of Con (i), in that ( 0 = 1 ) will fail to be provable. The A-statement (1) says that two Aoo-functions, that is two primitive recursive functions, are equivalent, that is they take the same value for the same argument. We might then consider a new rule which allows us to prove the equivalence of two equivalent primitive recursive functions. Primitive recursive functions are built up from the initial functions by the schemes of substitution and primitive recursion. The rule R 1 enables us to prove the equivalence of two primitive recursive functions which are built up from functions, whose equivalence has been proved, by substitution of functions, whose equivalence has been proved. For instance, if we have proved px = p'x and ax =
a[rf\ = /%] and r[£, £, y] = cr[£, £, y].
(4) (5) (6)
We require a rule which enables us to prove /)[£, y] = //[£, y]. Given such a rule we could then prove the equivalence of two primitive recursive functions which are built up from the same construction sequence but using functions of proved equivalence at corresponding places in the construction. From (5) and (6) and R 1 we obtain
Conversely from (6) and (7) we obtain (5). Thus our new rule should allow us to obtain ,[£,]-p'fc,] (8) from (4) and (7).
426
Ch. 10 Induction
Note that (8) might be A-valid but it might be impossible to find a, /?, T, a such that (4), (5) and (6). For instance the primitive recursive function whose value is zero for the argument v if (2. v + 6) is the sum of two odd primes and whose value is one otherwise might be such that we are unable to show that it is built up using the same construction sequence as the constant function zero, but using instead functions of proved equivalence at corresponding places in the construction. As a first suggestion consider the rule Qo[0,t)] = oft]) wo) (p[a,t)] = Tcr[a,t)])Vft> where fcr is defined by D 121, t) comprises exactly all the free variables in o*[t)]; g, £,t) exactly all the free variables in r[g, £,k)], (o is subsidiary and can be omitted; £ fails to occur free in cr[t)], r[I\, £, tf\, o) and p[Tt, t)]: a is free in p[a,t)] and in ro^a,*)]. Rule R 3' is more powerful when we allow any numerical term a instead of just a variable g. From (4), (7), R 3' and R 1 we can obtain (8) as desired. The system A with the new rule R 3' will be called the system Ar. In the system Ar we can prove the equivalence of two primitive recursive functions which satisfy the same recursion equations. As for the system A we can show that if a disjunction is provable then so is one disjunctant, we can find which and find a proof for it, we can also obtain an irresolvable A r -statement, thus T.N.D. fails in the system Ar. For if T.N.D. held in A r , then G V JV["G"], where G is the irresolvable A r -statement, would be an A^-theorem hence so would be one disjunctand, which is absurd. A particular case of rule R 3' is O»ftl= 0) V ft> (Plffg, fl = 0) V to This can be generalized to
^(o)
^ fi{} v 0)
where g is free in ^{g} and fails to occur free in
10.2 Possible ways of extending the system Ao
427
R3(iv> is properly stronger than R'" because we can R3(iv)-prove (0 + x = x + 0), but we are unable to R'"-prove this. The R 3
_ . 8(0+ x) = S(0 + x)Axl
JK 1
0+#=#+0
Hyp
S(0 + x) = S(x + 0) x + 0 = xAx3.l Sx + 0 = SxAxS.l Sx + 0 = Sx + OAxl W+*) = 8x Sx = Sx + 0 8(0+ x) = Sx + 0 0 + Sx = S(0 + x)Ax3.2 0 + Sx = Sx + 0 ' Thus we have
0 + Sx = Sx + 0
whence by R 3(iv) we obtain 0 + x = x + 0. If we have the premisses of R 3W then we have the premisses of R 3(iv)— if we can prove <j>{SQ v OJ then we can deduce
by induction; we can overcome this difficulty by introducing a new symbol -> of type ooo called the implication symbol and writing a one premiss rule j - as ((->$)&) which we usually abbreviate as (
1
suggests itself. With this modification rule R 3(iv^ becomes
or even
((^{0} v to) & ((^{g} V0))-> (
428
Ch. 10 Induction
with the same conditions on £ and a as before. This is an axiom scheme. The disadvantage of this method is that proofs cease to be direct, in Modus Ponens the whole build up of <j) disappears at one step. We have the possibility of defining (0->^)
for
#["#"] v ^ ,
and leaving the rules in their original form, and we should have a truthvalue for A-statements of the form (^->^ r ), but then in R 3(v) we again lose directness of proof. This definition of implication is called material implication. All these suggested rules have this in common: if we are able to prove the upper formulae then we can prove successively ^{0}V(0, 0(1}V0), 0(2}vw,... moreover the proofs are on a general plan which can be described and we are given a demonstration that the plan is correct. This leads us to consider the scheme
X'
10.2 Possible ways of extending the system Ao
429
which could otherwise arise. Rule R 3(vi) has an unbounded number of premisses, but this is unobjectionable when the appended conditions are added. It is possible to check an unending sequence of statements if the checking is on a general plan for which a checkable demonstration of correctness has been given. We have done this sort of thing many times before, for instance in the system AOo we showed that a + /? = /3 + a for any closed numerical terms a, /?; we sketched a general method whereby any particular case could be dealt with and gave a demonstration that the method was correct. The resulting meta-theorem failed to be an Aoo-theorem because the methods used in obtaining it were unavailable in the system Aoo itself. The rule R 3(vi) has conditions attached to it, this is harmless, we have had conditions attached to rules many times before, chiefly connected with variables, but it was so easy to see whether they were satisfied that we failed to require that a demonstration be given that this was so. But in the case of rule R 3(vi) it might be really difficult to see that such and such a plan was in fact correct, hence we demand that the demonstration of correctness be given. Then we are able to supply the proofs of ^{0} v (o, ^{1} v (o,... as far as we like and know that though we have to stop somewhere yet the ones we have left out would be correct if written down, but without the proof of correctness of the general plan we might doubt whether some of those proofs which we omitted to write down were in fact correct. The proofs which we have encountered anywhere as yet have given rise to proof-trees; these have axioms at the tops of the branches and at each node a bounded number of branches unite by an application of a rule. Each thread of the proof-tree contains a bounded number of nodes. We wish to keep this last feature of a proof-tree when we use rule R 3(vi). We shall require that the number of applications of rule R 3(vi) in any thread be less than a fixed number. The order of a proof using R 3(vi* will be defined as the maximum number of inductions—applications of R 3(vi)—in any thread of the proof. We then restrict ourselves to proofs of bounded order. Without this restriction we should have great difficulty in defining a proof-predicate. An example of a proof of unbounded order is obtained by taking a proof of bounded order and adding redundant parts, e.g. by II a, R 3 (vi) and I&. (We shall add the rule 16.)
430
Ch. 10 Induction
10.3 The system E We will now construct a formal system in which the demonstration of correctness required by rule R 3(vi) is to be carried out. The system E is an equation calculus, it will be used to prove the equivalence of primitive recursive functions. The symbols of the system E are 08 J X= x ()' where the prime is a generating symbol, the types are as in the system Aoo. The axioms of the system E are the Aoo-axioms 1 3.1 3.2 4.1 4.2 and the rules are the A00-rules R 1 R 3' without subsidiary formula. 1. We can find a primitive recursive function whose value is always one but it is impossible to JL-prove that it is equivalent to the constant function one. Thus the system E is incomplete. Let prfE be the proof-predicate for the system E, this is constructed as for other systems and is primitive recursive. Let sE|V,a;] be Godel's substitution function for the system E. Consider prfE[x\ sE[x, x]] = 1 and let its g.n. be K, let the g.n. ofprfE[x', SE\K, K]] = 1 be v, then v — S[K, K] is an Aoo-theorem and hence is an E-theorem. Suppose that prfE[x', v\ = 1 is an E-theorem, then by R 1 so is prfE[x'', SE[K, K]] = 1. There will then be a numeral n such that prfE[n, v] = 0, here n is the g.n. of an E-proof of prfE[x',sE[K,K]] = 1 which is the E-statement with g.n. v. But if prfE[x',v] = l is an E-theorem then so is prfE[n, v\ = 1, because E-theorems are A-valid hence prfE[n, v\ = 1 is A00-true hence is an Aoo-theorem and hence is an E-theorem. But this is absurd because a closed numerical term determines a unique numeral. Hence prfE[xf, v\ = 1 fails to be an E-theorem. We now show that prfE[xr, v\ = 1 is A-valid. Suppose that prfE[rr, v\ = 0 is A00-true for some numeral n, then the E-statement V is an E-theorem. But this E-statement is prfE[x',sE[K,K]] = 1 in which case prfE[x', v\ = 1 is also an E-theorem, and so prfE[n, v] = 1 is Aootrue. But this is absurd because a closed numerical term determines a unique numeral. Hence prfE[n, v] = 1 is A00-true hence is an E-theorem for each numeral n. Thus Xx.prfE\x, v] is the required function. PROP.
10.3 The system E
431
Prop. 1 shows that we must be very careful in saying that fx = 0 is an E-theorem when we have somehow shown that/zr = 0 for each numeral n. We should require something stronger than rule R 3' to prove this in all cases. To E-prove that fx = 0 we must show that / satisfies the same recursion equations as the constant function zero, and this we are sometimes unable to do. COR. (i) E-validity fails to be a primitive recursive property. In other words E-validity fails to be E-definable, because the only E-definable concepts are E-defined by an equation of the form fx = 0 where/is a primitive recursive function. Suppose that valE is E-definable so that valE\y\ = 0 if and only if V is an E-valid E-statement. Consider valE[sE[x, x]] = 1, let its g.n. be K and let the g.n. of valE[sE[K, K]] = 1 be v, then SE[K, K] = v is an AoO-theorem, hence is an E-theorem. Suppose that val^v] = 0 then V is E-valid hence VCLIE[SE\K, K]] = 1 is E-valid and so is valE[v] = 1 but this is absurd because a closed numerical term determines a unique numeral. Suppose that valE\y\ = 1 then V is E-invalid or fails to be an E-statement but V is the E-statement valE[sE[K, /c]] = 1, hence this is invalid and so VOIE[SE[K, K]] = 0 because valE is an Aoo-function whose value is either zero or one and fails to be both. But this again is absurd as before. Thus valE with the required properties fails to be E-definable. COR. (ii). E is essentially undecidable. The demonstration is exactly the same as for Ao. Prop. 5, Ch. 7. By Cor. (ii), Prop. 6, Ch. 7 E-validity is undecidable. COR. (iii). TL-valid statements fail to form a recursively enumerable set. If E-valid statements formed a recursively enumerable set then we could find a formal system in which they were the theorems by Prop. 15, Ch. 8. But then by the argument of Prop. 1 above we should be able to find an E-valid statement outside this enumeration. When we use R 3' ,o x pj3 = faj3
or the same with parameter we sometimes say 'by induction on x' with substitution, if /? is x we omit 'with substitution'. Similarly when we use R 3' to prove fx = gx.
432
Ch. 10 Induction
PROP.
2. The rule:
ax = 0
A
A
/o \
n
oo held constant
(9)
era = 0
is a derived E-rule. We say that # is AefoZ constant in a deduction if the deduction is without any induction on x. Write out the deduction of cr(Sx) = 0 from the hypothesis ax = 0 in tree-form and multiply both sides of each equation in the deduction by B^ax], The hypothesis becomes ax.B^ax] = 0, but this is an easily proved E-theorem. (xB±[x] = 0 by induction on x, whence ax.B±[ax] = 0 because we are allowed a substitution in the lower formula in rule R 3'.) The axioms become easily proved E-theorems ((x.a = x.fi) is easily E-proved from a = /?) and the rules become derived E-rules. For rule R1
We have by induction on x
BM'*l*n = - B X M . T ^ M . Z ' ] . Hence
b.p{a} = b.r{cc} b.p{b.a} = b.T{b.a}
(11)
b.a = b.p
Thus (10) is a derived E-rule. An application of R 3' P[O, Vi = gftfl
becomes
rL
?/J
P[8£,V\ = r[g, p\£, yl y]
L/J
~^L~*^ ~-'L*?rL*?'/J?7J. 6./o[a,7/] = 6.rcr[a,9/]
—
(12)
Using (11) this becomes b.p[0,7j] = b.a[7/]
b.p[S^,rj] = b.T[g,b.p[g,7/],r/] _____ 6.)O[a,7/] = b. r(6. cr) [a, 7]]
i A _;
10.3 The system E
433
since (see D 121) b.T(r[oc,7/] = b.r(b. a) [a, TJ] and since (13) is an application of R 3' then (12) is a derived E-rule. Thus we obtain an E-proof of B^croi]. a(Sa) = 0 and hence an E-proof of o-(Soc) = A1[ora].om(Sa), since A1[x] + B1[x] = 1 is easily E-proved by induction on x. Now consider the functions/which satisfy the scheme of primitive recursion:
/0 = 0, f(Sx) = A1[fx].*(Sx). One such function is Ox and another is cr[x]. Hence by R 3' cr[x] = 0, as desired, thus (9) P R O P . 3.
x + (x' — x) = x' + (x ~
x')
is an IL-tlieorem. This may be expressed as Max [x, x'] = Max [xf, x]. The result could be very eaily proved if we had the rule that two functions which satisfied the same scheme of double recursion are equivalent, but we require this E-theorem to show that the equivalence of two functions which satisfy the same scheme of double recursion is a derived E-rule. We have x' — x = 2 c" JL. x] by induction on x' and hence
x=(x^-l) + A1[ f
v
—
V
V1
by induction on x,
A (V A \
thus
= s similarly
^x"-x'],
the result now follows. The familiar properties of summations which we have used are easily E-proved.
434 COR.
Ch. 10 Induction
(i).
px = ax === x held constant pO = crQ p(Sx) = a(Sx) pa = era
is a derived IL-rule.
We have
(x~x') + (x'~x) = 0 X =
X = X
and
..
X
' =
(b)
are E-provable. The first follows from (x + x') —x' = x (by induction on x), thus if x + xr — 0 then x — 0 and #' = 0, this gives
whence by Prop. 3 x = x' as desired. The second follows from x = xf {x^-x') = 0 {x'^-x) = 0 From the hypotheses we obtain px-L-o'x) + (crx—px) = 0
()p()) whence by Prop. 2
=0
(pa-^- era) + (era —pa) = 0.
By (a) the result follows. 4. If two functions of natural numbers satisfy the same scheme (2), e£c, or course of values recursion or if two sets of functions of natural numbers satisfy the same scheme of simultaneous recursion or if two functions of natural numbers satisfy the same scheme of recursion with substitution in parameter or of double recursion or of simple nested recursion then their equivalence can be IL-proved. If p is a function satisfying the scheme of course of values recursion or of one of the schemes (2), etc., or if a set of functions satisfy the same scheme of simultaneous recursion then we showed in Ch. 5 that there was a primitive recursive function/such that/p satisfied a scheme of primiPROP.
10.3 The system E
435
tive recursion and hence fp can be E-proved to be unique; further we showed that there was another primitive function g such that gfpx = px, whence p is unique. For the schemes of recursion with substitution in parameter, double recursion or of simple nested recursion we need only consider the scheme of simple nested recursion because the scheme of recursion with substitution in parameter is a particular case of the scheme of simple nested recursion and the scheme of double recursion is settled when we have settled the scheme of substitution in parameter (see Ch. 5). As regards the scheme of simple nested recursion we need only show that two functions which satisfy the scheme p[0,
K]
=
CT[K],
p[8v,
= r[p[v9p[v9 *]]]
K]
(14)
are equivalent (see the account of simple nested recursion given in Ch. 5) because if pf satisfies a scheme of simple nested recursion then we can find a primitive recursive function/ such that/// satisfies (14) and we can find another primitive recursive function g such that gfp'x = p'x. Thus if the solution to (14) is unique then so is a function which satisfies a scheme of simple nested recursion. Let p be a function which satisfies (14) and let x be the primitive recursive function such that
[TMSv)[^[P) Kjj
if gv j s e ven.
#|>, K] = fty&l • Tit£> V\) KX>
Then
where
Tl[g,V]
X[8X>K]
so that
= e(8g)o-*[V] + (l^e(8£) = r
i\.x> Xlx>K]]
a n (
i Xl®9K]
= K
>
here ev = 0 if v is even, ev = 1 if v is odd, further x *s unique. LEMMA.
then
//
f[0,V] = a[rj], f[Sx,y] = T[x,f[x,V],V], f[x + x', 7j] = J(\&. r[x + g, f, V])f[x, y] x'.
Let g[x, x', 7/] stand for
436
Ch. 10 Induction
then and
g[x,Sx',7j] = T[x +
also and
x',g[xix'9ri]9ri],
f[% + 0,7}] =f[x,7j] f[x + 8x', TJ] = T[X + x',f[x + x'9 ri\,rf\.
Thus by R 3 ' we get g[x,x',y] =f[x + x',7}]> and the lemma follows. Applying the lemma to x we get *9 K] = also Now
g,£] = r 1 K,£]
for
hence . ^] ( 2 * Also
Thus We now show
^ [ 2 ^ , AC] = rtv[2*, ^[2*, /c]]]. p[8x, K] = x[2Sx, K] 2SSx
/o[SSa?,ic] = xl We have
X held constant.
K
> li-
yo[/Sf/S^, /c] = r\p[8x9 p[Sx, K]]] = r[x[2Sx, x[2Sx, K]]] = x[%88X9K]
We also have
as w e
l , AC] = ^ [ 2 , K] =
by hypothesis,
have just shown. T[(T2K].
Hence by R 3 ' we obtain p[Sx,K] = #[2®*,*], thus p is unique because X is unique. Thus two functions which satisfy the same scheme of simple
10.3 The system E
437
nested recursion can be E-proved to be equivalent. This completes the demonstration of the proposition. COR. (i). The recursive equations of a primitive recursive function are E-provable. For instance if p{0} = a, p{Sv} = y{v, p{v}} thenp{Sx} = y{x, p{x}} is an E-theorem. p{x} and y{x — 1, y{x — 1, p{x' — 1}}} satisfy the same scheme of primitive recursion. ?T = 0
P R O P. 5. IfrO = O is an Aoo-theorem and if - is an AoO-deductionform, then rx = 0 is an E-theorem. If 70 = 0 is an Aoo-theorem then it is an E-theorem. Note that if T{V] = cr{v} is an Aoo-axiom for all v then T{X} = a{x} is an E-axiom, and that if
frM}
foM}
M M
is a case of R 1 in Aoo then the same with x instead of v is an E-rule.
rv = o Thus if
- is an A00-deduction-form then the same with x r(M ) = 0 instead of F is an E-deduction. By Prop. 2 the result follows. COR. (i). / / rO = aO is an Aoo-theorem and if
TF ari
OT
TciW\ ^s
an
T\bl ) =
deduction-form then TX = ax is an E-theorem. The demonstration is similar to that of Prop. 5, but uses Cor. (i), Prop. 3, instead of Prop. 2. (ii). J/r[0,0] = 0, r[/ST,0] = 0 and T[O,AST'] = 0 are Aoo-theoremT\T ? F'l = 0 forms and if is an Aoo-deduction-form then T[X, X'] = 0 is COR.
an E-theorem. Similarly using Prop. 4.
438
Ch. 10 Induction
10.4 The system A7 The system A7 is a primary extension of the system Ao. It has additional symbols A of types O(OL), O(OU), O(OLU), ... and we use abbreviation D 224. The symbols A are called the universal quantifiers, they are introduced into Aj-proofs by a new rule R 3 called Induction. The system A7 fails to be a primary extension of the system A because free variables are forbidden in the system A7 and the rule for introducing the universal quantifier is different. The g.n. of A is to be 11, the type of A is then determined uniquely in a well-formed formula. The g.n. of x^d) is to be 12 + 0. We wish to formalize a rule of induction to the effect v (o for all n, where n is a Tr-tuplet, (A%) 0{ct{j}} {}} V (o' ' where h AAi is foft type 0(01... ( ) t), 1 n times
where the proofs of (j){v) v 0) are on a primitive recursive plan which can be E-proved to be correct. When this is so we say '
10.4 The system Aj
439
it down in detail, still less is it possible to write down in detail the A7proofs of the premisses, but it is to be possible to describe them, say by a proof-form, from which by substitution we can obtain the proofs in detail as far as we wish. An Az-proof is an unbounded tree-like figure and we shall be given sufficient instructions so that we can write down any bounded part of it, but we shall have to stop somewhere and are unable to write down the whole thing. Anyway it will be such that if <£{%} is an Aj-theorem in which each variable of the set r. is general in ^>{jc} and only these, then we can write down the full Ao-proof of 0*{n}, for any set of numerals n, where 0*{I\} is
from this by replacing TL by numerals in all possible ways we can obtain the Ao-proofs of the premisses
and
^{0,8v'} V o),
possible like (a) above, and the Ao-deduction-form
from which we can obtain the Ao-proofs of the premisses (j>{v,v'}vo) for
v,vf = 0,1,2,....
