STUDIES IN LOGIC AND
THE FOUNDATIONS OF MA THEMA TIC S
L. E. J. B R 0 U W E R E. W. BETH A. HEYTING Editors
~c ~
~ 1971
NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM· LONDON
MATHEMATICAL INTERPRETATION OF FORMAL SYSTEMS
TH. SKOLEM G. HASENJAEGER G. KREISEL A. ROBINSON HAO WANG L. HENKIN J. l.OS
1971 NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM· LONDON
© NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM - 1955
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NORTH-HOLLAND ISBN 7204 2226 4
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First edition 1955 Second edition 1971
PRINTED IN THE NETHERLANDS
TH. SKOLEM
PEANO'S AXIOMS AND MODELS OF ARITHMETIC Introduction More than 30 years ago I proved by use of a theorem of Lowenheim that a theory based on axioms formulated in the lower predicate calculus could always be satisfied in a denumerable infinite domain of objects. Later one has often expressed this by saying that a denumerable model exists for such a theory. Of particular interest was of course the application of this theorem to axiomatic set theory, showing that also for this an arithmetical model can be found. As I emphasized this leads to a relativisation of set theoretic notions. On the other hand, if one desires to develop arithmetic as a part of set theory, a definition of the natural number series is needed and can be set up as for example done by Zermelo. However, this definition cannot then be conceived as having an absolute meaning, because the notion set and particularly the notion subset in the case of infinite sets can only be asserted to exist in a relative sense. It was then to be expected that if we try to characterize the number series by axioms, for example by Peano's, using the reasoning with sets given axiomatically or what amounts to the same thing given by some formal system, we would not obtain a complete characterisation. By closer study I succeeded in showing that this really is so. This fact can be expressed by saying that besides the usual number series other models exist of the number theory given by Peano's axioms or any similar axiom system. In the sequel I will first give an account as short as possible of myoId proof of this, my exposition now being a little different in some respects. Mter that I intend to show how models of a similar kind can be set up in a perfectly constructive way when we consider some very restricted arithmetical theories.
2
§ 1.
TH. SKOLEM
Preliminary Remarks
We may set up a theory of natural numbers by adding to the predicate calculus of first order some constants namely the individual constant I, the predicate = and some functions namely the successor function, denoted by an apostroph, and addition and multiplication. Further we may assume the non-logical axioms x' ¥= I
(x'=y')
-?-
(x=y)
(y ¥= I)
-?-
(Ex)(y=x')
x+ I =x' x+y'=(x+y)' x·I =x
+x
xy' = xy x =x (x = y)
-?-
(U(y)
U(I)&(x)(U(x)
-?-
-?-
U(x))
U(x'))
-?-
U(y).
Here U denotes an arbitrary propositional function. It is most natural and convenient to let the propositional functions be those which can be constructed from equations by use of the connectives &, V and - together with the quantifiers extended over individuals, i.e. numbers. The two last axioms containing U are meant as axiom-schemes so that every individual case is an axiom. This formal system of arithmetic contains Peano's axioms. Other systems could be used as well, for example the system ZI' in Hilbert-Bernays [I], p. 293. Every proposition is equivalent to one which is built by use of quantifiers on an elementary expression, namely built by use of &, V and - on equations with polynomial terms on both sides when we replace x' by x+ 1. However, we may omit the negation, because x -=j::.y may be replaced by (Ez)((x=y+z) V (y=x+z)). Further any proposition constructed by use of & and V from
PEANO'S AXIOMS AND MODELS OF ARITHMETIC
3
equations is equivalent to a single equation because of the equivalences (1)
(a=b) V (c=d)
(2)
(a=b) & (c=d)~
(ad+bc=ac+bd)
~
(a 2+b2+c2+d2=2ab+2cd)
Thus every proposition is equivalent to an expression beginning with a sequence of quantifiers followed by an equation between two polynomial terms. As to the arithmetical functions many more are definable than the polynomials. Indeed let the proposition (3)
(Xl)' .. (x",)(Ey)A(xl , • • • • ,
X""
y)
be true. Here A is a propositional function which may be arbitrarily complicated. It may for example still contain quantifiers. The word true may mean either provable in the system or that the statement is assumed as a further axiom. Now it is well known that we can prove by use of the induction axioms that if (Ey)A(y)
is true, then (Ey)[A(y) & (z)(A(z) V (y ~ z))]
follows which means that every non void set of numbers contains a least element. Therefore from (3) follows (Xl)' .. (x",)(Ey)(A(x v ' .. ,
X""
y) & (z)(A(xv . . . , X"" z) V (y ~ z))).
Then one and only one y here exists corresponding to a given m-tuple XV"" X.... This y is therefore a function !(xI , . . . , x",). Using this function we may write (3) as a formula containing no other quantifiers than those which perhaps occur in A(~, . . , X.., y) namely as A(al , ·
.. ,
am' !(a v' .. , am))'
Repeating this introduction of functions one finds (see my paper "Ober die Nichtcharakterisierbarkeit der ZaWenreihe etc." [2], that every correct formula (Xl)' . (xm)(EYI)' . (Ey,,)(~)
. . (zJl)(E~)
. . (Eu q ) •
• •
. . .A (Xv •. Xm , YI" . y", ~, ....
»
4
TH. SKOLEM
may be written with free variables only A(€Lt, . . , am' fl(€Lt,· . , am)" . , f..(a1 , ·
•,
am), bi>' . , bp , gl (€Lt, .• , am' bi> .• , bp ) , •
• ).
Since for example we have the correct formula (x)(y)«x=y) V (Ez)«x=y+z) V (y=x+z»)
a function exists, usually written
Ix - Y!, such that
(a=b) V (a=b+la-bl) V (b=a+/a-bl)
is a correct formula. Let F be the set of all arithmetical functions in this sense. It is easily seen that F is closed with regard to the operation called nesting or substitution. Let for example z=!(x, y) and y=g(u) be respectively equivalent to A(x, y, z) and B(y, u). Then it is evident that Z= f(x, g(u» is equivalent to C(x, u, z), where C(x, u, z) is the propositional function (Ey)(A(x, y, z) & B(y, u».
It is clear after these preparations that every statement can be replaced by an equivalent equation between two elements of F containing only free variables. It is evident that all true formulas may be listed as an enumerated set S. To each of them we may find an equivalent equation whose both sides are functions belonging to F. Therefore in order to prove the existence of a model N' different from N for the set S of statements it will suffice to prove the following theorem. Let S be a set of equations whose both sides are elements of a denumerable set of functions closed with regard to nesting. Assuming the equations belonging to S all valid for the natural number series N we may define a greater series N* such that by suitable extension of all notions concerning N to corresponding ones in N* all equations in S are also valid for N*. In order to establish this I need an arithmetical lemma which I shall prove first. It ought to be added that this procedure is sufficient for our purpose because it will turn out that the equivalences we used above will remain valid in N*.
PEANO'S AXIOMS AND MODELS OF ARITHMETIC
§ 2.
5
An Arithmetical Lemma
We consider an enumerated sequence of arithmetical functions (4)
Let N(1), NCZ),
NCBI
be resp. the subsets of N for which
Mt) <Mt), Mt) = Mt), fl(t) > Mt).
One at least of NU), NIZI, N(S) is infinite. Let N 1 be that with the least upper index which is infinite. Then there are for each t E N 1 at most 5 possible cases for fs(t) in relation to Mt) and Mt), namely if for example N 1 is NU) fs(t)<Mt), fs(t)=fl(t), fI(t)
=
fz(t), fs(t) > Iz(t).
If N 1 is N(Z) so that fI(t) = Iz(t) for all t cases namely
E N1
we have only 3 possible
fa(t)
!J(t).
Let N(1) N(Z) NCB) and eventually N(() N (6 ) denote the subsets of l' I' 1 l' 1 N 1 for which the mentioned relations take place. Then again one at least of N~I), N~), .. is infinite. We let N z be the N~) with least r which is infinite. This procedure is continued so that we get an infinite sequence of infinite subsets of N N=No :::> N 1 :::> N z :::> •••••
For all t E N"_l the same relations < and = will take place between !J(t), . .f,,(t). Now let g(n) be the least number in N"-l' Then it is evident that if where n=max (a, b), then we have accordingly
for all t ~ n: Thus the following lemma is proved: If (4) is an infinite sequence of arithmetical functions, an arith-
6
TH. SKOLEM
metical function g(t) exists such that for any pair i, j the same relation <, = or > takes place between IM(t)) and l;(g((t)) for all t>max (i, j). The function g(t) is steadily non-decreasing. In our applications of this lemma we will assume that all polynomials, in particular all constants, occur in (4). Then one sees that the values of g(t) cannot possess an upper bound, because the intersection of all N, is the null set.