The position is somewhat similar to the use of axiom-schemes; rules normally are schemes. It is impossible to write down all the Ao-axioms, but we can describe them.
440
Ch. 10 Induction
Along with (a), (b), and any other method of description that we might use, we require a demonstration that the method is correct. We have decided that this demonstration take place in the system E. The proofs and deductions above are in the system Ao because they are without inductions. Let m be equal to the g.n. of an Ao-proof of 0{n} V o) and let be equal to the g.n. of §J{n}vw (if n is v\...9v(n) then is cr{Numv',...,Numv(7T)}) then we require an E-proof of prfo[T$9 crjjJhimj;]] = 0 to be given, this can be done by giving its g.n. From this g.n. we can recover o*[SKumj] and hence
10.5 Definition of an Aj-proof An A7-proof consists of a bounded number of levels, say 8A levels. Level one consists of a bounded number of g.n.'s of E-proofs of equations of the form
tf[[5R£|]
= 0,
where m is equal to the g.n. of an A0-proof of a closed A0-statement whose g.n. is equal to a"[-J£umn]. Let these A0-statements be
where n',..., n(A) are parameters. They indicate the variables which will be generalized in later levels. Then level one gives us the Ao-proofs of these Ao-statements for each set of numerals n',..., n(A) for all n E-correct. But we shall demand more. We want (15) to be E-correct for all n,n',..., y(A). (16)
be equal to the g.n.'s of the Ao-proofs of the respective Ao-statements (15). Let
i
[
, Wum v',..., Slum i*«], J
(17)
10.5 Definition of an A/-proof
441
be equal to the gr.w.'s of the respective statements (15). Then level one will consist of the gr.n.'s of the E-proofs of Sttum j \ . . . , 9htm x(A)]] = 0,) (18)
:
= o.J
^
A particular case is when £,..., £(A) are absent and we are left with a single Ao-proof. The production of level one consists of a bounded selection of A-statements of the forms n*}, n','..., n™} v w{/*>{n',..., n
}, n ' , . . . , n} v
}
A statement is in level $A if it is at or above the production of level SA but below the production of level A. An A r proof of level one is either an Ao-proof or ends in a single application of R 3. The idea is that if we have proved 0{n} for all 7r-tuplets of numerals n then we have proved 0{a{n*}} for all 7r-tuplets of functions a and all 7r*-tuplets of numerals n*, and that we have proved (Ar.) ^{a{£, n*}} for all similar tuplets of functions a and of numerals n*. Thus level one produces sufficient evidence that the A0-statements (19) are Ao-theorems for all appropriate sets of numerals n, n',..., n(A). The set n* of numerals may be any, it may overlap the sets n',..., n(A), or it may contain fresh numerals. The production may contain several different 7r-tuplets of numerical terms a',..., several different numerical terms a^°\ We say that the 7r-tuplet of numerals n is treated in level one. Having defined level d we proceed to define level SO. D 233 dedo[A, /i, v] = 0 if and only if /i is the g.n. of an Ao-deduction of the closed A-statement whose g.n. is v from hypotheses whose g.n. is A. Level S6 consists of the g.n.'s of E-proofs of equations of the form = 0,
442
Here
Ch. 10 Induction ft[n^,...3n
(A)
]
(21)
is equal to the g.n. of the sequence of those A-statements similar to (19) which form the production of level 6. These A-statements form the hypotheses of the deductions in level Sd, we may suppose that each deduction in level Sd is from the same set of hypotheses, because a deduction from a set of hypotheses O is also a deduction from hypotheses O and T , those in T being dummies. T^[n^...,#)]
(22)
for 1 ^ K ^ /iS0, is equal to the g.n. of the Ao-deductions of the A-statements (24) from the hypotheses whose g.n. is Po[n^\ ..., n(A)]. (23) for 1 ^ K ^ fiSdi is equal to the g.n. of the Kth statement (24)
[
(24)
J The production of level Sd consists of a bounded number of A-statements of the forms: ..., n w} v (25)
v G>
where n* is any set of numerals. We can arrange the notation so that n(d) is treated at the #th level. In (25) as in (19) we allow several different a',..., several different a{m\ The levels continue until all the variable numerals have been generalized. If new variable numerals are introduced by substitution at each level then the process fails to terminate and so fails to give an Aj-proof.
10.5 Definition of an A/-proof
443
The last level consists of a single Ao-deduction from the production of the penultimate level as hypotheses (case when £ is absent in (20) and (20) consists of only one line), or is a single application of R 3. In either case the closed A-statement thus produced is the Aj-theorem proved. A r proofs are complicated, but they can be checked. Suppose that an Aj-proof has SA levels, then we are given $A sets of numerals, the last set containing only one member. The members of the first set are gr.w.'s of E-proofs of equations of the form (18), the members of the other sets are gr.w.'s of E-proofs of equations of the form (20). From these gr.w.'s we can recover the E-proofs and from these in turn we can recover the A-statements Approved or A0-deduced, lastly from the hypotheses of level Sd we can find the production of level d. Thus we can see the cases of generalization and substitution that have been used. Moreover if the theorem proved is §5{£}> where each variable of the set £ is generalised in ^{j}, and only these, then we can write out in full the Ao-proof of 0*{n} for any set of numerals n, where ^*{x} is ${%} minus its general quantifiers. Note that a production of level 6 can be a production of any higher level, because ^ is a deduction from x a s hypothesis. In this case we say that x is carried forward. An A7-proof having only one level is either an Ao-proof, case when j is absent in prfo\j^
indicating the places which are going to be restricted as well as those which are to be generalized. In the resolved form of a theorem the restricted variables are replaced by certain functions of the superior general variables. The only functions available to us are primitive recursive functions, so that if we demand proofs of (15) for all v, v\ ...,y(A) E-correct then we restrict ourselves to cases when in the resolved form the functions required were all primitive recursive. Other types of functions required in the resolution can be replaced by their values which are expressed by numerals when the superior general variables are replaced by numerals, and so available to us; but we are without means of dealing with these cases in proving (15) for all v, v',..., v(X) E-correct. Furthermore the proof-predicate for Az becomes undefinable if we only demand (15) for all n E-correct.
We could depict an A7-proof as follows:
P.L.I
R3
n,v) v <*&, v) carried forward
(AE) '{£, ft, v) V ojjfi, v} V o>l{v}
P.L. 2
# { g , V, v) V wifo, v}) V
{£, V, v} v dfa, v}) v u)"{{v} v < P.L. 3 (AQ ((Aii) ((A£) &{£, y, Q v wl%. 0) V <{£}) V <
carried
carried
forward
forward
{£, r,} V ^{r,}) v
This depicts an A7-proof with 4 levels the last one being an Ao-deduction of \]r from the production of level 3 as hypotheses. Similarly with batches of quantifiers and with substitutions. <j)\ can only be a variant of $[ by R 1, etc. in other cases of <j>{£K The second displayed line consists of the hypotheses in the Ao-deductions of the statements in the third displayed line, etc.
10.5 Definition of an A/-proof
445
In this scheme level 1 gives the proof-schemes for ^1{AJ [i, v} v b>i{fi, v} for all A, /i, v, E-correct f° r a ^ ^9fa E-correct ^2{^> f4} V ^{f1} 03{A} v o)3 for all A, E-correct this is indicated by the column of dots above each. Written out in full it would be as in the example to follow. Level 1 ends with an application of R 3 giving the production (P.L. 1) of level 1. Similarly for the other levels. Of course we could apply II d to P.L. 1 obtaining an existential quantifier between (AT/) and (A£), etc. for 116, c. io.6 Theorem induction We have frequently in previous chapters used formula induction and theorem induction. We can still do so in an A7-proof. Formula induction is exactly as before because the formulae are as before except for extra symbols, namely the universal quantifiers. But Az-proofs are of a different character in that we now allow rules with an unbounded number of premisses. Theorem induction is as follows: If (i) Property P holds for axioms, (ii) if property P holds for the upper formulae of a rule then it holds for the lower formula of that rule, then (iii) property P holds for theorems. This requires amplification when property P is of the form if <j) is an A 7 -theorem then so is R[$4], where R|$] is an A-statement obtained from ^ by a recursive process. The amplification is required because in the case of R 3 we have the requirement 'for all n, E-correct', so that we should have to show that R[^5] for all n E-correct, then we can apply R 3 and the result will hold for the lower formula provided that (Ag) R[0]
is the same as B,[(Ag) 0],
which is easily checked, the process of forming R[^] being recursive. We deal with this situation as follows: (we confine ourselves to the introduction of a single universal quantifier). The Az-proof of ^ consists of a bounded number of E-theorems of the types prfo[rx, a[Numx]] = 0, dedo[px,rx, cr[Nwnx]] = 0.
446
Ch. 10 Induction
If property P is of the kind under consideration then each statement i/r in the Az-proof-tree is replaced by R[^] and possibly new ones are intercalcated. An application of R 3, say W \
f° r a ^ v> E-correct,
R \x{v} V o)],
as far as the upper formulae are concerned, we require to show that R[#M v ^] ^ or a ^ v> E-correct. This amounts to obtaining E-proofs for prfo[T*x9
dedo[p*x, T*X, cr*[Num x]] = 0,
where the asterisk denotes what these functions have become after application of the operation R. Let R'TT be the ordered set whose components are the g.w.'s of the result of operating R on the formulae whose g.n.'s are the components o f 7T.
D 234 (v)K for the ordered /c-tuplet consisting of the first K components of v, or v if K ^ wv. By hypothesis we haveprfo[TX, cr[Num x]] = 0 in an E-theorem, hence we prfo[(Tx}K, Pt[rx, K]] = 0 is an E-theorem. This follows at once from the definition of prf0, namely 2
(Axo[Pt[n,Z]]
n and the E-theorem which is easily proved.
S Px> ^
S
10.6 Theorem induction
447
We now show
whence by Prop. 2 we obtain prfo[R'[(™>A Pt[n'[(Tx)x.], s7R'[
E-prove this to be zero exactly as fov prf0 and our hypothesis that P V is an Ajr-theorem if V is in the production of level A, for then Hyp[Pt[x,x'-\] = Hyp[-R'[(Pt[x,x'])]l which, by our induction hypothesis is an Aj-theorem. In rule R 3 we have allowed the simultaneous introduction of a batch of universal quantifiers all at once. This is done to make Aj easier to use, see the worked example to follow. It is probable that the system A7 with rule R 3 restricted to the introduction of a single quantifier at a time is equivalent to the system proposed, but this investigation is omitted. The intention is that once a batch of quantifiers has been introduced with scope O then all other batches of quantifiers that are introduced by R 3 have a scope Y which properly contains O and its batch of quantifiers or the intersection of O and T is void. Thus between the introduction of a batch of quantifiers to O and the introduction of another batch of
448
Ch. 10 Induction
quantifiers having O in its scope there will be some building schemes to build up O and its quantifiers to T which properly contains O. It would be possible to introduce other batches of quantifiers. For instance from 2[y/2] ^ v < 2[y/2] + 1 for all v, E-correct, we might allow
)(2x^xr
^ 2x + 1).
We omit further discussion on these lines. Our method of introducing quantifiers by batches is without loss of generality. Because if we introduce a batch of quantifiers in two subbatches, possibly with other building schemes between (these must act so that our original batch is in the subsidiary formula, except for R 1), then we could have introduced them all at once either at the first place or at the second place. The notation we have used in (15)-(25) inclusive is to be interpreted as follows: in level 1 we generalize the 7r-tuplet n. In level 8d we generalize the 7r(^-tuplet n{6\ Thus we require (15)-(25) to be theorems for all n, ...,n(A). But in level 1 the tuplets n', ...,n(A) may be absent or only partly present, but the tuplet n is present, normally in full. If one or more of the places in the tuplet n is always void then, when we generalize, we get a vacuous generalization. If level SA is a single Ao-deduction from the production of level A as hypothesis, then the tuplet n(A) is always absent. If fresh numerals are introduced by substitution, then it is intended that these be ultimately generalized (in one notation others are just omitted). If fresh numerals n* are introduced, then those of them that are already in one of the tuplets n',..., n(A) will get generalized when that tuplet gets generalized. Thus the notation must be such that all the introduced numerals are in one of the tuplets displayed in an Aj-proof of 8A levels. 10.7 The Arproof-predicate We wish to define an A0-statement x{K>v} such that x{K, v) is A0-true if and only if K is the g.n. of an A7-proof of the closed A-statement whose g.n. is v. Suppose that the A r proof of V has 8A levels, then we shall require that K has 88A components
10.7 The A/-proof-predicate K® = <*<*• «,..., K^ S ^ ,
where
449
1 < 6> < £A.
Thus /c tells us the number of levels in the A r proof, namely there are TDK— 1 levels. i6d) is to give us the deductions in level 6. Thus /c(1> °f) is to be the g.n. of the E-proof of the 0'th line of (18), and K ^ ^ for 1 ^ 6 < SA, is to be the g.n. of the E-proof of the 0'th line of (20), K^SX) is to have only one component, namely the g.n. of an E-proof of a line of (20) with A instead of 0, or the g.n. of an Ao-deduction from the production of level A as hypotheses, and the last component of K is to be v. We can easily tell which is the case with level SA, because we can decide of a sequence of formulae whether they form an E-proof or an A0-deduction or neither. D235
(*S) for (K?)
for
pt[K,7T,d], Pt[K90],
(l.C. K) for Pt[K,VJK]. We frequently omit the outer parentheses. D 236
@AC = 0
if (#c|°)£ is the g.n. of an E-proof of an equation (18), for 1 ^ 0 ' ^ TDK™, and (Ksd)f> is the g.n. of an E-proof of an equation (20), for 1 ^ d ^ TDTC—2 and 1 ^ 0 ' ^ tn(/c^), and K £ ^ I has only one component and K^K^1 is the g.n. of an E-proof of an equation (20) or is g.n. of an Ao-deduction and (I.e. K) = v, 1 otherwise.
Clearly 0 is a recursive function because each of the clauses can effectively be tested. In an Az-proof the production of level 6 forms the hypotheses of level 86, and the hypotheses in the various deductions in level SO are all the same. We have to incorporate this into #{/c, v). /i 0
if 8 = g.n. of dedo[n, \9/i\, otherwise.
n if S = g.n. of dedo[n, A, /i], \ 0 otherwise. Clearly S and A are recursive functions. Note that dedo[0, A,/i] is the same as prfo[A, ju,]. 15
SML
450
Ch. 10 Induction
The hypotheses of the various deductions in level 6, 1 < 6 ^ WK— 1 are the same is expressed by ) = A(Z.c. K ^ ) ) ,
(26)
for 1 < 6 < VJK. We say that a line of (19) and a line of (15) are related if both are formed using the same (j>0 and a)0, similarly we say that a line of (25) and a line of (24) are related, if they are both formed using the same $S0 and o)se. A line of (24) may be related to several lines of (25), but a line of (25) can be related to only one line of (24). We are naturally supposing that the lines of (24) are without duplicates and similarly for the other cases. 0 if 77 is the g.n. of a line of (19) or (25) and n' is the g.n. of a related line of (15) or (24) respectively, 1 otherwise. Clearly H is a recursive function, because given two numerals we can effectively decide whether they are related in the manner described above, or otherwise. We now require
1
)fiE(lc.K^v))] = 0),
(27)
for 1 < 6 < &K ~ 1, where 8 = 6JA(1.C. K^0*1*) and 8' =
(27) says that each hypothesis in a deduction in level S6 is related to one of the statements deduced in level 6. This allows some statements deduced in level 86 to be redundant, but this is immaterial. We still require a clause to the effect that V is related to the statement proved in level SA or is the statement deduced in level SA in case level SA is an Ao-deduction. This is r / IV00 1\1 / FA-00 1\ H\V,(I.C. fel ) vv=kc. fei ).
(28)
Division by 2 is required because the components of K, except the last are considered as ordered sets and the penultimate set is a singleton. The Aj-proof-predicate is 0/c = O& 2 1<6<WK
(26) &
2
(27)&(28),
0
this is complicated but nevertheless it is recursive.
(29)
10.7 The Aj-proof-predicate
451
6. A 7 is consistent with respect to its truth definition.
PROP.
We have to show that A r theorems are A-true. It is clear that the A r axioms are A-true and that if the upper formulae of an A-rule are A-true then so is the lower formula. Thus A7-theorems are A-true. io.8
An example of an Arproof
EXAMPLE.
= x' v x 4= x') is an Artheorem.
(AX, X') (X
We give a one level proof without using 16. This example shows the advantage we gain by introducing batches of universal quantifiers all at once. First we have to give a description of A00-proofs of v = v' v v 4= v' for all numerals v, v' on a general plan and then we have to show that this plan is E-correct. That is to say we have to give the E-proof of prfoo[T[x, x'],
a
r~
>
T
0=0 Ax 1 OVO = O = 0v04=0 8v + 0
T
a b c
Ax 2.1
d[Numv] e[Numv]
0 {Q,Sv'} 11a
0*Sv' Ax 2.2 o = 8vr V 0 =}= Sv'
{Sv, Sv'}
v
= v'vv*v'
x a
m n
Sv = Sv Ax 1 T
la p
Sv 4= Sv' V Sv = Sv
Hyp. h • •
v^v'vv _ la
f[Numv'] g[Num v']
= vt
i
Sv 4= Sv' V v = v'
j
Sv = Sv w Sv + Sv' v = v' V Sv 4= Sv' Sv = Sv'v Sv ± Sv'
k J
15-2
452
Ch. 10 Induction
From these schemes T[V, V'] can be found, this is the g.n. of an Aoo-proofof , , r v = v v v =(= v . Let the g.w.'s of the various statements in the above schemes be as shown at the sides, h-p inclusive should be in full h[Numv, Numv'], etc. We haTC
r[0,0]=, T[SV, 0] = (d[Num v\9 e[Num v]) = r[Num v], say, r[0, Svf] = (i[Num v']9 g[Num v']) = s[Num v'],
say,
T[SV, Sv'] = T[V, /] n = T[V, v'ft[Num v, Num v']9
say.
This defines a primitive recursive function r\y, v']. Let cr[Num v, Num vf] be equal to the g.n. of v = v' V v =f= vf. Then we have = 0, prfoo[T[Sv}0],(r[Num8v,Num0]] = 0, prfoo[T[098v'],cr[Num0,Num8v']] = 0, and
pvfoo[Tiv9 v']> cr[Num v, Num v']] = 0 prfoo[T[SvSv'lcr[NumSv,NumSv']]
= 0.
We require to show that prfoo[r[Q, 0], a[Num 0, Num 0]] = 0,
(i)
prfoo[r[Sx, 0], 0-[JV^m fe, Num 0]] = 0,
(ii)
prfoo[T[0, Sx], cr[Num 0, NumSx]] = 0,
(iii)
are E-theorems, and that: prfoo[T[x9 xf], a[Numx, Numx']]
= 0
r[Sx9Sx']9 or[NumSx,NumSxf]]
(iv) = 0,
is an E-deduction. Then by Prop. 4 we shall have prfoo[T[x,x'], cr[Num x, Num x']] = 0
as desired.
10.8 An example of an A/-proof
453
(i) is an A00-true equation, hence is an E-theorem. To deal with the others, we note that prfoo[K, V] = A2[Pt[K, TDK], V]+
£
(Ax00 Pt[K, £]
that is to say; the last component of K is v, and each component of K is either an axiom or arises from an earlier component by a one-premiss rule or arises from two earlier components by a two-premiss rule. Using the easily proved E-theorems: 0 + 0 = 0 and
2 PX' = °>
0xa; = 0 and if px = 0 then
it suffices to show that
A2[Pt[K, VJK], V] = 0
and that one of the factors in the product is zero for each 1 ^ £ ^ VJK. Ad (ii). We have to E-prove prfoo[r[Numx],disj[d'[Numx],d[Numx]]]
= 0,
where d'[Num v] is equal to the g.n. of Sv = 0, now, r [Num x] = ( d [Num x], disj [d' [Num x], d [Num x]J). Thus A2[Pt[r[Numx], wv\Numx]], disj[d'[Numx], d[Numx]]] = A2[disj[d'[Num x], d\Num x]], disj [d'\Num x], d[Num x]]] = 0, in virtue of the easily proved E-theorem A2[x,x] = 0 and substitution. Also we have Ax2-1\Pt\Y\Numx~\, 1]] = 0 because ^a; 2 ' 1 ^] is v = <8, 8, 3, 8, 1>V<9, 9, 0, 9> &tm,K or
Ax^v = n (A2[v, <8, 8, 3, 8, 1>T<9, 9, 0,
where trriK for ' K is of type t'. In our case we have Pt[r[Numx], 1] = d[Numx] 2A
and Ax [d[Numx~\\ = 0 is an E-theorem, because d[Numx] = <8, 8, 3, 8, l> n iV r ^m^ n <9,9,0,9)&tmNumx. This follows since Numx =
454
Ch. 10 Induction
and
x
n
(AJLK,
<8,8,7,8,6> n f<8,6>TV <9,9,9»'n<9»"n<9>]
We have
£m Num 0 = tmO = 0,
, and
tmNumx = 0 tmNumSx = 0 iVwm $# = <8, l)nNum xn(9), hence if ^m iV^m a; = 0 then tm Num Sx = 0, since the second clause in the definition of tm is satisfied. Thus from Prop. 4 we have tmNumx = 0. Thus Ax*-1\(i\Numx~\~\ = 0. In our case there are only two components in T[SX, 0], viz. d[Numx] and disj[d'[Numx], d[Numx]] and we have RlUa[d'[Num x],disj[d'[Numx],d[Numx]]] = 0. This follows since: i?pi«[/C,77]= n
A2[7T,disj[£,K]].
In our case this is n
l^£^disj[df[Num &[Numx]\
A2[disj[d'[Num z], d[Num z]]9 disj[£, d[Num x]]], x],
Now d'[Numx] < disj[d'[Numx], d[Numx]] as follows from D 139, also we have the E-theorem
n PX' = n p%f x p%" x n p%' f ° r %>" < xX*<X"<X
Thus
RllIa[d'[Numx], disj[d'[Numx], d[Numx]]] = 0,
and so RV-\Pt\T[Numx], 1], Pt[r[Numx], 2]] = 0. Hence (ii). (iii) Similarly. (iv) This follows from Rlla[h', i'] + Rlm[i', j'] + RlIa[]', k'] + ^ ^ m ' + RllIa[m', n'] + A2[Pt[T[8z, Sx], TUT[SX, Sx]], V] = 0. (v)
10.8 An example of an Aj-proof
455
So that (iv). Here i', ...,1' are the same as i, ...,1 but with x instead of v and x' instead of vr. It remains to show that each of the summands in (v) is zero. i^«[h^'] = ( ^ 2 [ i ^ ^ for some a, /?. The remainder of the products refer to cases when subsidiary formulae are present. This is easily seen to be an E-theorem in virtue of the E-theorem ^42[#,a;] = 0 and substitution. The remainder follow similarly, Ax1 m follows as did Ax2'1\&\Numx~\~\, it involves the E-theorem tmNumx = 0. This completes the E-proof of prfoo[T[x,x'], cr[Numx, Numx']] = 0. Thus (Ax, x') (x = x' v x # x') is an A7-theorem. We use the terminology *
PROP.
are Ao-rule-forms then x{^]for a^ *t E-correct. The demonstration is similar to the cases treated in the example. There we dealt with one axiom in detail and one rule in detail. The others are dealt with similarly. COR. (i). / / 0{©t} is an Ao-theorem-form then 0{n} for all n E-correct. We put down the A0-proof-tree-form for 0{© J. By Prop. 6 the axioms hold for all n E-correct and if the upper formulae of an A0-rule hold for all n E-correct then so does the lower formula, thus by a simple induction the same holds for the theorem at the base.
456
Ch. 10 Induction
COR. (ii). If we have an Ao-deduction-form of
prfo[rTL,or[NumTL]] = 0 = 0*
COR. (iv). / / ^{n} and ft{n} are deduced from &>{n} and if the deductions hold for all n, E-correct, and if^j-^or
^ *, ^
* are A0-rules, then #{n}
is a deduction from o){n}for all n, E-correct. The demonstration is similar to that of Prop. 7. This corollary shows the advantage we get from dividing an A r proof up into a sequence of A0-deductions. For instance the deductions in level 86 are all Ao-deductions from the productions of level 6 as hypotheses. The fact that these hypotheses are themselves Aj-theorems appears only in the requirement that they are related to the results of Ao-deductions in level 6 in the manner described in D 239. The A r proof of 0{n} for all n, E-correct is the E-proof of prfo[rx, cr[Numx]] = 0. This drops out and is replaced by the relation requirement of D 239. Otherwise we would be getting the g.n. of prfo[rx, cr[Numx]] = 0 occurring in the A r predicate, then the g.n. of this g.n. and so on. From these corollaries we see that if {A£) ${£} is proved in the system Ao plus the principle of Mathematical Induction then (A£) <£{£} is an Aj-theorem. We now show that the substitution in the induction rule is redundant. LEMMA. / / ^r{v) is an Ao-theorem for all v, E-correct, then i^{oc{v}} is an Ao-theorem for all v, E-correct, where a is a primitive recursive function. Let TP be equal to the g.n. of an Ao-proof of fr{v} and let cr\Num v\ be equal to the g.n. of ty{v}.
10.9 Relations between Ao-theorems and E-correctness
457
We are given prfo[rx, a[Numx]] = 0 is an E-theorem, hence so is prfo[ryx,cr[Numyx]] = 0,
(i)
where y is a primitive recursive function. Now prfo[ryv, a[Num yv]] = 0 says that ryv is equal to the g.n. of an Ao-proof of \jr{S*yv}, where S*0 = 0, 8*8v = SS*v. S*yv is the g.n. of the numeral determined by' yv'. It is S... SO. We have 'yv' -times
yv = S*yv, generally yx = S*yx is an E-theorem, Cor. (i), Prop. 4. The identity function and S* obey the same scheme of primitive recursion. We require to show prfo[T'x, or[fi[Num x]]] = 0 is an E-theorem, for some r', where fi[Num v] is equal to the g.n. of oc{v}. Numot{v} is equal to the g.n. of 8... SO; from oc{v} = 8*a{v} we obtain Eq[j3[Numv], [Numot{v}]] = 0; 'O^'ti
we require prfo[r"x, Eq[fi[Nunix], Numoc{x}]] = 0 for some r",
(ii)
to be an E-theorem. Take a for y in (i), then from (i) and (ii) we obtain the E-theorem: ) , a[j3[Numx]]] = 0, as required. It remains to find r" and to show that (ii) is an E-theorem. To do this we refer to the s.p. of a{v} = 8*ot{v} as given in Ch. 4. We have three cases: (a) a{v} = v, (b)
*{v} = 8*'{v},
(c)
*{v}
Here OL'{V) is of lower order than a{v}; y{v}, 8{v}, p{v) are of lower rank than a{v}. We suppose that the result holds for primitive recursive functions of lower order or rank than cc{v). Case (a). We have to show prfo\r"x, Eq[Numx,NumxJ] = 0 is an E-theorem for some r". Clearly r"v — Eq[Num v, Num v\ and we are finished.
458
Ch. 10 Induction
Case (6). We are given, by hypothesis prf0[T"x,Eq[/3'[NumxlNumoi'{x}]] = 0, where fi'[Num v] is equal to the g.n. of ot'{v}, is an E-theorem for some primitive recursive function r". We want to show:
prfo[rfrfx,Eq[(S, iy^[Numx]\9),NumSaf{x}]]
=0
is an E-theorem for some primitive recursive function T'". By hypothesis we have an Ao-proof of oc'{v} = S*a'{v} for all v, E-correct. To this we add Sx'jv} = S*'{v}
Axool, }
88*a'{v} = S*Sa'{v} S'{} S*S'{} Soc'{v} = S*Soi'{v} Cor. (i), Prop. 4
all these hold for all v, E-correct, hence using Prop. 7, Case (6). Case (c). y{v) and 8{v) are of lower rank than oc{v}, hence by the induction hypothesis we have: and
y{v) = S*y{v}, for all v, E-correct,^ 8{v} = S*8{v}, for all v, E-correct.J
<#pS*y{v} S*8{v} is of lower rank than oc{v} unless y{v} and 8{v} are both numerals. We get three subcases: Subcase (C-L). One of y{v} or 8{v} is different from a numeral, Subcase (c2). y{v}, 8{v} are AC, 0 respectively, Subcase (c3). y{v}, 8{v} are K, Sn respectively. Subcase (cx). We have (iii) and by R 1, Jp{y] y{v} 8{v} = Jp{v) S*y{v) S*8{v) for all v, E-correct, also by induction hypothesis, Jp{v) S*y{v} S*8{v} = S*Jp{v) S*y{v} S*8{v} for all v, E-correct, hence using Prop. 7, Subcase (c^). Subcase (c2). We have (K might be 8 ...Sv) ] KO = K Ax00 3.1,
10.9 Relations between Ao-theorems and E-correetness
hence by R 1,
459
Jp{y\ KO = S*Jp{v) #c0,
these hold for all v, E-correct, hence, using Prop. 7, Subcase (c2). Subcase (c3). We have Jp\y) K(8n) = p{v) n(Jp\y) KTT) AX 0 0 3.2, Jp\y)KTT is of lower order than oc{v}, hence by induction hypothesis J>P{V}KTT
hence, by R 1
= S*Jfp{v}Kn
for all v, E-correct,
Jp{v) K(STT) = p{v) nS*(yp{v} KTT),
this is of lesser rank than oc{v}, hence by induction hypothesis p{v}n8*(Sp{v}Kn) = S*P{V}7TS*{JP{V}KTT) = S*Jrp{v}K(Sn). These hold for all v, E-correct, hence, using Prop. 7, Subcase (c3). This completes the demonstration of the lemma.
IO.IO
Some properties of the system A 7
8. The system A 7 contains an irresolvable statement. The A7-proof-predicate is of the form: (Ex") P[K, V, X"]9 where P is primitive recursive, because ' K is the g.n. of an A r proof of the closed A-statement whose g.n. is v' is recursive, and any recursive statement can be expressed in this form. Consider (Ax')N["(Ex*)P[x'98[x9x\,xT], PROP.
where s is Godel's substitution function, let its g.n. be K, and let v be the g.n. of (Axf)N[(i6Ex")P[xf,s[K,KlxfT], then V = S[K,K] is an Aootheorem. Now consider (Ax')N["(Ex") P[x\ v,»"]"], if it is an A7-theorem then so is (Ax') N["(Ex") P[x', S[K, K], X"]"], whence by the properties of the A r proof-predicate (Ex")P[n, v,x"], for some numeral n, is an Ao-theorem, and so (Ex") P[7T, S[K, K], X"] is also an Ao-theorem, here the numeral n is the g.n. of an A r proof of (Ax') N["(Ex") P[x', S[K, K], X"]"], but if this is an Az-theorem then so is N["(Ex") P[TT, S[K, K], #"]"] for each numeral n.
460
Ch. 10 Induction
But {Ex") P[TT, S[K, K], X] is recursive, hence so is its negation. Hence only one of them is correct, but under the supposition that (Ax')N["(Ex")P[x',v,x"Y'] is an A7-theorem both of them must be correct. Thus it must be that {Ax') N["{Ex") P[x', v, x'T] fails to be an A r theorem. We now show that {Ax') N["{Ex") P[x\ v, x"]99] is A-valid, thusitsnegation fails to be an A7-theorem, this means that it is irresolvable. If {Ax') N["(Ex") P[x', v, x"]"] fails to be A-valid then it will take the value f and so {Ex") P[n9 v, x"] will take the value t for some numeral n. But this is an Ao-statement and if it takes the value t then it is an Ao-theorem, hence n is the g.n. of an A r proof of the A-statement whose g.n. is v. Thus {Ax')N["{Ex")P[x'9v,x"]99] is an Aj-theorem, and so is N["{Ex")P\n,v,x"]"] and so {Ex") P[n, v, x"] takes the value f, but it is recursive, so we have an absurdity, thus {Ax') N["{Ex") P[x'9 v, x"]"] is A-valid but fails to be an A7-theorem. We have thus found an example of a universal A-statement {Ax)
for
77 = 0,1,2,...
(30)
while {Ax') {Ex")Q[x\ v,x"] fails to be an A r theorem. Thus the conditions for an induction fail to be satisfied, in fact we omitted to show that the proofs of {Ex") Q[n, v, x"] for n = 0,1,2,... are on a primitive recursive plan E-correct, and this must fail to be the case. In obtaining (30) we used arguments of quite a different kind, the consistency of Ao, etc. This proposition shows that we must be very careful in saying that {Ax) <j){x} is an A r theorem when we have somehow shown that
10.10 Properties of the system A/
461
down that the proofs of
a{n} = /?{n} V oc{n} 4= /?{n} for all n E-correct. This comes from the example by the lemma. Thus Case (a). (b) <j> is a disjunction, say
N["f{n}"] v {f{n} v ^{n}) Jy["^{n}"] v (^{n} v ^'{n}) ^["(^{n} v ^'{n})"] v (f{n} v ^'{*}) Thus by Cor. (ii), Prop. 7, the result holds for all n E-correct. (c) 0 is a conjunction, say 0{n} is ^{n} & ^'{tl} and the result holds for i/r{n} and for i/rf{n}. Then ^{n} V N["i/r{n}"] for all n, E-correct, ft'{n} v ^["^'{n}"]
for all n, E-correct.
We have
v JV[(^{n} & ^ { n } ) ]
„,,
(^{n} & f {n}) v ^ [ " ( ^ Thus by Cor. (ii), Prop. 7, the result holds for ^{n} & ^'{n} for all n E-correct.
462
Ch. 10 Induction
(d) ^ is a general statement, say (Ag) ${£}, and the result holds for Then \]T{K, n} V N["fr{K, n}"] for all K, n, E-correct. We have
{K}.
JT{K, n} V (Eg) N["fr{g, n}??] for all n, E-correct by Prop. 7, Cor. (ii) (Ag) f{g, n} V N["(Ag) f{g, tt}"] by induction, for all n E-correct. Thus by Prop. 7 the result holds for (Ag) i/r{g, n}. (e) <J) is an existential statement, the demonstration is similar. This completes the demonstration of the proposition. P R O P . 10. If cj> is a closed I^'G-theorem
and if i/r is obtained from (j) by replacing each predicate n in <j> by an Aoo-equation with exactly the same number of free variables and if the same Aoo-equation or a variant thereof is used to replace other occurrences ofn, so that n and its replacement always have the same set of free variables, then ifr is an Aj-theorem.
T.N.D. is an Aj-theorem and all the I^F'c-r\ile8 are A r rules or are derived A r rules, except l i e ' . In the /J^-proof-tree of (j> make the replacements as described in the ennunciation of the proposition, add the Aj-proofs of the cases of T.N.D. used, and we obtain an Az-proof-tree except that we have l i e ' instead of induction and there may be free variables. Replace the free variables by numerals in all possible ways, we obtain an A r proof-tree when we replace applications of lie', say {£} V o (A£) <{>{£} vco by
<j){v) v o)
for all v, E-correct.
v* This follows since the proofs of the upper formulae of the above inductions are exactly the same except for change of numeral, hence by Cor. (i), Prop. 6, they are E-correct. This completes the demonstration of the proposition. 11. The rule Ib is independent. Consider the A-statement
PROP.
(Ex) (Axf) (Ex\ x'") (Ax™) (N["<j>{xff/, x\ z iv }"] V tfx, x\ x"%
(i)
10.10 Properties of the system A/
463
from results in Ch. 3 and Prop. 10 above we know that (i) is an A r theorem. If we are denied the use of rule I b then the only possible way of proving (i) would be from T.N.D.:
by use of R 3, lid twice, R 3, lid. But this would involve generalizing on one occurrence of the variable £, and this is impossible. Thus 16 is independent. Note that Prop. 5 of Ch. 8 fails in A 7 , for if G is an irresoluble A r statement then G V JV["G"] is an A r theorem but neither disjunctand is an Aj-theorem. io. 11 Reversibility of rules 12. Rule 116' is reversible. We have to show that if (<j) & i/r) v o) is an A7-theorem then so are 0 v o) and xjry co. We may suppose that the main formula of II a is atomic. First consider an Aj-proof of (^ & i/r) V o) of level one and which ends in a single induction. An Ao-proof is dealt with as in Prop. 2, Ch. 2. We have then the E-theorem: prfo[TX, or[Numx]] = 0, where TV is equal to the g.n. of an Ao-proof of ( ^ f t f ) v w ' v 0)"{v}, whose g.n. is equal to cr\Num v\. PROP.
Here o) is o)' v (Ax) (o"{ot{x}}. In the A o -proofs of (
sider the corresponding occurrences of &, these can only enter the proofs at applications of II bf. Suppose that one of these is )*{v}
M Let the g.n. of the Ao-proof of the left upper formula be T[V and let TXV be T[vn...nTivv, where the rj^'s refer to all the Ao-proofs of left upper formulae of IIV that contain corresponding occurrences of &. Note that (^' & ^') can only be a variant of ($ & ijr) by R 1. Similarly define T2V using the right upper formula. Let TSV be TV less the components of rxv and T2V. Then T*V = T1vnr2vnTzv is equal to the g.n. of an Ao-proof of (0 & i/r) v o), it may differ from rv by having some repetitions. Now let T'ZV be the g.n. of the result of substituting <}>' for (^' & ft') at all occurrences of this conjunction which are related to (
464
Ch. 10 Induction
want to show that: prfo[r'x, cr'[Numx]] = 0 is an E-theorem, then <j) V co will be an A r theorem of level one. By hypothesis prfo[TX,
(0)
We have
. A ^ « , * xoox (r*xf for (r*x)f) prfo[T*x, a[Num x]] = A2[lc. r*x, a[Num x]] +
n where si, = WT±X9 b = a, + wT2x, c = b + mT3x. This follows from the definitions of TXX, T^X, TZX. We have four similar summands for prfo[r'x, cr'[Numx]] which we call (ii). We wish to show that prfo[Tfx, cr'[Numx]] = 0. We have I.e. T'X = cr'[Numx] by definition, hence A2[1.C.T'X,
cr'[Numx]] = 0,
from the easily proved E-theorem A2\x, x] = 0 and R 1. Also {r'x\ =
x^T*
and (0). Similarly for the third summand. For the fourth summand of (ii) we have r'xf = disj[<x,fi] and rxf+h = di8J[<x,/3']
for b < g ^ b + c.
But we have Ax0[disj[oc, ft]] = Axo[disj[cc, ft']],
Rll[disj[oc,j3ldisj[7,j3]] =
) I
Rllldijlj'ldijlj']]
(iii)
10.11 Reversibility of rules
465
and similarly for Ell. The terms a and p can be explicitly defined in terms of T'X, and similarly for a and /?'. To show that (iii) are E-theorems it suffices to show that Axl[disj[a,/3]] =
i
Ax*o2[disj[x,/3]] = Ax$2[disj[oc,j3']], ]] =
RP"[disj[ot,/}'ldisj[y,/]']l
(v) Rlllb'[disj[a, /?], disj[7) pi disj[8, /?]] = Rlllb'[disj[x, P'l disj[y, P'l disj[S, /?']]• In showing that (ii) is zero we are dealing with the only application of R 3 which occurs in an A r proof with only one level. Take the various axioms in turn. The term /? is of the form conj[P"f, /?*] and the term /?' is /?f, or else both are null formulae, this follows from the construction of P'. If both are null then the first axiom becomes ^4#J[a] = ^t#J[a], which is an E-theorem. If one of /?, /?' is present then so is the other, and since axioms are atomic, then both sides of the first equation in (iv) are equal to one. This is somewhat heuristic, what we require is Axl[disj[x, conj[xf, x"]]] = Axl[disj[x, x']] if x' 4= 0, and
Axl[x] = Axl[x],
are E-theorems. Even this is insufficient because the two cases are run together in (iv). This is effected by proving that if x + 0, Axl[disj[x, A±[x'] x conj[x', x"]]] = Axl[disj[x, a?']],
(vi)
if #' is replaced by zero this becomes ^4#J[a;] = ^^[o;] (see D 174), while if x' is replaced by something different from zero both sides are equal to one. Actually we need only E-prove (vi) under the hypothesis stat[x, x',x"], but it will suffice to E-prove the more general statement. Now Axl[x] = Ax[A2\x, Eq[oc, a]] + tmoc], the term a can be explicitly defined. To E-prove (vi) it suffices to E-prove AJx] x Axl[disj[x, x']] = A±[x] x (AJx'] + B^x'] x Axftx]), and
A-^x] x Axl[disj[x, A-^x'] x conj[x', x"]]] = AJ[x] x ( 4 ^ + B i M x Axl[x\).