§ 3.
The Proof of the Existence of N*
Let F be an enumerated set of arithmetical functions of one or more variables containing besides the successor, addition and multiplication all functions occurring in the left and right terms of a set S of equations with only free variables supposed true for N, F further being supposed closed with regard to nesting. Let F 1 be the denumerable subset of F consisting of the functions I,(t) of one variable. Then relations < and = can be defined between the elements of F 1 in the following way. According to the lemma a function g(t) exists such that for any two i and j one of the three relations
holds for all t>max (i, j). I put respectively
t.ct; 1.=/i' t.>t, in these three cases. It is easy to see that the relation = thus defined is an equivalence relation and that the relation < is asymmetric and transitive. The different equivalence classes of the elements of F 1 defined by = shall then constitute the diverse elements of N*. In a very simple and natural way every function I(xv' ., x,,) in F can be extended to mean a function in the domain N*. Indeed, if every X,(t) for T= 1, .. , n is E F v then y
is
E
=
I(X1(t), . . . , X,,(t))
F v because F 1 is closed with regard to nesting. Further one
PEANO'S AXIOMS AND MODELS OJ' ARITHMETIC
7
easily sees that if X,.l(t) and X".(t) for r= 1, .. , n are the same elements of N*, then and
Yl
f(Xl,l(t),
, X",l(t))
Y 2= f(X1,2(t),
, X".2(t))
=
denote the same element of N*. Thus f also defines a function in the domain N*. Clearly all elements of N also belong to N*. Indeed they are furnished by the f in F l which are constants. Further, since the relation = has been defined in N*, an tke equations constituting S have a meaning in N*. It remains to see that they are all valid in N*. Let us consider an equation in S. It has left- and right-hand terms with some free variables, say a v all' . .. Replacing these by arbitrary elements of N* we get an equation in N*. Since this is valid for every value of tin N, it is valid for every t' when t has been replaced by g(t'). A fortiori the equation takes place for all t' > the maximum of the indices which the left- and right-hand sides of the equation possess in the sequence fl(t), f2(t), . . .. Thus, remembering the definition of = in N*, we see that the equation holds good for arbitrary elements in N*. Now I will prove that the equivalence (1) remains valid in N*. The correctness of the implication (fX={3) V (y= 15) -+ (fXt5+{3y=fXy+{3t5)
for arbitrary elements fX, {3, y, 15 of N* is seen so easily that I may confine myself to the treatment of the inverse implication. Let fXt5 + {3y = fXy + (3t5. This means that for all t' > h say and t = g(t') (5)
fX(t)15(t) + (3(t)y(t) = fX(t)y(t) + (3(t)b(t).
The number h may be chosen as the maximum of the indices of the left- and right-hand sides of (5) in the sequence (4). According to (1) this implies that for every t'>h and t=g(t') (fX(t) = (3(t) V (y(t) = <5(t)).
8
TH. SKOLEM
Now let hI be the maximum of the indices in (4) of the functions a(t), f3(t), y(t), <5(t). Then if t' is > max (h, hI) either a(g(t')) = f3(g(t')) and then we have 1X=f3 or y(g(t'))=<5(g(t')) so that 1'=<5. Hence (1) remains valid in N*. Similarly we can show that (2) remains valid. Let us now assume that la - bl E F. Then if IX, f3 are E N*, we have for all t (lX(t)=f3(t)) V (lX(t)=f3(t)+IIX(t)-f3(t)l) V (f3(t)=a(t)+ + IIX(t)-f3(t)l). Putting t=g(t'), the last formula is valid for all t'. Let h be the maximum of the indices of the two sides of the three equations. For a value of t' > h one of the equations is fulfilled. Then just this equation holds for all greater values of r. Therefore we have either a=f3 or 1X=f3+ la-f31 or f3=a+ 11X-f3I. A consequence of this is that the equivalence between IX *-f3 and (Ex)((IX=f3+x) V (f3=IX+x)) holds good in the theory of N*. Let A(a, f(a), b, g(a, b)) be a free variable formula equivalent to the true formula (x)(Ey)(z)(Eu)A(x, y, z, u)
in the previously explained sense. Then as I have just shown A(a, f(IX), f3, g(lX, (3))
is true in N* and hence also (x')(Ey')(z')(Eu')A(x', y', z', u'),
where x', y', z', u' are variables extended over N*. This means that also every formula containing quantifiers and true concerning N is also true for N*. In particular we may remark that the induction scheme remains valid for N. Now let us start with the general formula (x')(Ey')(z')(Eu')A(x', y', z', u')
and as previously explained find an equivalent free variable formula A(a,!(IX), f3, g(a, (3)).