(vii)
466
Ch. 10 Induction
Both sides of the first equation of (vii) take the same value if x = 0 or if x' = 0 also because
A2[disj[Sx, Sxf], Eq[a, a]] = 1
^ 2 [<8, 8, 5>n/?n<9>nyn<9>, <8,8,2>nan<9>V\9>] = 1
is an E-theorem, from the E-theorem A2[xnx',xn(x' + Sx")] = 1. The second equation of (vii) is dealt with similarly. Thus the first equation of (iv) is settled. The other equations in (iv) are treated similarly. The equations in (v) are E-theorems because alteration of the subsidiary formula fails to affect a rule; thus RlIb[disj[x,x'],disj[x",x']] = RlIb[x,x"] is an E-theorem, thus RlIb[disj[x,x'ldisj[x",x']]
= RlIb[disj[x,x'"]disj[x",xff']]
and similarly for the other rules. The set (v) should also include the cases when J3 and ft' occur in the main formula of la, I&, R 1. In the case of a cancellation of {(j)r & ft') by 16 we have: Rllb[disj[disj[x', x']9 x"\ disj[x', x"]] = RlIb[disj[disj[x, x], x"], disj[x, x"]]. Similarly in the other cases. The full details are somewhat lengthy but enough has been displayed to show how they can be obtained. This completes the case of A7-proofs of one level. Now suppose that the result holds for A r proofs having A levels and we will show that it holds for A r proofs having $A levels. An A r proof having 8A levels finishes with an E-proof of dedo[px, TX, cr\Num x]] = 0.
pv is the g.n. of a sequence of A r theorems, some of which are of the forms o) v {(j>f & ^ ' ) , the remainder are without a related occurrence of (
10.11 Reversibility of rules
467
The details of this E-proof are almost the same as in the case of an A7-proof having only one level. The main difference is that we have another conjunctive clause to add to (ii) namely: Hyp[r'f]9 with an obvious notation. To deal with this we have extra equations to add to (iv), namely
Hyp[disj[x, /?]] = Hyp[disj[x, /?']].
But this is n
AJ[px2, disj[a, /?]] =
n
AJip'xS, disj[a, /?']].
It suffices to show that AJj>x%, disj[ot, /?]] = A%[p'x%9 disj[oc, /?']].
(viii)
is an E-theorem. But this follows at once by definition because px$ = disj[y,/3] and p'x% = disj[y,/3'l so both sides of (viii) are equal to A2[a, y], and the result follows. The full details of the rest of this case are as for the case when there is only one level. This completes the demonstration of Prop. 12. COR. / / ((j) & ft) v o){v}for all v, E-correct, then (j> v o){v}for all v, E-correct. 13. Rule R3is reversible. By this we mean that if (AE) <£{£} v o) is an A7-theorem then so are cj){y} v (o for all v, E-correct. Note that if (A^)rf>{oc{E,}}v (o is an A7-theorem then <j){v) v o) for all v, E-correct may be wrong. We proceed by induction on the number of levels in the A r proof of (Ag) ^{g} v co. First suppose that there is only one level. (A£)
\If{v\ v o) for all v, E-correct
'
,
,
,c
f
^ - ,c ^
' whew *{«M} IB tfW,
hence the result follows at once from the lemma after Prop. 7. If OJ contains K and the lower formula holds for all K, E-correct then the upper formula holds for all K, V, E-correct.
468
Ch. 10 Induction
Now suppose that we have obtained the result for all K, E-correct for A r proofs with A levels and we will show that it also holds for all K, E-correct for A r proofs with SA levels. The $Ath level is either an Ao-deduction from the production of the Ath level as hypotheses or else it is a single application of R 3. In the first case the hypotheses are of the forms:
where (^4£)0'{£} is a related occurrence of (^4£)0{£}, and where to" is without related occurrences of (Ag) ${£}. We modify the Ao-deduction which constitutes the $Ath level as follows: replace the hypotheses and their related descendants by
for all v, E-correct,
The structure of the deduction is otherwise unaltered. The result is an Ao-deduction of (j){v) v o) for all v, E-correct. This follows from Prop. 7 and our induction hypothesis. By our induction hypothesis (j>'{v} V (of holds for all v, E-correct, because the production of level A are Ar theorems having A levels. (Ag) x{£} V o)" V
r a v f° f° r ^ > E-correct, where ^{g} is ^
and as in the first case we have finished.
10.12 Deduction theorem
469
10.12 Deduction theorem 14. The deduction theorem holds in A 7 . We have to show that if ^r can be A r deduced from hypotheses
(At;) \If{£\ v o) and
is an E-theorem, where a is the g.n. of the sequence
(ii)
n
here T'X = r1x oi ^T^x, where /? is the g.n. of the A r proof of JV["^>"] v ^ > , and T*X is the result of replacing each component of T2X, say 8, by disj[8,a']. Referring to the definition of ded0, which is similar to (i) in Prop. 12, we see that the gist of the matter we wish to E-prove is: Rl\x, x'] = BP[disj[x, x"], disj[x', x"]],
(iii)
but we have already seen that this is an E-theorem, it says that an application of a rule remains an application of that rule when something is added to the subsidiary formulae throughout. Thus the first case is settled.
470
Ch. 10 Induction
Now suppose that we have demonstrated the result for A r deductions of A levels and we will show that it holds for Az-deductions with $A levels. An Aj-deduction with SA levels is either an A0-deduction from the production of level A as hypotheses or it is a single application of R 3 from the production of level A as hypotheses. In the first case if we replace the productions of level A by their disjunctions with N["ftSK)"] then they are, by our induction hypothesis, A r deductions from $\ <$>",..., ftK) as hypothesis, thus if we alter the A r deduction of ty as in the first case then we are left with an A r deduction ofN["ft&KV9] V ^from <j>', <}>",..., ftK) as hypotheses, thus this subcase is settled. If the A7-deduction of level SX is a single application of R 3, then we proceed as in the first case. We first have to show that if i/r{v} is deduced from the hypotheses ^', 0",..., 0(/c) and that this holds for all /c, E-correct, then N["ftW] v f{v} can be A r deduced from $', $\ ..., fir) for all v, E-correct. This is dealt with as in the first case but writing j instead oix. Thus we may suppose that the productions of level A, namely x{v}> e ^ c hold for all v, E-correct and that the same holds for N["ft8K)"] V x{v}' The only difference between this case and the first is that instead of (i) we now have dedo[pxn<xn{a'),TX, a[Numx]] = 0 is an E-theorem, we modify this as before by adding on the disjunctive clause N[( 'ft8**9'] to px, TX, cr[Numx]. We then require dedo[p'xna, T'X, (rf[Numx]] = 0 to be an E-theorem, again the gist of the proof that this is so is (iii). Thus the second case is settled. This completes the demonstration of Prop. 14. 10.13 Guts with an A00-cut formulae We now come to the problem whether the system A z is weaker than the system A 7 plus the cut, call this system A JC . That is to say, is there an A-true statement which is an AJC-theorem but lacks an A7-proof ? An Aj-proof of level 1 is either an Ao-theorem or else is a single application of R 3, i.e. is of the form ^{n} V o)
are Ao-theorems for all n, E-correct.
Now M.P. can be eliminated from an Ao-proof (the cut formula must be an Aoo-formula because its negation is required in the cut). Thus if
10.13 Cuts with an Aoo-cut formula
471
§5{n} V o) can be Ao^-proved then it can be Ao-proved. Here Aoc is the system Ao plus the cut. Thus if we have
(i)
then we shall have 0{n} V o) can be Ao-proved for all n.
(ii)
The question whether this is E-correct remains open. If the cut is eliminable from AIC then to apply R 3 we must have (ii) E-correct. If we have (i), then the Aoc-proofs of 0{n} V 0) are on a primitive recursive plan, E-correct. Removal of cuts in these Aoa-proofs so that they become Ao-proofs (which would be on a certain plan) might destroy the primitive recursiveness of the plan, though it might be general recursive, or it might upset the possibility of proving E-correctness, if the plan remained primitive recursive. An Aocrproof is easily converted into an Ao-proof. The cut formulae are Aoo-formulae, hence they are A00-true or A00-false; this is a recursive decision, unfortunately it violates the required condition of being primitive recursive. In the Aoa-proof we omit all false ancestors of false cut formulae and all false ancestors of negations of true cut formulae. This will convert a cut into a case of II a (or rather a sequence of cases of II a if we require that II a only dilutes with atomic formulae) other rules remain applications of the same rules. If we do this and also remove all branches above A00-true ancestors at places where branches join, then we are left with an Ao-proof of the formula produced by the original cut. (Removal of these branches can be dispensed with, they are merely redundant, it is technically simple to retain them.) It is easily seen that if we do this elimination of a cut then higher cuts have their cut formulae unaffected. Thus simultaneous removal of false ancestors can take place, or they can be removed in any order. A false ancestor of one cut is distinct from any false ancestor of another cut. Since the test for A00-truth, though recursive, fails to be primitive recursive, we must seek some other test. Let us remove all ancestors of cut formulae and of their negations. The two upper branches above a cut will become two new trees, by the completeness of Ao one of them at least will be a proof-tree of its final formulae, namely the one above the cut formulae if this is A00-false, otherwise the other one. From the original proof-tree we obtain the new tree by a primitive recursive
472
Ch. 10 Induction
process. It is a primitive recursive process to decide if a number is the g.n. of an Ao-proof. Thus we can decide, on the information available to us by a primitive recursive process, which branch to take. Before, when we were held up by a decision, though recursive, failing to be primitive recursive, we were using only part of the information available to us. Having removed the cuts we are left with an Ao-proof-tree of the same formula as before, the g.n. of this Ao-proof is a primitive recursive function of the g.n. of the original Aoc-proof. Let rn be equal to the g.n. of the original Aoa-proof of
10.13 Cuts with an A -cut formula
473
where $' is (0" V ft"), since $' is A00-false then both (j>" and <j>'" must be Aoo-false. Then cf>" is to be the false ancestor of
-,—^, is except for remodellings a one premise rule or is R 3
in the A r proof of OJ{U} V oc = /? and if the occurrence of oc' = /?' in the lower formula is an occurrence of a false ancestor of oc = ft, then the occurrence of oc' = /?' in the upper formula is an occurrence of a false ancestor of a = /?; 1
j
fv'/?' is except for remodellings a case of rule II b' and if the occurrence of a' = /?' in the lower formula is an occurrence of a false ancestor of oc = /? then both occurrences of a' = /?' in the upper formulae are occurrences of false ancestors of oc = ft; if
is except for remodellings a case of rule R 1 and if the occurrence of oc' = J3' in the lower formula is an occurrence of a false ancestor of oc = fi, then if a" = /?" is A00-false it is an occurrence of a false
474
Ch. 10 Induction
ancestor of a = /?, but if a" = ft" is A00-true, then y = S must be Aoo-false, and is an occurrence of a false ancestor of a = /?. Again it is a primitive recursive decision to decide which is the case. Note that it is impossible for related occurrences of a = /? to occur in the main formulae of II&', d, otherwise they would occur in the theorem in the scope of & or E. By 1b a = ft might have two false ancestors in the upper formula. We now omit all false ancestors of a = JS in the A0-proofs of o){n} V oc = /?. By this r% becomes r ' j and
clflmaT'x^
for
0 < x' ^ mrx,
so we can disregard these terms in the definition of Rl\ and Rl%. Note that by retaining redundant proofs we have WTX = wr'x. We have the E-theorems: (a) Rt*[Tz29TX$]
=
BIX[T'X$,T'X%]XA2[T'X$,T'X£1
where
l i s a
one-premiss rule, (b) (C)
These follow because we show that they hold in each of 500 subcases; these come about as follows. We have
and
T'X%9 = disj[Sp[[x9 x']9 Sp'2[x, x']]
or TX£ fails to be a disjunction. (See D 179.) NDv
-c
TO if v fails to be g.n, of a disjunction, .. ., . otherwise.
10.13 Cuts with an A^-cut formula
475
In the first case we have the subcases: NDTX% = 0,
px\_x9 x'l = pfx[x, x']
and p2[x9 x'] = p2[x9 x']9
Pi[x, x'] #= pi[x, x']
and p2[x9 xf] = p2[x9 x']9
px[x9 x'] = p[[x9 x']
and p2[x9 x'] + p'2[x9 x'\
p±[x9 x'] + p[[x9 x']
and p2[x9 x'] + p'2[x9 x']9
there are similar cases for x" giving 25 cases for iWx\TXX>9TXX»'\9 X has 5 possible values thus 125 subcases for one-premiss rules. The cases are mutually exclusive and exhaustive. We show for each X
in each of the 25 cases, whence we obtain for each X
without condition. The 25 cases are distinguished by: Let
oc^x.x'] = ^i[^ a I>i[^^iPi[^ a2[x9x'] = ^ x t ^ l ^ ! ^ , ^ ] , ^ ^ ots[x9x'] = ^ i ^ a ^ i t ^ ^ ^ i t ^
Let
fijx,
x\ x"] = B&ajx, x'] + a±[x, x"]]9 02[x, x'9 x"] = BJiaJx, x'] + a2[x9 x"]],
Then j3e\x9 x', x"] = 1 when the ^th condition is satisfied. We now show that fie[x, x'9 x"] x Rlf[rx%9 TX?\ = pd[x9 x'9 x"] x RI*[T'X%9 T'X%-\ x AJ/xZ, for 1 ^ 6 ^ 25, and for each rule. And
2 fio[x,x'9x*] = lf
T'X$1
(iii) (iv)
476
Ch. 10 Induction
since the conditions are exhaustive and mutually exclusive, from (iii) by addition for 6 = 1 to 6 = 25 and (iv) we get %, TX?] = Rlf[r'x%, r'a#] x A2[T'X%, T'a£].
(v)
This is (a); (b) and (c) are obtained similarly but require 125 cases each. From (a), (b) and (c), and the E-rule, if
px = p'x
we get
then
2 Px> = S
P'x
prfo[T%, cr[Numx]] =
as desired. It remains to show that the equations (iii), (iv) are E-theorems. The situations in the 25 cases can be described as follows: we here speak of TX™' as the upper formula and TX£< as the lower formula. We speak of Sp±[x, xr] as the main formula of the upper formula and Sp2[x, x'] as the subsidiary formula of the upper formula, Sp-^x, x"] and Sp2[x, x"] similarly refer to the lower formula. We use this manner of speaking because BI^[K, V\ = 0 if and only if K is the g.n. of the upper formula of rule X and v is the g.n, of the lower formula of that rule. We use the following abbreviations: f.a. l.f. n.d.
false ancestor, lower formula, a formula other than a disjunction, i.e. atomic or a conjunction or an existential statement, m.f. main formula, s.f. subsidiary formula, r.s.f. right subsidiary formula.
We now give a table of the 25 cases and against each we note the only possible rules that can be satisfied in such a situation, in the last column we add conditions which may have to apply if the rule is to hold. The asterisk denotes that the formula concerned contains an occurrence of a false ancestor of a = ft. Take, for instance, (10). Here TX$ = disj[Sp-^x, x']9 Sp2[x, #']], the first component of the disjunction contains a false ancestor of a = /?, while the second component is without occurrences of a false ancestor. TX™» is either atomic or is a conjunction or is an existential statement. The rule (h* V &
then looks like
— where the lower formula fails to be a disjunction. X
10.13 Cuts with an A 00 -cut formula
477
The only possible rule is II d without subsidiary formula. But then this 6* V l/f
would be — _ .=-, but S* contains an occurrence of a false ancestor, (Eg) (<j>* V ^ ) but a false ancestor once governed by E would have an occurrence governed by E in the theorem and this is impossible. The bar denotes a substitution of a variable for a numerical term. Hence this situation fails to arise and so filo[x, x', x"] = 0, and so (iii) holds for 6 = 10. Similarly in the other cases. no.
u.f., u.f., If., If., m.f. s.f. m.f. s.f. * * *
* n.d.
poss. rules any Ila none none lid, 16 none la, 6, lid la, I l a la
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
*
n.d. * * * * *
* *
* *
* n.d. *
* * *
* n.d. n.d. n.d. n.d. n.d.
* *
* n.d. *
* *
* n.d.
lid none la any Ila lid none la 16 any lid R2, Ila, d Ila none none lid, K2
remarks l.f.9 m.f. is the/.a. 16 without s.f. and m.f. is n.d. lid is impossible, an/.a. fails to occur governed by E la with right s.f. missing, I l a with/.a. in secondary /• only l a with right s.f. missing, f.a. in left s.f. and 2nd m.f. lid impossible as in (7) l a with r.s.f. missing lid impossible as in (7) l a with r.s.f. missing 16 with s.f. missing lid impossible as in (7) R 2 with s.f. missing
both with s.f. missing
This argument is somewhat heuristic, what we want is E-proofs. Let us first find the E-proof of (iv). For this purpose it is convenient to use y and z as extra symbols for variables of type i to avoid a plethora of primes. We require a generalization of Cor. (ii), Prop. 5, namely:
478
Ch. 10 Induction
If p[090,0,0,0,0] = 0 and if p[oc, a', /?, /?', y, y'] = 0 whenever at least one argument is zero and if p[x,x',y9y',z,zf] = 0 p[Sx, Sx', Sy, Sy', Sz, 8z'] = 0 then p[x, x', y, yr, z, z'] = 0. We shall define a certain function y and show that y[x9x'9y9yf9z9z']
=0
using the above generalization; we will then make substitutions. Replace x
'
by
y
by
ND.rxZ,
y'
by
ND.TX$9
z
by
z'
by (p2[x,x'']
(p2[x9 x'1 -z-p'%\x9 x']) + (p'2[x9 x'] -p2[x,
x']),
(Pl[x, x"} ^p[[x, x"]) + (p[[x, x"] ±Pl[x, x"])9
Define: a*[x,x',y] = AJ
a* [x9x'9y] = A ^ <**[x9x'9y] = AAA = BlV,
fit[x,xr,y,y',z,z'] = BJof [»,«',yl + oj^z',^]]
(1 ^ 6 ^ 5),
A*[^, xf9 y9 y'9 z, z'\ = jBxtof [a?, a?', 2/] + agL 6 [s,»', y']]
(6 < 0 ^ 10),
^lo[z9z',y']] ^^y']]
(11 ^ 5 ^ 15), (1« < 0 ^ 20),
O[z,z',y']]
Then
yt^^y.y',»,«']=
S Pt\*rf>y>y\z9%'\. 162
(21 ^ 5 ^ 25).
10.13 Cuts with an Aoo-cut formula
479
We wish to show that By computation Similarly
y[0,0,0,0,0,0] = 1. y [0,Sx', Sy, Sy', Sz, Sz'] = 1,
similarly
y[Sx,Sx',Sy,Sy',Sz,O]
= l,
y[0,0, Sy, Sy', Sz, Sz']
= 1,
and so on, in fact it is easily seen that as long as one of x, x', y, y', z, z' is replaced by zero then exactly one of a*[x, x', y] is zero and exactly one of a*[z, z', y'] is zero. Thus the first requirement of our generalization of Cor. (ii), Prop. 5, is satisfied. The second requirement follows at once because exactly one of OL*\SX, SX', Sy] and exactly one of a*[8z9 Sz', Sy'] is zero so that exactly one of f!*[Sx, Sx', Sy, Sy', Sz, Sz'] is one. This completes the E-proof of (iv). We now return to the E-proof of (iii). If J30[x, x', x"] = 0 then (iii) holds, if /3e[x,x',x"] = 1 then there are some relations between rx^, rx™*, r'x%, T'X%. For instance if fax, x', x"] = 1 then rx% = r'x% and TX$ = r'x% and so (iii) holds for 6 = 1. But we want an E-proof that (iii) holds. What we have just said becomes in the system E:
fax,x',x"] = 0
fax,x',x"]
=l
(iii)i (iii)i where (iii)! d e n o t e s (iii) w i t h 0 = 1 . W e n o w g e t a n E-proof of (iii) x a s follows: we h a v e or ,„-,,-, , „, „ nv
fij[x, x', x"] x (1 - A[x, x', x"]) = 0,
and
0i[x, x', x"] + (1 —/?a[x, x', x"]) — 1, are E-theorems,
thus we get from the above fl[s, x', x"] x (1 ^./jjjx, x', x"]) = 0 (iii)! x fijjx,x',x"] (iii)! x {l^ftjjx,x\x"]) (iii)! x (fijx,x',x"] + {l-^fijx,x',x"]))
480
Ch. 10 Induction
For 6 = 2 the conditions are: TX™' is equal to the g.n. of ^ v o) and is without f.a. 's, TX™" is equal to the g.n. of $'* v o/ and 0'* contains an /.a. but to' is without/.a.'s, T'#^ is equal to the g.n. of (j) v w, T'2$> is equal to the gf.%. of 0' vfc/where $' is §J'* after removal of/.a.'s. The figure to be tested for being a rule is: -JJZ
;
before removal of f.a.'s, after removal of/.a.'s,
\LJ
V U/
here ^, w, 0', a>' are all present. The only possible rule is II a (6 v OJ)
~rrL—7~7
1 r
v belore
{Xs}, Xx. {x, y] &
.X ^J.u.b.(*X.
Chapter 11 Extensions of the system A7
11. i The system A' We have seen that the system A 7 is incomplete and that any extension of it which remains a formal system will also be incomplete. We could add, as extra axioms, some A-true but A7-unprovable statements so as to obtain more A-true statements as theorems in the resulting extended system. But as long as we have a formal system it will still be incomplete and an irresolvable statement can be constructed on the same lines as before. We can do this programme in a systematic manner as follows: We have an effective method for constructing an irresolvable J§?-true JS?-statement in a formal system j£? which contains recursive number theory and negation. Call this j£?-statement (?{J§?}. We first form 0{Aj} and then the system G' which consists of the system A 7 with the extra axiom 6?{AZ}; having formed 6?(A) we construct 6?(fifA) by adding the extra axiom 6?{6?(A)}. Having formed the systems A 7 ,G',..., 6?(A),... we then form the system 6r* as the union of all the systems A 7 , G',..., i.e. the system A 7 with all the extra axioms we added in forming the systems G',...; this system again will be formal, hence we can form G{G*} and the system G*' which is the system G* plus the extra axiom G{G*}. So we can proceed through the constructive ordinals. But we shall have to stop before we come to the end of the constructive ordinals, otherwise we shall cease to have a formal system. Another way of extending the system A 7 is to add property variables, i.e. variables of type 01. But we shall need care in doing this. A property is a term of type ot, for instance X£.^{£}. Suppose that we have introduced property variables and quantifiers for them, then we can form the property 'k£>.(AE)
]
512
Ch. 11 Extensions of the system A/
character. Secondly in order to be able to prove more A-true A-statements in the resulting system we shall have to have some method for ejecting property variables from a proof. If we continue to keep proofs direct so that anything which enters a proof will remain in that proof till the end, except that duplicates can be removed and equals replaced by equals and a term may get lost by introduction of an existential quantifier, then we shall have to have an extra rule which will enable us to eject a property variable. One such rule is Modus Ponens or the cut. In this rule the whole of the cut formula is ejected in one step of the proof. We have shown that the cut can be eliminated from the system A7, but we shall show that this is no longer possible when property variables are introduced. The cut thus provides a method by which property variables can be ejected from a proof. We now construct a hierarchy of formal systems, each is a primary extension of its predecessor, and has direct proofs and predicative properties. A° is A 7 with closed properties X£.^{£}, X££'.0{£, £'},..., these can only occur as (X£. <£{£,}) a, and so, can be replaced by
Type
Name
0
i
S x J = ~ V A Xol X0ll, Xolu,... A0l
u Successor function i Variable for a natural number ut(ut) Iterator operator on Equality predicate oo Negation ooo Disjunction o(oi) Universal quantifier for natural numbers oi One-place property variable on, oiu,... Two-, three-, ... place property variables o(o(oi)) Universal quantifiers for one-place property variables o(o(ou)), Universal quantifier for two- three-,... place, o(o(oin)), property variables
Aou, A0Ul,..., X (
Zero
Abstraction symbol Left parenthesis Right parenthesis Generating symbol.
11.1 The system A'
513
We use S with or without superscripts or type subscripts for an undetermined property variable of the type shown by the subscript, (if a type subscript is absent then the context should indicate the type or the type is immaterial), we similarly use A for an undetermined property. Atomic statements are: (a = /?), (S4a),
(Euafi)9
..., (\a),
...,
where a, fi are numerical terms. Properties are without bound property variables. D259
(OC + JS) for
D 260
(^ & i/r) for
D261
(mm
for
~(a = /3), - (~ 0 v ~ f),
~(A£)~
(EE)
- (AE) ~ ^{H}.
We write (AE)
(a = a), /8a 4=0, ^#2.2 0 4= Sot, JfcaO = a, ^ x 3.2 J?pot(S/3) = (X£. a{g}) ^ . . . /?(»> = a { / ? } ^ ' . . . /?<*>,
here a,a{0},/?,/?',...,/?(7r) and/> are closed. £, ^' fail to be free in a{PJ. Remodelling schemes. la w ' v ^ v f v w t
16
permutation.
cancellation.
Building schemes. Ila
^
dilution, ^ atomic
116 ~(j)\/(i) ~i/r\/(o composition
lie double negation
~ ^{a} v OJ , where a is free in ^{a}, 17
514
Ch. 11 Extensions of the system A/
existential dilution for natural numbers, 11/
~ ^{A} v o) , where A is free in E)0{3} v existential dilution for properties. Arithmetic schemes. R 1
R2
substitution
(a =t= ft) v o) 08a *S/3)VG) progression
R 3 ^{0} v (i) jS{l}vw,...,
for all v, E-correct,
induction
R 3* $4{A'} V o), 0{A"} v a),...
for all properties A', A",..., E-correct,
property induction (we allow a substitution)
R4 abstraction
alteration of variable
a free in i/r{a}, £, y\ fail to occur free in ^{I\}. In the above o; is the subsidiary formula and can be absent, ^ is a secondary formula and must be present, the others are main formulae and must be present. Properties are without bound property variables. All formulae in axioms or rules are to be closed. Property induction goes as follows: D 262 prop v for v is the g.n. of a property without bound property variables. Let TV be equal to the g.n. of an A'-proof of ^{A} v (o when v is the g.n. of an A'-property A, and let crv be equal to the g.n. of
11.1 The system A'
515
be the g.n. of an A^-proof of 0{S} v o), and let q be the g.n. of ^{E} v co, then if v is the g.n. of S and K is the g.n. of A, then Subst [K, V, p] is equal to the g.n. of an A'-proof of ^{A} v co and Subst [K, V, q] is equal to the g.n. of 0{A} v (o. Thus to obtain R 3* we have to E-prove: (1 -^- propx) x prfA>[8ubst[x, v,p], Subst [x, v, q]] = 0.
(i)
Now p = (p',p",...,pin)} and we have prfA>[p,q] = 0, p^n) = q. Now prfA>[p, q] = 0 splits up into a set of Aoo-provable clauses of the types: Ax[pW] = 0, = 0,
= o, for certain values of A, A', etc., which can be found from prf^[p, q]. So it suffices to E-prove a set of statements of the forms: (1 ~ propx) x Ax [Subst[x, v,p{X)]] = 0, (1 _^propx) xRP[8vbst[x9v,pM],8ub8t[x,v,pVV]]
= 0,
(1 - propx) x Rl2[Subst[x, v,pM]9 Subst[x, v9p^]9 Subst[x, v,p^]] = 0 for the same values of A, A', etc. In the first of these there is nothing to do because axioms are without property variables in A'v. Thus Subst[x, v,p{X)] = jp(A), and since we have Ax[pM] = 0 then we also have (1 — propx) x Ax [Subst[x,v,pW]] = 0. Suppose we have
= o,
which is the same as A2[piX"\ disj [p*,p(y)]] = 0, where p* can be explicitly defined in terms of p^V). Then this becomes (1 — propx) x A2[Subst[x, v9p^]9 disj [Subst[x, v,p*], 8ubst[x, v,p^f)]]] = 0 which follows from the E-theorem A2[x, x] = 0 by substitution and pW) = disj[p*,pW]. The other cases are dealt with similarly. Hence we can obtain (i). Thus we have a case of R 3*. An A'-proof, like an A7-proof, is divided up into a number of levels. Level one consists of a bounded set of single applications of R 3 and R 3* these are called the production of level one. When level A has been defined then level SA consists of a bounded set of single applications of R 3 and R 3* from the production of level A as hypotheses. The last level con17-2
516
Ch. 11 Extensions of the system A/
sists either of a deduction, without using R 3 or R 3*, from the production of the penultimate level as hypotheses, or of a single application of R 3 or R 3* from the production of the penultimate level as hypotheses. The full definitions are exactly as for A7, except that now we have a rule of generalization for property variables as well as for numerical variables. 11.2 Remarks (a) We could dispense with many-place properties by using ordered sets. Thus we could use A(a',..., a(7r)> instead of Aa'... a(7r). (b) We prove (.4£)^£{£} by proving
11.2 Remarks
517
But a class may be represented by its characteristic function, and a function may be represented by a class of ordered pairs, such that when the first member is given then the second member is unique. Also the distinction between properties and classes is largely notational, e.g. write Aa as (aeA) or ((eA) a), where e is a symbol of type ot(ot). (e) If we add T.N.D. as an extra axiom and Modus Ponens as an extra rule then we can dispense with Ax 1 and rule R 1 if we define {oc = j3) for
(AX)(Xoc-+X/3).
We have (AX) (~ Xoc v Xoc); for any property A (~ Aa V Aa) is an axiom, since we are having T.N.D., now apply R3*, (the required E-proof is easy), but this is Ax 1. For R 1 we have:
(Xg.fl#)avw
(Xg.flg}a)->JXg.ffg}/?) (Xgflg})av&>.-MXg.flg})/?Vft>
We have assumed the reversibility of R 3* ! (/) Remark (e) raises the question whether we can dispense with any of the other symbols. For instance in the system Aoo we could dispense with the propositional connectives since: a = 0 v/? = 0 may be replaced by
a x / ? = O,
We have already used these in forming characteristic functions. We could also replace (AE,) 0{£} by Max[oc{^}] = 0, where a{£} is the characteristic function of ^{£}, similarly (E£) ${£} may be replaced by Min [a{£}] = 0. All our statements would then reduce to the form a = 0, so we could omit ' = 0' and just transform numerical terms according to certain rules. This procedure would identify truth with zero and falsity with unity. We prefer to have some duplication, it seems to facilitate reading, after all each numerical term has many others equal to it, and we should only
518
Ch. 11 Extensions of the system Aj
get each numerical term uniquely represented if we discarded the iterator symbol and the abstraction symbol, so that the only terms of type i were the numerals themselves. But then we would have taken the whole interest of the system away. Thus, however we go about it, some duplication is going to occur, in the sense that some concepts are going to be represented in many equivalent ways, it is a matter of taste how much duplication of this sort we allow. 11.3 The hierarchy of systems A(p) We now define a hierarchy of formal systems starting with the systems A°, A' which have just been constructed. We add new symbols XKl of type (01), XKU of type (ou), etc., called property variables of order K. The system A^Sp) will contain these symbols for K = 0,1,2,..., v and the only properties allowed in A^Sv) will be those in which XKl, XKU, etc., occur, free or bound, for K = 0,1,2,..., v. The passage from A(p) to A^Sp) then consists in adjoining the property variables Xvl of types (pi), XVU of type (ou), etc., the property quantifiers of A' may be applied to them. The only way a property variable can enter an A(i;)-proof is via II a, /, R 3*. The order of a statement or a property in one of these systems is the greatest of the orders of the free property variables which it contains, zero if it contains none, and of the successors of the orders of the bound property variables which it contains, but zero if it contains none. Then the system A ( ^ contains properties of order v and statements of order 8v. The axioms of AM are all closed and the rules preserve closure, hence A(i;)-theorems are closed. In R 3* the properties run through all closed properties of order ^ K while the variable in the lower formula is of order A:, for 1 < /c < J>. In 11/ the property variable in the lower formula is of the same order as the property displayed in the upper formula. (So in AM we must avoid properties with bound property variables of order v.) 11.4 Properties of the systems 1. If $ is a closed A-statement which is an A^Sv)-theorem, then $ is an Aj-theorem. In other words the systems A(Sv) are useless for proving A-true A-statements which are A r unprovable. PROP.
11.4 Properties of the systems A(i;)
519
Suppose that
P R O p. 2. The systems A(j;) are consistent in the sense of Con (i), undecidable and incomplete. By Prop. 1 if we could Approve 0 = 1 then we could A r prove 0 = 1 , and hence we could A00-prove 0 = 1 , which is absurd. If A(v) were decidable then so would be A7, which is absurd. If A("> were complete in the sense that for a closed A(i;)-statement ^ either ^ or ~ <j> is an A(v)theorem, then by Prop. 1 the same would hold for A 7 which again is absurd. 3. Rules l a , 16,116, c, R 2, 3, 3*, 4, 4', 4" are reversible. The results for rules la, 6, R 2, 4, 4', 4" are trivial. The results for 116, c, R 3, 3* are dealt with as for the similar results for A7. For l i e we omit all the ancestors of ~ ~ in ~ ~ ^ v o), and then proceed on the lines of Prop. 12, Ch. 10. The details are left to the reader. PROP.
11.5 The system AM* We are unable to show that T.N.D. holds in A<-8v\ or that Modus Ponens can be eliminated. If we try to use formula induction in A^Sv) then when
and
AVLot and
Avtlocfi, etc.
520
Ch. 11 Extensions of the system A/
the latter cases are of the form })«P,
etc.
but these, if we apply R 4, can be any A^-statements. We can get over this difficulty by using formula induction on formulae to which R4 is inapplicable. But then when we come to formulae of the form (AE) §S{3} in our formula induction then we would be referred to formulae of the type ^{A} where A is any property of the same or less order than that of the variable S. Now ^{A} must be of the form ^{(X£. ^{£}) a}, and this by R 4 becomes 0{fr{ot}}, where ^{a} is any formula form of the same or less order than the variable 3. Let (^13) §J{3} be closed and of order 8v9 then this is so either (a) because 3 is of order v or (b) because (j) {Tvt} contains an occurrence of a bound property variable of order v, or contains a free property variable of order Sv. In case (a), ^{^{a}} contains one less variable of highest order, and if case (b) fails, it is of lesser order, but in the case when (6) holds and (a) fails,
$ is called the cut formula, co is subsidiary and can be absent, % is secondary and must be present. If both were absent we should end up with the null formula. We take T.N.D. in the general form (j) v ~ <j>, where <j) is any closed A(y)statement. This is different from the other axioms for they are atomic statements. But, as discussed above, any attempt to deduce the general case from atomic cases seems doomed to failure. We call the system ASv) plus T.N.D. and Modus Ponens the system A(")H\ P E O P . 4. / / (j) is a many-sorted ^Q-theorem and if i/r is obtained from <j> by replacing many-sorted atomic predicates by atomic A^*-predicates with exactly the same number and sort of free variable and if the same atomic
11.5 The systems A<">*
521
\(v)* predicate is used to replace a given atomic & ^-predicate at all its occurrences {except for change of free variables), then \jr is an A^*-theorem.
The J^o-proof of (j) can be laid out in levels like an A(y)*-proof. The axioms in the J^2C-proof are all cases of T.N.D. which translate into cases of T.N.D. in A(y)*, and these are A^*-axioms. All the J^-rules except He are A^-rules. A case of l i e can be replaced by a case of R 3 or R 3*, according to the sort of variable, as explained in Prop. 10, Ch. 10. The final result is an A(i;)*-proof. COR. (i). A^*'-tautologies are Aw'-theorems. COR. (ii). A^* is regular. COR. (iii). The Deduction Theorem holds in A(")H\ 11.6 The definition of A-truth in A'* 5. A-truth can be defined in the system A'*. We have to define an A'-statement x{v}> such that x\y] is a n A'*-theorem if and only if v is the g.n. of an A-true A-statement. We defined A-truth by the following scheme (A) in Ch. 8; PROP.
(Ax)(P[x,0]~Q{x}) (Ax, x') (P[x, Sx'] <-* R{x, x\ P[r'{x, x\ x'% x'],...9
]
J
1 (A)
where the variable x" is bound by a quantifier in R. This scheme defines a property P. This scheme resembles the scheme of recursion with substitution in parameter, but differs in having bound variables. This is the cause of the undefinability of A-truth in A itself. We can cast scheme (A) into a scheme for defining the characteristic function of the property P, if we do this then the bound variable x" will give rise to an infinite sum or infinite product or to an unlimited least number operation. In either case we have a case of definition by a form of induction quite different from any we have encountered before. To put the scheme (A) into a scheme for definition of the characteristic function we proceed as in Prop. 3, Ch. 5, except that we add 'replace (EE) (j){£) by Ila{£} = 0, where a{£} is the characteristic function of ', or we could replace it by Mina{g} = 0 or by a{(Jig[a{£} = 0]} = 0;
522
Ch. 11 Extensions of the system A/
having dealt with E we can deal with A. This scheme of definition is more general than that of recursion and fails to be constructive, because we are in general unable to decide the unlimited least number operation. Note that scheme (A) defines a sequence of properties
but the numeral K fails to occur in i^.[#], it only appears indirectly as the number of quantifiers in PK[x]9 so that Px>[x] fails to be A-definable (it being impossible for an A-formula to have x' quantifiers, cf. the discussion on Val in Ch. 5). Instead of the general scheme (A) consider the simplified scheme
(Ax)(P[x,0]~Q{x})
|
(Ax, x') (P[x, Sx'] <-> R{x, x', P[r{x, x',x"), x']}), J where the variable x" is bound in R. We can put scheme (B) into the equivalent form: (Ax, xf) (P[x, x'] <->: x' = 0 & Q{x}. v (Ex") (xf = Sx" & R{x, x\ P[r{x,x'',x>»},x»]})), (C) where x'" is bound in R. We then wish to define a property P which satisfies (C), i.e. such that (C) is an A'*-theorem. If (C) is an A'*-theorem, then, by the reversibility of R 3 so is P[v, qv]<->:qv = O& Q{v} .vqv = Sv'& R{v, v\ P[r{v, v', xrff}, v%
where v is the g.n. of a closed A-statement and qv is the number of quantifiers in the prenex normal form of' v'. Hence if P[v, qv] is an A'*-theorem, then: if qv — 0, i.e. if 'v' is an AOo-statement, then Q{v}, i.e. cv' is an AOotheorem, (see scheme (A) in Ch. 8), if
qv = Sv'
then
R{v, v\ P[r{v, v\ x"% v%
Now assume, as induction hypothesis, if P[v, v"] is an A/sH-theorem and v" ^ v' then e p' is an A-statement with v" quantifiers in its prefix when in prenex normal form then 'i>' is A-true, then if qv = Sv' and P[v,qv] is an A'*-theorem we shall have ' vJ is a closed A-statement and if v is the g.n. of (AE) <j){£} then
11.6 The definition of A-truth in A'*
523
Ch. 8). Thus the result holds for Sv' if it holds for v'. Thus it remains to A'*-prove scheme (C). The gist of the argument is as follows: We wish to define a property P such that (Ax, x') (P[x, x'] <->H[x, x\ P]), for given H. We say that a property A satisfies condition C at v, v' if A|>,y'] <->#[>, i/, A]. We want to define a property which satisfies condition C everywhere. Define K{x\X) for (Ax,x")(x" ^ x'->.X[x,x"]
AO, v"] <-> H[v, v", A] for
v" ^ v' and all
v,
i.e. if and only if A satisfies condition C for v" ^ v' and any v. Thus if K{v\ A} then A[V, v"] is the same as the property we wish to define up to v1 inclusive and all v. We first show that there are such properties and that they are unique up to v' inclusive and that they can be A-defined. Now define p ^ ^
for
(AX)(K{x',X}-+X[x,x']),
then P will be a property of order one, and P[v, v'] will hold if K{v',A}->A[v,v'] for all properties A of order zero. In particular if A[v, v"] satisfies condition C for v" < v' and all v, then K{v', A} holds and P[v, v'] will be A|>, v'] and so P[v, v'] will satisfy condition C, for all v, v'. Note that P[v, v'] is the intersection of all zero order properties which satisfy condition C up to v' inclusive and all v. The definition of the property P requires a bound property variable of order zero, hence P is a property of order one. Strictly we should speak of the statement P[v, v']. (i). (EX)K{0, X} is an A'*-theorem. From T.N.D. and R 4 we obtain LEMMA
(7ax'.Q{x})vO-+Q{v}9 by Ax. 1 and dilution by II b'
(\xxf. Q{x}) vO -> 0 = 0,
(\xxf. Q{x}) vO -> . 0 = 0 & Q{v},
524
Ch. 11 Extensions of the system A/
by dilution, and writing A for (kxx'. Q{x}) A[v, 0] -» : 0 = 0& Q{v}. v (Ex) (0 = Sx&G{v,x, A}), where
G{x,x',X}
for
R{x,x',X[r{x,xf
(1)
,x"'},x']}.
Again from T.N.D., R 4 and dilution,
From Ax. 2.2, dilution, R 3 and dilution again (Ax) (0 4= Sx v - (?{*>, a, A}) v A|>, 0]. By IIV
0 = 0 & <2{y}. v (Ex) (0 = Sx& G{v, x, A}): -» A|>, 0],
(2)
From(l), (2) by 116' A[>, 0] <->: 0 = 0 & Q{v). v (Ex) (0 = Sx& G{v, x, A}). FromProp. 16, Ch. 10, R 3 and 11/for property variables of order zero, (EX) (Ax', x") (x" = 0 -> :. (X[xf, x"] <->: x" = 0 & Q{x% V (^») (xfr = Sx&, G{x', x, X}))), whence, the result. Note that A[y, 0] is A-defined. (ii). (EX) K{v, X} -> (EX) K{Sv, X) is an A'*-theorem. From R 4 A'[0, K] <-> i?{0, /c, Z}, X of order zero,
LEMMA
(3)
where we have written A' for Axx' .H{x, x', X}, then A' is of order zero. We have the tautology: K ^ v -> . X[d, K] <->fl"{0,AC, X}: -»:. A'[0, /c] ^ ^{0, /c, Z}. -> : #c < v where Z is a property variable of order zero, hence by II d, R 3 (Ax,x')(x < y^.Z[a;^^]^^{x^a;,Z})^.^,a;O(A / [^ / ,^]<-> ->(^», a?') (x ^ v -> . X[x', x] <-> From the derived rule
. \ J° ,f ,1^
{Ag)${{Q}v
*
the last clause may be replaced by (Ax,x1,x",x'") (x^8v&x = 8x".-*.X[p[x,x",x"%x"]
11.6 The definition of A-truth in A'*
525
whence by regularity (Ax, x') (x^Sv->
.{Ex") (x = Sx" & G{x'9 x", X}) <-> {Ex") (x = Sx"
whence by 3FC (Ax,x')(x ^Sv->
-A'[X',X]<->H{X',X,X}.^.A'[V,x]^H{xf>,A
again by ^c (Ax, x') (A'[xf, x] <-> H{x', x, X}) -> (Ax, xf) (x ^ Sv -> . A'\_x', x] ~H{x',x,A'} Altogether so far we have (Ax,x')(x <
v-*.X[x',x]^H{x',x,X})^.(Ax,x')(A'[x',x]^H{x', ->(Ax,x')(x ^ Sv->.A'[x',x]<->H{x',x,A'})
using the cut to omit the A/:|{-theorem obtained from (3) by R 3 and then applying 11/and R 3* the result follows. We have used the free property variable X so that the application of R 3* is clear. Note that A[x,Sv]~H{x,v,A[x,v]} isA'*-defined. (iii). (Ax) (EX) K{x, X) is an A.'*-theorem. This follows from lemmas (i) and (ii) by R 3, the A'*-proofs have been written out in full, so that we can apply Prop. 5, Ch. 10 to rx where rv is the g.n. of an A'*-proof of (EX) K{v, X). LEMMA
LEMMA(iv). (Ax,x')(x^ 0&K{0,X}&K{0,X'}.-+.X[x',x]<^X'[x',x]) is an Af*-theorem. We should have added 'where X and X' are zero order properties. We have K{0, X} <-• (Ax, x1) (x = 0 -> :. X[x', x]<-+:x = 0& Q{x'}. v (Ex") (x = Sx"&G{xf,x,X})) -> (Ax') (X[xf, 0] ^>: 0 = 0 & Q{x'}. v (Ex") (0 = Sx" & G{x', 0, X})) by Prop. 16, Ch. 10 -> (Axf) (X[x', 0] <-> Q{x'}) by the cut, Ax. 2.2, ->(Ax,x')(x ^ 0->.X[x',x]*+Q{x%
526
Ch. 11 Extensions of the system Aj
Hence by the predicate calculus K{0, X} & K{0, X'}. -> (Ax, x')(x^0->: X[x', x] <-* Q{x'}. & . X'[x'. x]
w u We have Hence
(v). (Ax,x')(z ^ v&K{v,X}&K{v,X'}.-+.X[x',x]~Xf[x',x]) (Ax,xf)(x ^ Sv&K{Sv,X}&K{Sv,X'}.->.X[xf,x]^X'[x'~~xf\) is an A'*-deduction (see remark after lemma (iv)). K{8v,X} c y . K{v,X) K{8v,X} K{Sv,X'} K{v,X}&K{v,X'} Hyp. (Ax,x')(x^ v->.X[x',x]^X'[x',x])
since
by regularity,
(Ax, x'){x^v->. G{x\ x, X} <-+ G{x', x, X'}) (Ax,xf)(x < Sv&cx = Sx".
then (Ax, x') {x^Sv^.
{Ex") (x = Sx" & G{x', x\ X})
(Ax,xf)(x ^ Sv^1H{x',xiX}<->H{x',x,X'}) K{Sv,X}&K{Sv,X'}&(Ax,x')(x
by reguiarityj
^ Sv->.H{x',x,X} *^W>*>X$
(Ax,x')(x ^
by 0>c,
Sv^:X[xf,x]^H{x',x,X}.&.X'[xf,x~\ ^Hix^
x X
> ')'&
- Hix'> x>X) " Hix'> x>X^ \AX,X')(X ^SV-> :X\xf,x'\^Xf[xf,x']).
by
^ c,
The result now follows from the Deduction Theorem. LEMMA(vi).(Ax,x')(x < x"&K{x",X}&K{x",X'}.->.X[x',x] +->X'\xf, x]) is an A'*-theorem. This follows from lemmas (iv), (v) by R 3; this can be applied since we have written out the full A'*-proof for each numeral v on a primitive recursive plan, E-correct (see Prop. 5, Ch. 10).
11.6 The definition of A-truth in A'*
(vii). (Ax,x')(K{x,X}->.P{x,x'}<^>X[x,x']) (see remark after lemma (iv)).
LEMMA
527
is an A'*-theorem
From lemma (vi) by SFC\ where X and X' are of zero order,
(Ax,x')(K{x,X}&X[x',x].->.K{x,X'}^Xf[x',x]), hence by the reversibility of R 3, then using R 3* and R 3 we get (Ax, xf) (K{x, X) & X[xf, x]. -> (AX') (K{x9 X'} -» X'[x', x])). Thus, by definition of P, (Ax, x') (K{x, X} -> . X[xf, x] -> P{x', x}).
(4)
Again, from J^r and lemma (vi) (Ax,x') (K{x,X) -> :K{x,X'} ->
X'[xf,x\.->X{x\x\),
and so by the reversibility of R 3, using 11/, R 3 and the definition of P, (Ax,x')(K{x,X}->.(AX)(K{x,X}->X[x',x])->X[x\x]) -+.P{x',x}->X[x',x]. The lemma now follows from (4) and (5) by ^ . LEMMA
(viii).
(Ax') (P{x'9 0} ++Q{x'})9
(Ax, xf) (P{x\ Sx}^R{xf,
x, P[r{x'9 x, x"), x]}),
are A''*-theorems. By definition we have (Ax') (K{0, X} -> .X[x'9 0] ^ Q{x% by lemma (vii)
(Ax9) (K{0, X} ->.P{x', 0} <-> X[x'9 0]), K{0, X) -> . (Axf) (P{xf, 0} ^ Q{x'}).
hence by J ^
By R 3*, lemma (i) and the cut, the first part of the lemma follows. Again, by definition (Ax, x') (K{Sx, X} -> . X[x', Sx] <-> G{x', x, X}), by lemma (vii) (Ax, xf) (K{Sx, X} ->. P{xf, Sx} +->X[x', Sx]),
(5)
528
Ch. 11 Extensions of the system A/
whence by 3FC (Ax, x') (K{Sx, X} -> . P{x\ Sx} <-> G{x', x, P}) ->. P{x', Sx} ^R{x',x,P[r{x',x,x"},x]}), using the definition of G. By the reversibility of R 3,11/, II d and using R 3 we get (Ax) (EX) K{Sx, X}. ->. (Axx') (P{x\ Sx} ^ B{x', x, P[r{x', x, x"}, x]})9 the lemma now follows from lemma (iii) using the cut. This completes the demonstration of Prop. 6. COR. (i). We can A^-define a property which satisfies scheme (A) when Q, R contain bound property variables of order K. The demonstration is similar. We just need to keep careful track of the orders of properties. For instance A in lemma (i) will now be of order SK. X throughout the other lemmas will have to be of order SK. Finally P will be of order SSK, and all the lemmas are A^^-proved. D 263 qv for the number of quantifiers in the prefix of' v', when in prenex normal form. COR. (ii). P{y, qy] <->
Now this is (Ex)prf00[x, v] <-»0. It suffices to Az-prove: (i)
(Ex)prf00[x, v]-*
0 -> (Ex)prfoo[x, v\.
These follow easily if ^ is A00-true in which case it is an Aoo-theorem and so (Ex)prfoo[x,v] is an Ao-theorem. Similarly if <j) is A00-false, in which case ~ (j) is an AOo-theorem the second of (i) is easily proved.
11.6 The definition of A-truth in A'*
529
There remains the first of (i) when
~ (ifo) prf00 [x, Neg v] -> (p.
Now
(jgg, a;') (fff/oofo, v] & prfoo[x',Neg v]) (Ex, xf) prfoo [xnxfn {conj\y\Neg v] >, conj\[v, Neg v]] (Ex) prf00[x, conj [v, Neg v]].
By the Deduction Theorem in A 7 (Ex, xf) (prfoo[x, v] &prfoo[x', Neg v]) -> (Ex)prfoo[x, conj [v, Neg v]], but (d> & **> (b) —> 0 = 1 , whence (Ex)prfoo[x,conj[p,Negv]] -> (Ex)prfoo[x,g.n. of (0 = 1)]. But(^4#) ~ prfoo[x,g.n. of (0 = 1)] and so ~ (Ex)prfoo[x, conj[v, Negv]]. Thus ~ (Ex, x') (prfoo[x, v\ &prfoo[xf, Neg v]) and so (Ex) prf00[x, v] -> ~ (Ex) prfoo\xf, Neg v] and so finally (Ex)prf00 [x, v\ -> (j) as desired. What we have done so far can be put down as a detailed A7-proof of 6 y
and
~6 r — P{V ,()}+->
From these by Deduction Theorem and the cut we obtain P{v, 0} <-> (j). Now if in this we take ^{f} instead of (j) and if cr[-Jhmt t] is equal to the g.n. of 0{f} then we obtain a detailed A 7 -proof of P{v, 0} <-> ^{1} for all I, E-correct. Now take as our induction hypothesis: 1
P{o-[9htm I], qcr[3lum !]} <-> ^{f} for all I, E-correct'.
Then we have just shown that this holds for closed A-statements
530
Ch. 11 Extensions of the system A/
for all f, 77", E-correct, hence we get Picr'lWumI], q
(Ax)(^(Exf)Prfz[x\x]v
~(Ex')Prfj[x',Negx]).
(1)
Prfj [K, V] is the proof predicate for A7. We shall take A 7 in an equivalent form; namely that sub-system of A' which is obtained from A' by omitting all property variables and all rules in which they occur. This amounts to taking negation, disjunction and universal quantification as primitive and defining conjunction and existential quantification in terms of them. Let T be the truth-function for A7, then by Prop. 5 T can beA'*-defined. In order to A'*-prove (1) we first give detailed A'*-proofs of
and From (2) we get
(Ex)PrfJ[x,v]-+T[v},
(2)
T[>]-> ~T\Negv].
(3)
- T[Neg v]-+~ (Ex) Prfz[x, Neg v\.
(4)
From (2) and (3) by the cut we get T[Neg v]^>~ (Ex) Prfj[x, v].
(5)
From (4) and (5) by the cut we get - (Ex) Prfj[x, v] v - (Ex) Prfj[x, Neg v\.
(6)
11.7 Consistency of A/
531
If we can display these A'*-proofs in detail then we can use R 3 and obtain (1), which is what we want to do, we can also get: (Ax)((Ex')PrfI[x',x]-^T[x]) which is the consistency of A 7 with respect to its truth-definition. Note that this A'*-proof of (1) contains property variables which occur in T and are ejected by the cut. Similarly from (3) we get: (Ax)(~T[x]v ~T[Negx]) which expresses the consistency of the A-truth definition. In order to carry out this programme we shall have to formalize a great deal of what we have said informally about A-truth. We start with (3), we have T[v,0]^(Ex)prfoo[x,v], T[V,SK]<^:
clstat v&(prv)% = (g.n.oiA)&(Ax')T[st[m[prv],
(g.n.oix),
Numx'], K] . v .(pr v)% = (g.n. of E)
(D)
&(Ex')T[st[m[prv], (g.n. ofx), Numx'],K]. In this replace v by Neg v and note that clstatv& (pr v)% = (g.n. of A).<^>.clstatNeg v& (prNeg v)% = (g.n. oiE), and
the same with E and A interchanged,
(7) (8)
and
clstat v «-> clstatNeg v.
(9)
Thus T[Negv,SK]<-+clstat v&((prv)£ = (g.n. of A) & (Exf)T[st[m[pr[Negv]], (g.n. oix), Numx'], K]). V .(prv)£ = (g.n. of E) & (Ax') T[st[m[pr[Neg i/]], (g.n. of x), Numx'], K].
Suppose that we have shown that T[Negv,K]-> then since
(10)
~T\V,K\9
clstat v-+st[m[pr[Neg v]], (g.n. ofx), Numx'] = Neg [st \m\jpr v], (g.n. of
x),Numx']]
532
Ch. 11 Extensions of the system Aj
we get b y / J ^ T[Neg v, 8 K] -> clstat v& ((pr v)% = (g.n. of E) & - (Ex') T[st[m[pr v], (g.n. of x), Numx'], K] . v .{pr v)% = (g.n. of A) & ~ (Ax') T[st[m[pr v], (g.n. of x), Numx'], K]) -+~T[V98K\.
(11)
Note that we have ~ ((prv)% = (g.n. ofE)&(prv)? and
= (g.n. of A))
(prv)% = (g.n. of A) v (prv)£ = (g.n. of E).
The Aj-proofs of (7), (8) and (9) can be written out in full and that of (11) from the hypothesis (10). We wish to do the same for T[Negv,0]~> ~T|>,0] and then we can apply R 3 and obtain (Ax)(~Txv ~T[Negx]), the consistency of the A-truth definition. Now T[>, 0] could have been defined as clstatQQ v & a{v} = 0, where a{v} is the characteristic function of ' v\ Then T[Neg v, 0] <->. clstat00 v & a{v} = 1. Thus we have to show ~ (clstatooV & a{v} = 1) v ^ (clstat00 v & a{v} = 0), i.e.
clstatooV -» . - (a{v} = 0& a{v} = 1).
(12) (13)
But ~ (a{v) = 0&a{v} = 1) is an Aoo-theorem for all v, E-correct, hence so is (12) and hence so is T[Negv,0]->T[v,0]. Thus we have (3) for all v, E-correct. (13) is Corifa) Aoo which is thus A7-proved. We had already noted that it fails to be an A-theorem. It requires T.N.D. for the above proof, but T.N.D. can be eliminated from an A r proof.
11.7 Consistency of A/
533
Now we want to give an A'*-proof of (2) for all v, E-correct. To do this we formalize the demonstration we gave in Prop. 6, Ch. 10. To do this it suffices to give A'*-proofs for all v, v', v\ E-correct of the following Ax[v]->Tv, 1
(14)
Rl [v\ v] & T|V]. -> Tv,
(15)
Rl [v\ v\ v] & T[v'] & T[v"]. -> Tv,
(16)
2
clstat [Gen v] & (Ax) T[s[v, x]]. -> T[Gen v],
(17)
where D264 Then
Genv
for
(g.n. of (Ax)f v.
Prf[K, v] -> T[v]
(18)
has an A'*-proof for all v, E-correct. For (14) we have Ax[v] ->. OL{V) = 0 & clstat [ot{v}]
To deal with (15), (16) and (17) it suffices to give A'*-proofs for all vy,v\ E-correctof ^ T M (19) T[v'] v T[v"]. <-> T[disj[v', v"]],
(20)
(Ax) T[s[v, x]] 4-> T[Gen v],
(21)
We have already done this for (3), in a similar manner we do it for (19). For (21), in scheme (A) write Gen v for v then st[m[pr v]] = v and (prv)2 = (g.n. of A)
and clstat v
so we get
T[Gen v, 8K] <-> (Ax') T[s[v, x']],
whence
T[Gen v] <-> (Ax) T[s[vy x]].
For (20) we have to formalize Prop. 3 of Ch. 4. We give A'*-proofs for all v\v'\ E-correctof (22) T[i/, 0] v TK, 0]~T[disj[v', v"l 0], and if we have A'*-proofs for all v\ v", E-correct of T[i/, K] V T K , 0] ~T[di8j[v', v"]9 K]
(23)
534
Ch. 11 Extensions of the system A/
then we show how to obtain A'*-proofs for all v\ v", E-correct of T[V, SK] v T|>", 0] <->T\disj[v\ */'],SK],
(23')
and if we have A'*-proofs for all v', v", E-correct of T|V, K] v T [ / 3 K']++T[di8j[v'9 *>"], K + K']
(24)
then we show how to obtain A'*-proofs for all v', v", E-correct of T|V,SK] v T|>",SK']<->T[disj[v', *"],SK + SK'],
(24')
(22) is clstat [i/, v"] & (oc{v'} = 0 v a{v") = 0) ^ dstat[disj[vf, v"]] &ot{v'}xa{v"} = O. (25) We can easily give A'*-proofs for all v\ v", E-correct of clstat [v\ v"]^> clstat [disj[v\ v"]] and of
*{v'} = 0 v a{v"} = 0 <-• a{v'} x a{v"} = 0,
whence (22) follows. For (23') we have T|V,SK] v T[v",0]~B{v', tc9T[r[v'9 K,X*]9 K]} VT[V",0]
isj[v'9 v"]9 K, T[r[disj[v', v"\ K, x"l K]} by tautology, using the special construction of R and the hypothesis „-. a n (23)hence m n . .r, <-*T[di8j[v , v ],SK] as desired. We proceed similarly for (24'). Thus we have obtained A'*-proofs of (19) for all v, E-correct. Hence we can obtain A'*-proofs of (14)-(17) inclusive for all v, E-correct, and hence of (18) for all v, E-correct. Thus we have A'*-proofs of (2) for all v, E-correct, and hence similarly of (6). Thus, finally (1) is an A'*-theorem, as desired. This completes the demonstration of Prop. 7. Note that the property variables which are in T in (2) and in (3) get eliminated by Modus Ponens. COR. (i). It is impossible to eliminate Modus Ponens from A7*. More precisely A' is strictly weaker, than A'*. Suppose that we could eliminate Modus Ponens from A'*, then we could do so in the A'*-proof
11.7 Consistency of A/
535
of CoW(ii)Aj in Prop. 7. The result would be an A r proof of Con^ A7, because it would be entirely without property variables, for once a property variable enters such a proof then a property variable remains in the proof from that place to the end, but Con^Aj is without property variables. The proof may use T.N.D., but cases of T.N.D. which are free of property variables can be A7-proved. Thus we could obtain an A2-proof of Co%i)Aj which by Prop 19, Ch. 10 is absurd. C O R . (ii). The A 7 irresolvable statement can be A'*-proved. We had found in Ch. 10 §17 that if we could Aj-prove the consistency of Aj then we could A r prove the A7-irresolvable statement. Thus if we can A'*-prove the consistency of A7 then we can A'*-prove the A r irresolvable statement.
II.8
Definition ofA^-truth
Now let us consider a truth-definition for A w *. This is the same as for A(*\ We know from Prop. 14 and Prop. 11, Cor. (i), Ch. 8 that a formal system which contains negation and recursive function theory fails to contain its truth-definition, provided the system is consistent. The two propositions just referred to give two different ways of framing a truthdefinition. Let v be the g.n. of the closed JSf -statementtf>.Then we frame the two truth-definitions as follows: (a) T is a truth-definition of j£? in JS?' if Tv <->
536
Ch. 11 Extensions of the system A/
jS?'-statement T which intuitively possessed the required properties, but such that TV<->0 failed to be J5f'-provable. On the other hand it might be impossible to construct a one-place closed J2?'-statement T which even intuitively possessed the required properties, we would then, of course, be without anything to 3?'-prove. We have had various cases of truth-definitions before connected with the various formal systems we have constructed. Aoo foils to contain its own truth-definition, because this is a general recursive property which fails to be primitive recursive, and AOo can only deal with primitive recursive properties. Ao contains its own truth-definition and is consistent. Ao is without a negation, so the contradiction which would be forthcoming in a formal system containing negation and recursive function just fails to arise. A, A 7 both have the same truth-definition, but this fails to be expressible in them, because the set of A-true A-statements is a productive set and such a set fails to be arithmetically definable. We shall find that A^-truth can be A^-defined. This we will now discuss. The definition of A(Ac)-truth is by induction on the construction of the statement. (i) the A^-truth of 0{(Xg.^{£})a} is the same as that of ^{^{a}}, so we may suppose that rule R 4 has been applied as long as possible, we then say that the statement is in \-normal form. (ii) closed Aoo-equations are A(/c)-true if and only if they are A00-true. (iii) ~ <j> is A(/c)-true if and only if ^ fails to be A(/c)-true. (iv) <j>f V <j)ff is A(lf)-true if and only if ^' is A(/c)-true or if $" is A^-true. (v) (A g) {£} is A(*>-true if and only if
hence ^{A} will be of the form where ^{FJ is a closed A^-statement-form of order less than or equal
11.8 Definition of A^-truth
537
to that of the variable S, thus any property variables there may be in ^{FJ- are bound and of order at least one less than that of the variable S. All the clauses in the definition of A^-truth are the same as in the definition of A-truth except the first and the last, the last clause is far more complicated than anything we have encountered so far. In it we pass from (AE)
which, in general, is a more complicated statement. According to our rules of formation the following will be A^-statements AX0l,A(X0llg)9A0l{-kX0i.AX0l),
Eol(kXot.(~
AXot&EX'J)etc.
of these the third is closed, the others open. According to our A^-truth definition we see that the third one is A(A:)-true if and only if -4(X£. ${£}) is A(lc)-true for every
is without free occurrences of S. Thus ^{A} is of the form
we then put this into X-prenex normal form. Suppose that the original X-prenex normal form was (QE)(£ifdc/)(pf{E,dcf}, then the new one is:
538
Ch. 11 Extensions of the system Aj
") 0"{£', $"}, where (&'$') is a prefix of quantifiers all of order less than K or numerical and (&"$") is another prefix of quantifiers all of order at most that of the variable S. The second prefix is, in general, longer than the first prefix, but, in a sense to be described, it is simpler. The prefix of
where ji is the greatest order of one of the new batch of quantifiers. Now let us order #$AC-tuplets by last differences, so that {0, d',6",..., 6^)} < {77, 77', 77", ..., 77^)} if and only if 6M < n^ and #(A) = 77(A) for /i < A ^ SK. We then say that the tuplet on the left is inferior to the tuplet on the right. Thus in removing the initial quantifier of (j) we pass from an ordered tuplet to another one which is inferior in the order of the removed quantifier. Thus the process will eventually stop. I t is thus a process that we would expect an oracle to do for us. 11.9
Scheme for an A^K)-truth-definition
We will now find a scheme satisfied by the A^-truth-predicate. Let T[77(*>, T ^ D , 77^ 2 ), ..., 77', 77; V]
hold if and only if v is the g.n. of a closed A(Ac)-statement in X-prenex normal form which is A^-true and has nM initial quantifiers up to and including the right-most one of order K, the highest order allowed in A^K\ and has 77(Af~1), initial quantifiers after the right-most quantifier of order K and up to and including the right-most quantifier of order (K — 1), and so on, till 77 is the number of numerical quantifiers to the right of all the property quantifiers. Then we shall have D 265 LPK v for v is the g.n. of a closed A(/c)-statement in X-prenex normal form. D 266 l.p. v for the g.n. of the X-prenex normal form of v. mv is defined in D 232.
11.9 Scheme for an A^-truth-definition
539
v & (((!# = (g.n. of AJ) & (Ax) 1
(
(prop^x-> Tiy^ ),..., TT^\ n ^» + h^x, v],..., n + /^|>, j/]; ^ , j/j?, mv]]) v ( ( ^ = gr.^. of Efl) & (JEte) (propfx 1
^ ),...,77^),TT^""1) + h1[z,v]9...,n + h[x,v]; /i can be explicitely denned, T f l , ^ 1 * , ...,n; v]->LPKv&({Ax) (v% = g.n. of AK)& (prop^x -> T ^ ^ 1 ) + ^[a?, p],..., TT + hK[x, v]; l.p.subst[x, vf, mv]]) (&) 1
V {({vf =flf.n.of # J & ( ^ ) T ^ ^ ) + fcifo !>],..., *r + AJa;, p]; Z.^). subst [x, v£, mv]]))),
here h2[x9x'],...,hK[x,x'] are primitive recursive functions of x,x'. Scheme (E) is much more complicated than any scheme we have had so far but clearly we ultimately reduce T O r ^ , ^ 1 * , ...,TT; V] to T[TT; V], where T[n; v] is the truth definition for A7. 7. A^-truth can be A{SK)*-defined. We proceed by induction on K. We have by Prop. 5T[x;y] can be A'*defined, we assume that A(/c)-truth, i.e. T[x^K\ ...,x;y] can be A^***defined and show that A ( ^-truth can be A(SSK)*-defined, i.e. PROP.
satisfying scheme (E), but with SK instead of K, can be We proceed as in Prop. 5, the main difference is that here we have a set of parameters instead of just one, there we had P[x, x'] with the single parameter x, here we have T[x,x', ...,x(Slc);y] with the parameters x,x'9...,xf-SK\ We want to define a predicate T[x,x', ...,xf-SK);y] which satisfies: where T is bound in H. As in Prop. 5 we define K[x,x',...,«<«>,X] T[x,x', ...,afi\y]
for (Ay,z)(z ^ x-> for (AX)(K[x,x',
.X[z,x',...,x
...,xl«\X]-+X[x,x',
...,afi\y\),
540
Ch. 11 Extensions of the system A/
where X is of order 8K. We then proceed to A(Sfiflf>*-prove 8 lemmas following those in Prop. 5, the final result is that T, as just defined, satisfies scheme (E) in A(Sffif/c)*. We use Cor. (ii), Prop. 5 to keep track of the orders of property variables. For instance, since T[x;y] is A'*-defined, then T[x,x'\y\ is A"*-defined, and so on. As in Cor. (ii), Prop. 5, we can show that T as just defined is an adequate truth-predicate for A^K\ In a similar manner we can define OoW(ii) A(^ and show that this can be A(S/c)*-proved, following Prop. 6. Now let us look at A(w), the union of all the systems A(/c)for K = 0,1,2,.... It will be convenient to alter the notation. Instead of T[x,x;...,afi*;y]
write
T[(x9x'9...,aP>),y]9
and similarly for K. Note that the scheme that T is to satisfy is of the form (h and s are primitive recursive): T[(Sx)nx', y\~H\y,
T[(x)nh[x, y], s[x, y]]],
(i)
where x is bound in H. Then we define K[(x)nx', X] ~ (Az, y)(z*zx-*.X[(z
>n x',y]~ H[y, X]),
T[
(ii) (iii)
The orders of X and y are the same. Now an expression with a property variable of undetermined order is absent from our symbols. However we can always raise the order of a property variable in the definition of T provided K[(x}nx', X] fails for all X of order greater than ord y. We could then replace the property variable X in the above definitions of K and T by X^ the property variable introduced in forming A(w)' or A(w+1). But let us make the definitions of K and T above but with the variable X^ of order a), then we can carry through an A(o>+1)*-proof that T as defined by (iii) satisfies its defining scheme (i). We can also A^+D^-prove that it is an adequate A(w)-truth-predicate. And so we can go on through the constructive ordinals. I I . i o Truth-definitions in impredicative systems Now let us look at the situation in impredicative systems, that is systems in which we allow impredicative properties. Here, in the simplest case,
11.10 Truth-definitions in impredicative systems
541
we need only one kind of property variable, and a property can contain bound property variables, so that a property can depend, for its definition on all properties including itself, e.g. A£. (AS) $£{S£}. To define truth in an impredicative system we should require: (AE)(fi{Eot} is true if and only if 0{Aa} is true for all closed properties A; but such a property could be \£.(AE) 0{S£} itself. Thus to decide the truth of (A 3) ^{Ha} we should be referred to, among other things,
11.11
Further extensions of the systems A(/c)
Further extensions of the systems A(/c) may be obtained by adjoining variables for properties of properties, that is variables of types o(ot), o(ott), etc. We already have a constant of type o(ot) namely A, the universal quantifier, so that AAL is a statement in AM. AAt is read ' At is a universal property', i.e. every natural number has property A r The advantage of this type of extension is that we can then make statements about properties of a more general kind than we could before. Formulae of these types would be: XX.^{X}, XXX'.(j){X,X'}, etc. The resulting system would allow us fuller use of the abstraction symbol. In fact,
542
Ch. 11 Extensions of the system A/
we could add variables for any type that we can get from the means at our disposal. So far our systems have been limited in that we have only used a few of the types at our disposal. The difference between predicative and impredicative systems would still arise, and to avoid it we should require the concept of order for properties of properties. But one fails to reach finality. If we introduce an enumerated sequence of variables of every possible type available with the symbols at our disposal then we shall still only have an enumerable set of properties of type oi, these we can enumerate and by a diagonal process obtain a property of type oi different from any in the enumerated list. It is our form of introducing universal quantifiers that restricts the possible things of a given type to being enumerable. If we want to talk about all possible properties of type oi whether we had a formula for them or were unable to find one, in a given system, then we should require a different method for introducing the universal quantifier, probably somewhat as in the system A. But it seems somewhat odd to want to talk about things which are unnameable in the system (in the language used). This introduces us to the idea of starting with an unbounded set of symbols, perhaps one for each possible thing of a certain type. But this seems a bit absurd, because we are unable to use an unbounded set of symbols in any case, though we might be able to describe certain processes to do with them. But if we want to construct a language, we want to do things in that language itself and we want to avoid the use of another language to assist us to express what we want to express. Any extra equipment that we need in order to describe processes using an unbounded set of symbols should have been incorporated in the original construction of the system. Thus it would seem that formal systems with an unbounded set of symbols are impossible. Compare our discussion on Turing machines when we argued in favour of a displayed set of symbols, instructions, etc. g.n.'s are impossible for an uncountable set of symbols. The hierarchy of systems A(/c) for K = 0,1,2,... can be continued into the constructible ordinals. Thus A(Sa) is obtained from A(a) by adjoining new sorts of variables X(ca), X ( a u ) ,... of type (ot) and order a, and quantifiers for them. But if a is a limit number then A(a) consists of the union of all the preceeding A<# for /? < a. When we adjoin a new sort
11.11 Further extensions of the systems A(/c)
543
of variable in these ways we also adjoin new rules II d, R 3* for their use. Useful systems are A(w) and A(eo>, where e0 is the first e-number. 11. 12 Incompleteness of extended systems To show that the extensions of A 2 are incomplete and contain an irresolvable statement we require a slightly different argument to that we have used so far. P R O P . 8. AM* contains an irresolvable statement, if it is consistent. Let Pr/(/c)* [A,/£] be the proof-predicate for the system A(/cr)*. Then M* [A,/^] is a general recursive function of A and JLC. Consider:
(i) {Ax') ( - Pffd* [x\ s[x,x]] v {Ex") {x" < x'&Prf^* [x",Negs[x,x]])), let its g.n. be d, and let v be the g.n. of: (ii) (Ax') {~ PrfM* [x\ s[0,6]] v (Ex") {x" ^ xf & PrfW* [x",Negs[d,d]])); (iii) (ii) is equivalent to (Ax') (- Prf<*)* [xf, v] v (Ex") (x" ^ x' & Prf^* [x", Neg v])), since v = s[d, 6] is an AOo-theorem. (iii) says that if (ii) is an A(^*-theorem then there is an A(/f)*-proof of the negation of (ii) and this A(/c)*-proof has a lesser g.n. than the A(^*-proof of (ii) itself. Hence if A(/c)* is consistent with respect to negation (iii) fails to be an A(/c)*-theorem. If (iii) is an A(/c)*-theorem then so is (ii), since v = s[6,6] is an Aoo-theorem. Hence (iv)
PrfM* [T7? p] for some numeral n, and since AM* is consistent
(v)
(Ax") - PrfM*[x*,Negv]. Hence from (iii) - Pr/M* [n, v] v (Ex") (x" ^n& Prj™* [x"9 Neg v]), but this is the negation of ((iv) & (v)). This is absurd, thus if A(/c)* is consistent then (iii) fails to be an A(/c)*-theorem.
544
Ch. 11 Extensions of the system A/ Now suppose that the negation of (iii) is an A^-theorem. Then the negation of (ii) is an A(/c)*-theorem, thus:
(vi) PrfM* [n, Neg v] for some numeral n. From this we get {Ax') (x' > n->(Ex") (x" ^x'& Prf*» [x", Neg v])). If A(*>* is consistent with respect to negation then ~ Pr/W*[0, »],..., ~PrfM*[n - 1, v] (/f)
are A *-theorems, whence: (Ax1) (x' < 7T -» - PrfM*[x'9 v])
(vii)
is an A(/c)*-theorem. From (vi) and (vii) we get (Axf) ( - Pr/W* [>', y] V (Ex") (x" ^x'& Prf[x", Neg v]))9 which is (iii). This again is absurd, hence the negation of (iii) also fails to be an A^*-theorem. This completes the demonstration of the proposition. This gives us a standard method for producing an irresolvable statement from a given consistent system.
11.13 Real numbers The systems A(Ac) give rise to various orders of analysis. We have already explained how the rational numbers, positive and negative, can be represented by ordered triplets of natural numbers, and so can be discussed in Aoo. Properties could have been called classes of natural numbers, it is a matter of taste whether we use the word ' property' or the word' class', thus we could consider classes of rational numbers, because in our construction of ordered triplets there is a (1-1)-correspondence between natural numbers and ordered triplets of natural numbers, thus a property could be considered as representing a class of rational numbers, or a class of natural numbers, the context should make it clear which was intended. In the system A(*> we introduce real numbers thus: D 267
Sect X
for
(Ax, xf) (Xx & xr < r x. -> Xxf) & (Ex) Xx & (Ex) - Xx & (Ax) (Xx -» (Ex') (x
Xx')),
11.13 Real numbers
545
this defines a Dedekind section; here < r is the relation of order between ordered triplets (see Ex. 7, Ch. 5). The last clause in the definition limits us to sections without last member, this is a matter of taste, it means that a section is uniquely represented. If Sect A then A is called a real number of order K, where K is the order of A. We shall find that in a first course in analysis we can get on without using bound variables for sets of real numbers if we state our theorems as meta-theorems. We did this in Aoo where we stated the commutative law of addition as : a + /? = /? + oc, where a and /? are numerical terms. We can define a set of real numbers by abstraction thus: XX. ({X} & Sect X), the set of sections which satisfy
l.u.b.'kX.(
for
Xx.(EX') (X'x&
The union of all the sections which satisfy
^sl.u.b.(kX.(0{X}&
SectX)),
(ii) (AX) ((X < 8 l.u.b. (XX.
SML
546
Ch. 11 Extensions of the system A/
^ is a theorem of the required order and is without free occurrences of X. Thus the real numbers of order K are incomplete. This difficulty is overcome by using the system A(w). A set of real numbers in A(w) will be of some order given by a natural number, say /i, the l.u.b. of the set (if bounded) will also be represented by some greater natural number A hence it will be in A(A) and so in A(w). Thus A(w) will be complete. In general, analysis in A(a), where a is a limit number, will be complete, but if a is a successor then A(a) will be incomplete. Classical analysis is impredicative. There we have only one sort of variable for classes and so the class given by l.u.b. depends (on account of the bound property variable) on all classes including the very one defined. This is a blemish in classical analysis and is avoided in our case by having various orders of properties. The resulting analysis is naturally more complicated. Having defined the l.u.b. of a non-null bounded set of real numbers we can then define Urn and Urn and so limits, but in general these will be of higher order than the members of the set or sequence. Weirstrass' theorem will follow in A(w), but will fail in A M for certain sets. Functions of real variables will be given by 7>x. (j){x, X}, the domain of the function is given by the set of X for which SectX&8ecfkx.
PROP.
-*{Ev) (AXS) (EYS) (0, ^SXS ^sls: -> .Xs.#a, Ys}
where Xs is a variable for a real number of order K; 0S, ls are the real
11.13 Real numbers
547
numbers zero and one respectively <s, +8i — s are order, addition and s
reciprocal for real numbers, we are supposing that (AX.) (Os^sXs^sls.-*.Sectlx.
<j>{x, Xs) & Sect Xx. f{x, Xs}).
(AXS)
(EX)(SectX&
for
Xx.({x,X}St8ectX).
{ Write HJiUvVa] for
(AXs)(EYs)(Us^sXs<sVs.->.Xx.
this is only wanted when; 0s ^ s Us <SVS < sls.
Hp -> (AX8) (0s^sXs<s hence
Mins[ls, \x. f{x, Oj]. -> . \x. <£{x, Oj
Hp -> (EVa) (Ev) Hv[0s, Vs],
i.e. the set of real numbers \x. (Ev) HV[Q8, V8], If two overlapping intervals have property H then so does their union, in fact if one has property H for v' and the other has property H for v" then their union has property H for Max [V, v"]9 thus: 0s ^sU's<sU:<sV's<sV"s^sls.^:
(Ev') HV,[U'S, V's] & (Ev") HAU"s,V"s\^ (Ev)Hv[U's,Vs\.
Again Hp -> (^Fs) (o, < SVS < . ls -> (E/i, v) Hv \vs-„ ]f s, Vs + , ^ s 1) , ^
-> (AVa) (0, <SFS < s 1.. -> (.4Fs)(0s <SVS <sls. 18-2
548
Ch. 11 Extensions of the system Aj
Thus
->(Ev)Hv[Os,ls] as desired. Note that the demonstration is in A^SK) on account of the use of l.u.b. The ordinary first course of analysis can now proceed without difficulty, series, limits, Biemann integral (best done by dividing interval of integration into equal parts), differential calculus, still without the need to introduce variables for properties of properties or any higher order variables, these would correspond to variables for classes of classes, etc., in our case sets of real numbers. This is because in the elementary parts of analysis we need only use free variables for sets of real numbers, so that our theorems can be stated as meta-theorems. We did this in Aoo, as mentioned before. In the ordinary development of analysis, having developed the theory of series and limits, we go on to the Riemann integral and the differential calculus, and then to the Theory of Functions of a Complex Variable. This requires the construction of complex numbers. These are defined as ordered pairs of real numbers with special definitions of equality, addition and multiplication; an ordered pair of real numbers is represented by: D269
{{X,X'}} for *x.(X[)X')9
i.e. \x.{Xx v X'x),
this is only required when Sect X & Sect X'.&.{Xx-+
Evx). &. (X'x -» Odx).
One property gives rise to two other properties, the one applying to the even numbers and the other applying to the odd numbers, similarly from two properties we can get one. D270 Evx for 2/x, i.e. 2 divides x, D271 Odx for 2^x, otherwise. Having defined the complex numbers we can proceed to develop the theory of functions of a complex variable until we come to the definition of a simply-connected domain. This requires quantification over functions of real variables. The crucial clause is: c any two points of the set can be joined by a continuous curve lying in the set'. This requires 'there is a
11.13 Real numbers
549
function...' the function, of course, gives the curve, and to express the crucial clause we require quantification over function variables. Thus at this point we are forced to introduce variables of more complicated types. A function of a real variable is of type OL(OL). Particular functions are given by D272
{E\X)${X}
\X .\x.(E\X') for
(
(a)
{EX) (&{X} & {AX') (0{X'} -> (Ax) (X'x *-> Xx)))9
read 'there is exactly one X such that 4>{X}\ The same with a suffix s on all the X's would read 'there is exactly one real number Xs such that {Xs} & Sect Xs\ We want (a) with the suffix s on all the X's. As one develops analysis every now and again one will require to introduce variables and constants of more and more complicated types. One can try to keep the development predicative or one can take the easier course offered by the impredicative approach. But sooner or later one comes across something that can be done using Mult. Ax. and maybe fails without its use. In the predicative case, given, say a set of real numbers of order K, we can always pick out a member of the set, though the choice fails to be defined in the system we are using. In the predicative case there is an enumerated set of real numbers of order /c, enumerated say by increasing g.n., so we can always pick out the one in the set with least g.n. But this process fails to be definable in any of the systems A(/c). Also there may be cases when we are unable to decide whether a given real number belongs to a given set of real numbers. Thus in any case we should require a function of type i(ot) which associates a real number with each property, and an axiom for its use, e.g. X(!FX), ^X being the natural number picked from those having the property X, etc. for real numbers. Having developed the theory of functions of a real variable to a certain extent we next require the Lebesgue Integral. In any of the systems A(a) there are only an enumerable set of real numbers, in fact we can enumerate them by increasing g.n. This is possible because a real number is a well-formed formula of type oi and we have an effective method for testing this; the well-formed-formula (w.f.f.) must further satisfy the condition of being a section, this might be difficult to decide; but, if m kx.^>\x\ denotes a w.f.f. of type oi then, provided (Ex)
550
Ch. 11 Extensions of the system A/
as we get by enumerating those of Ax
l.u.b.'kx
for
(
and _i
S -Kx((f>{x, X} & Sect [X, \x<j){x, X}]) g.l.b. S
for
l.u.b. -kx (${x, X} & Sect [X, \x
and similarly for S , the Riemann integral is then expressed by means of o sequences. When we come to the Lebesgue integral we want to define the inner and outer measures of arbitrary sets of real numbers (to take the one dimensional case). Now all the sets of real numbers that we can define are enumerable and as such should have measure zero. So it looks as if the whole theory of Lebesgue integration will be unexpressible in any A(a). Now a set of real numbers is of measure zero if and only if it can be covered
11.13 Real numbers
551
with a set of intervals whose total length is arbitrarily small. We can cover up all the real numers in A(a) by the set of intervals
., + «^ ! J where v is any numeral and A^ for /i = 1,2,... runs through all real numbers. But this set fails to belong to any A(a) and so we are unable to use it. Similarly for the set of real numbers given by J£s
mXs(f>\Xs} el =
lf
X
for
s\js<sXs<s
l.u.b. 8 J ^ Xs9iXs}>
—^ s
0 otherwise. In order to keep within our formal system A(a) the numerical term must be explicitly defined in that system. Thus we shall have to define eM as follows
where 6 is the g.n. of Xs (^S<SXS<S g.n, of a ^ > proof of ed\
^ ± M c Xs
552
Ch. 11 Extensions of the system A/
The outer measure is defined by D274
mXs
ls]-Xs
for a set in the interval [0s, 1J, similarly for other intervals. Note that in the definition of e^ we are unable to replace prf by truth because the truth-definition of A(*> is outside A(/f). As we pass through the hierarchy of systems A(AC) so we obtain more and more real numbers. Thus an interval XS[AS < s Xs < s A^] should be given an order, namely the order of the variable Xs. The order of the set 3£s(f){Xs} is the order of the variable Xs even though the order of
so that the open interval (As, A8) of order K lies in the set Xs
(6)
Now this A(SK) free variable proof (i.e. a proof in the system obtained from A(SK) by allowing free variables) will be without bound variables of order K because Xs(f){Xs} is of order K. Thus we will still have a free variable A(7r)-proof if we replace the free variable Xs everywhere by one of higher order TT. Thus if we can A ( ^-prove (a) then we can A^-prove the corresponding statement when the variable Xs is replaced by one of higher order TT. Similarly for any two ordinals (constructive, of course). Thus if an interval of order K is contained in a set of order K, then the corresponding interval of order n greater than K lies in the corresponding set of that higher order. Where the set %8<j>{X^ with Xs of order K corresponds to the set Ys
11.13 Real numbers
553
to lie in the corresponding interval of that higher order then that real number also can be proved in a suitable order to lie in the corresponding set of higher order. 11.14 ^ e analytical hierarchy We have already noted that instead of property variables we could have used function variables. Set variables are much the same as property variables. In the theory of Turing oracle machines we used functions rather than sets. We could build up a similar theory using sets instead of functions, the oracle would then tell us, on demand, whether a natural number was in a certain set or otherwise. But the theory of Turing oracle machines seems to go nicer using functions rather than sets, and most of the literature uses functions rather than sets. In the same way the analytical hierarchy, about to be described, seems to go more easily using functions rather than sets. In Ch. 9 we defined a hierarchy of arithmetically definable sets of natural numbers, and noted that we could define a similar hierarchy of /-definable sets of natural numbers. We wish to extend this hierarchy to sets whose definition involves bound set or function variables. In general these will fail to be arithmetically definable. In our account of the extensions of the system A 7 we introduced property variables of various orders. Similarly we shall consider functions of various orders. In our systems only the primitive recursive functions get explicit definition. General recursive functions were given an implicit definition as
is A-true. For the function / we could take \xiy
where
(Ax) (Ey) {x,y)& (Ax,y,z)
(${x,y]& ${x,z}.->y
= z).
Thus a function defines a set of ordered pairs. Similarly functions of order 8K are defined where
554
Ch. 11 Extensions of the system Aj
In the literature most of the work about the analytical hierarchy has been done using function variables rather than set variables (or property variables). We define systems A(fK) as for the systems A(Af) except that we use function variables instead of property variables. The sets of natural numbers that we are concerned with are given by statements in A(/} with n free numerical variables and have all function variables bound. These statements can be put into prenex normal form. We consider first the system A'f9 by using ordered tuplets we may consider that all functions are one-place functions. Write A1, E1 for function quantifiers and A0, E° for numerical quantifiers. Then the prefix of an A/-statement in prenex normal form will consist of a sequence of ^415s, Ev&, A°'s and JE7°'s, in some order. In this sequence omit all numerical quantifiers and the resulting sequence of function quantifiers is called the reduced prefix. A H/Sv-prefix is one whose reduced prefix contains v alternations of quantifiers and begins with Ef. A U'Sv-prefix is one whose reduced prefix has v alternations of quantifiers and begins with A'. A ^-statement is one whose reduced prefix is a statement with aSJprefix. A Restatement is a statement with a n^-prefix. 2^ is the class of all 2^-statements, 11^ is the class of all II ^-statements. SQ and Wo are statements with empty reduced prefix, thus they are A-statements. We now generalize what we did in Ch. 9 P R O P . 10. SJ u Urv cz S^n
11^.
The demonstration is as for Prop. 2, Ch. 9. P R O P . 11. The following prefix transformations are permissible. I.e. in each case, for any statement with a prefix in the left column there is an equivalent statement with a prefix in the same row but in the right column, and there is an effective way of finding this statement.
f...A°A°...
...A0...
11.14 The analytical hierarchy
r...A1A1... *
'
I
T7T1 77T1
...A1... T7T1
...JDJ
0
(iii)
{
A ,E°...
555
...
1
A ...E1...
t:
iv
(i) (ii) where (iii) (iv)
We have already done this in Ch. 9. Replace ...(Af)(Ag)...
Similarly for EOAX. Or take negations using ~ (p instead of
(Eg) (DVi£...,i[0, v, m, f£, £, 0] = 0)
556
Ch. 11 Extensions of the system A/
for some 6 (see Prop. 1, Ch. 6). Thus PROP.
12. A YS^-statement is equivalent to = 0),
where 0/ begins with E' and ends with A', and consists of alternate kinds of quantifiers. A^2V+1-statement is equivalent to
m)(A^x)(Unl,mmtl[e\x9\)9^'lx'l0]
* 0),
where jQ' begins with E' and ends with E' and consists of alternate kinds of quantifiers. Similarly for ^-statements. The first part follows at once on contracting a couple of JS7°'s. The second part follows since ~ ^{f, v, m} is recursive in f thus
is equivalent to
~(E°x')(Uni^tl[d,x,t),W,x',0]
(E°xf) {UnJl_lt[df,x,\)9fa',x'9O]
= 0)
= 0) for some 6'.
Using this we see that we get a prefix (Q/f)(A^x){A^x')(Un[...'\ + 0). Contracting the two J.°'s the result follows. Prop. 12 allows us to enumerate ££- and IlJ-statements. We denote the set in T\v given by (&f)(^)(ff»£...,1[0,...] = O) by SU0], and the set in S^+i given by (£l'f)(^a;)(C/^)...>1[e',...] = O) by ZL+1[0']. Similarly for IXj-sets. The corresponding A'-statements with free variables shown are denoted by S i [6; /,..., JA>], Sln-iP';"'.-." 6 *]. Similarly for IlJ-statements. P R O P . 13. The following prefix transformations are permissible in the sense of Prop. 11: C...E' ...E° (i) ...A' ...A"
x) (E'g) (ii)
(E'g) {A'f){E»x)...
11.14 The analytical hierarchy
557
where in each case (i) the quantifier changed is the final quantifier in the prefix, and in (ii) in each case the matrix of the statement inprenex normal form contains f only in the form fj. (i) Suppose we have ...(E'f)(BPx)(Unlmmm9l[d9v'9...9fx9x9O]
= 0)
as an equivalent statement, this in turn is equivalent to ...(Ey)({Wx)
Unlmmm9l[09v'9...9y9x9O]
= 0)&wy = x).
Now contract the two existential quantifiers. Similarly for ...A'. (ii) Suppose we have where the only occurrences of/are marked. This is equivalent to ...{E'g) {A'f)(EH) ...{.,fi,.,x,.,-Kygifi,y),.}. Similarly for the other case. Note Mult. Ax! This proposition sometimes enables us to reduce the place in the hierarchy of given 2J- or IlJ-sets or statements. P R O P . 14. There exists a hierarchy of A'-definable sets of lattice points in &K defined by A'-statements of the forms:
= 0,1,2,... having K free numerical variables. Each type contains a set which fails to be equivalent to any set in any of the earlier types or to any set in the type in the same column if any. The demonstration is similar to that for Prop. 4, Ch. 9. P R O P . 15. 2} n 11} contains a set outside the class of arithmetically definable sets. Consider the set consisting of the g.n.'s of true closed A-statements. This set is a II}-set by Prop. 5, Ch. 11. (This proposition was demonstrated using property variables, but could equally well have been done using function variables. ) If T[v] is the A^-statement which says that v is the g.n. of an A-true closed A-statement, then Clstat v & ~ T[Neg v] is also a truth definition for the system A, but this is a 2}-statement. Thus the set ofg.n.'sof closed A-true statements is a set in 2} n II}. But we had already found that this set is outside the class of arithmetically definable sets.
558
Ch. 11 Extensions of the system A/
This is one point where the analogy with the @P$, 2lKv breaks down. D275
AJ for Sjn IIJ.
Instead of using variables for functions or sets or properties we could have used variables for real numbers in [0,1). Because the binary expansion of a real number in [0,1) defines a set, namely the set whose characteristic function is/{r} where the binary expansion of the real number is ./{l}/{2}.... Because of this we use the term 'analytical 5 , the final ' a l ' is added to distinguish between 'analytic' as used in mathematics. II.15
On the length of proofs
P R O P . 16. For each primitive recursive function / , there exists an A7theorem
Prff[y,x]
for Prfz[y,x]& (Az)y ~ Prfj[z,x],
so that if Prf¥\0, n] then 6 is the least g.n. of an A7-proof of (n \ Similarly for A'* and Prf'*M[y, x\. Clearly Prff[y, x] & Prff[z, x]~y = z. Define
Prf[y,x]
for
(i)
M
Prf'* [y,x]&(Ez)fy+1Prf¥[z,x].
If the Prop, fails then for some primitive recursive function f Prf gives a proof-predicate for A z . Now consider (Ay) ~ Prf[y,s[x,x]], let its g.n. be K, let v be the g.n. of (Ay) ~Prf[y,s[K,K]], then v — S[K, K]. AS before cv' is an irresolvable Aj-statement, but each of ~ Prf[n, v] is an A7-theorem for n = 0,1, Also from Cor. (ii), Prop. 6 of this chapter this irresolvable statement is A'*-provable. Let its minimal A'*-proof have g.n. n. We have \-Pff**[n,v] (ii) \~AMV)
(y < fr -+ ~ Prffy, v\).
Also hAj(Ax) - Prf[x, v] <-> (Ax) (Prf'*[x, v] -> (Ay) (y
(iii) ~Prf[y, v]))
+-> (Ay) (y
11.15 On the length of proofs
559
Whence from (iii) \-Aj (Ax)Prf[x, v], which is absurd, because it is our A7-irresolvable statement. C O R . For each primitive recursive function f there exists an A 7 theorem <j) such that the A 7 - and A'*-proofs of $ of least length A7, A'* respectively are such that Xj > /A'*.
We proceed as before except that we define Prff'[y>»] for Prfz [y, x] & (Az) ((wz < my v (mz = my -» z < y)) -> ~Prf[z,x]). H I S T O R I C A L REMARKS TO C H A P T E R
11
The method of extending a formal system by continually adjoining a true irresolvable statement was first done by Turing (1939), he was trying to get a complete system. The introduction of variables of higher type is due to Russell (1905), he introduced type theory as a method for eliminating the paradoxes which were appearing in Cantor's set theory. He also introduced the ramified theory of types, wherein each type was divided up into orders. Here we have only one type of property but an unending series of orders within the type. We close the book before introducing higher types, viz. properties of properties, etc. Rule R3* has been used by Lorenzen (1955) and others but without the proviso that the general proof be demonstrated correct in some other system. Something similar to Prop. 1 has been found by Wang. The form of definition by induction used in Prop. 5 to show that Atruth may be defined in a higher system is due to Wang (1953). The proof of the consistency of A 7 in a higher system is due to Tarski (1933) and Bernays (see H-B 1934-6). Predicative and impredicative systems were first discussed by Poincare (1905), he strongly objected to impredicative systems. Russell tried to keep his work to the predicative case. The formation of languages with an unbounded set of symbols was first considered by Godel (1946) and Henkin (1949); see in this connection work on languages with statements of unbounded length by Karp (1964) and Barwise (1968), where further references are given. The existence of an A(*>*-statement, given in Prop. 9 is related to Rosser's version of Godel's construction of an irresolvable statement.
560
Ch. 11 Extensions of the system Aj
Real numbers defined as sections were first thought of by Dedekind (1892, 1893), see also (1909) and Wang (1963) p. 73fif. Other constructions of real numbers as fundamental sequences, binary decimals, etc. are less suitable for our purposes. Various writers have written out real number theory analysis in full in the notation of some formal system. The first of these was Peano (1897), followed by Whitehead and Russell (1910-13), since then there have been several others notably Hilbert and Bernays (1934-6), Rosser (1953), Lorenzen (1955), and from the constructive point of view, Goodstein (1961) and Klaua (1961). The development of real number analysis as we now know it is due to Cauchy, Weierstraus, Abel, Lebesgue, etc. The analytical hierarchy has been studied by Kleene (1955, 1959, 1962, 1963), Addison (1955, 1959, 1962a, 19626) and Spector (1958, 1959). Prop. 16 on the length of proofs is due to Godel (1936), we follow the account given by Mostowski (1952). EXAMPLES
11
1. Devise a system of g.n.'s for A(/c) in such a way that there is a (1-1)correspondence between symbols and g.n.'s. Do the same for Aw. 2. Give the full definitions D 263-266 inclusive. 3. Obtain a proof-predicate for A(/f)*. 4. Discuss the consistency of A(/c)* on the same lines as the consistency ofA z . 5. State and investigate the analogue of Prop. 18, Ch. 10 for A(/c)*. 6. Find the truth- values of (l? ot X ot ).4X oi , and generally of (PX0l)QX0l, where P, Q are either A or E of appropriate types. 7. Find the truth values of (P0lXJ (P'X0l vP"XJ, (PmXJ (P'0lX'0t) (Q'XO&QX'O), where P', P", Q', Q" are A or E. 8. Prove Sectl.u.b.{'kX.^{X}&,SectX). 9. Prove (i)
Examples 11
561
10. Express the axiom of choice in AH 11. Define Gon^A^ and show that it can be A(8/c)-proved. 12. Discuss Con{n)A^\ 13. Develop extension of AOo using variables for classes and the esymbol. Discuss predicative and impredicative classes. 14. Develop extensions of A z using function variables. Define predicative and impredicative extensions. 15. Complete the demonstration of Prop. 3. 16. Define (X + 8Xf)
for
Ax.(Ex',x")(x = x' + x
Now define successively \X\s, —SX,X — SX', — ,
then
XxsX'
for I^ + ^ I S - ^ W
Show that all these are sections, and prove the commutative, associative and distributive laws of addition and multiplication. 17. Prove Wierstrass' theorem that any bounded set containing an unbounded number of members has a limit point. If the set is of order K what is the order of the limit point ? 18. Prove that a /cth order function which is continuous in a closed interval is uniformly continuous in that interval. In which A(7r) is this proved? 19. Prove that a /cth order function which is continuous in a closed interval is bounded in that interval, attains its bounds and also takes each value between its bounds. What restrictions, if any, are there on the order of the intermediate value? In which A(7r) does your proof take place ? 20. Define lim f{Xs}. Here f{X8} is of the form -kx'(Ex)(k{Xs}x&x'
< x).
21. State and prove Rolles theorem for /cth order functions. 22. Define the upper and lower Riemann integral for a bounded function over a bounded interval by using subdivisions of the bounded interval into intervals of equal length only. This method of subdivision gives a sequence of upper sums and a sequence of lower sums, and is
562
Ch. 11 Extensions of the system Aj
logically easier to handle than the normal method of using all possible subdivisions. If we used any subdivision then we should require some variables of higher types. 23. Define the exponential function in the complex plane, and prove its functional equation. 24. A set of points in a plane can be defined as a set of complex numbers
thus l.(EY, Y')(
Y'y)&(Ay)(Yy.^Evy) &(Y'y.->Ody)).
Using D 270, 271 define the closure of a set of points in a plane and prove that the closure is closed. 25. Define the interior, exterior and boundary of a set of points in a plane. If the set is of order K what are the orders of its interior, exterior and boundary? 26. Prove Wierstrass' theorem for a sequence of distinct points in a square. 27. Define the distance between two sets of points in a plane. Prove that the distance between two closed bounded sets of points in a plane is zero if and only if the sets have a point in common. 28. Show that a nested sequence of bounded closed sets of points in a plane have a point in common.