Then according to the determination of !(IX) we have !(a) ~ f(IX). On the other hand the definition of finN yields for every t that f(a(t)) ~!(IX(t)), whence f(a) ~!(IX). Hence the formula A(a, f(IX), f3, g(a, (3));
PEANO'S AXIOMS AND MODELS OF ARITHMETIC
9
but according to the determination of g the relation g(lX,fJ) ~g(lX,fJ) takes place for arbitrary IX and fJ in N*. On the other hand we have for all t according to the determination of g(a, b) in N that g(lX(t), fJ(t)) ~ g(lX(t), fJ(t)) with the consequence g(lX, fJ) ~ g(lX, fJ). Thus the functions f and g retain after the transition from N to N* their roles as least elements with the considered properties. The transition from N to N* may of course be repeated so that we get a model N* * and so on. § 4.
Some more Special Results
I would like to add some remarks on the setting up of models of certain fragments of number theory, in particular fragments of recursive arithmetic. In these simple cases the definition of nonstandard models can often be established in a perfectly constructive way. Let us for example consider the following theory T. The statements of T shall be built by &, V and - from the propositional functions x < y and x = y assuming the classical propositional calculus and the axioms concerning =. The recursive definitions of addition and multiplication shall belong to T, the successor being here simply denoted by addition of 1. Further we shall have the recursive definition of < namely a
(a
Also the substitution rule shall be valid, i.e. from a correct formula we always obtain a correct formula by substitution of a variable (variables) by a numerical term (numerical terms). Finally we assume the induction scheme U(O,b,c .. ) U(a, b, c, .. ) ~ U(a+1, b, c, . . )
Uta, b, c, .. ) It is well known that we can derive in T the ordinary laws of addition and multiplication, namely the associative, commutative and
10
TH. SKOLEM
distributive laws, and the ordinary laws concerning the relation <, namely asymmetry, transitivity and the theorems (7) (8)
(9)
(a
b) (a
On the other hand many formulas expressible in T and valid for the natural numbers are probably not provable in T. Thus for example nobody has ever succeeded in proving on this basis the statement (10)
which means that V2 is irrational. Now let N' denote the set of polynomials j(t)=a,,t"+ . . . +ao, all a r integers,
with all a, ~ 0 and a..> 0 resp. ao ~ 0 when n=O. If g(t)=bmt m+ + ... + bo' I write j = g when and only when n = m and, for all r, ar=b r. Further I write jg according as j(t)
Ili«, /3, y, .... ) V tu« + I, /3, y, ... ) be correct, the variables ex,
/3, y,
... ranging over N'. Then they
PEANO'S AXIOMS AND MODELS OF ARITHMETIC
11
are correct by restriction of the range of variation to N which is contained in N', the constants being some of the elements of N'. We may write this U(O, b, C, .• ) U(a, b, C, •• ) V U(a+ 1, b, c .. ), where a, b, c, .. have N as domain of variation. Then the induction scheme for N yields U(a, b, c, .. ). Let us here replace a, b, c, .. by variables iX, (J, y, .. ranging over .N'. Since U(iX(t), (J(t), ... ) is obviously true for all t, it is true for values of t so great that the atomic relations < or = building up the expression U remain settled, and that means that U(iX, (J, y, ... )
is true. Thus the induction scheme remains valid for N'. As a consequence of this every provable formula in T must also be provable in the theory T' with the variables ranging over N' instead of N and obtained by taking into account the definitions above connecting T' with T. I will insert the following remark. If instead of only the polynomials with integral coefficients we take all those having integral values for integral values of the variable we get a domain N~, where not only addition and multiplication can be carried out with retention of the usual algebraic rules, but also the general summation .E can always be performed. When F(T) is a function in N~, then ..4.
.E F(T) is a function G(A) in N~.
Or in other words a uniquely
T-O
determined G(T) exists such that G(O) = F(O) and G(T+ 1)=G(T)+F(T+ 1).
If for example F(T)= 1, then G(T)=T, and if F(T)=T, then G(T) = 1/2·T(T + 1). It is clear that when TEN~ also 1/2T(T + 1) EN~. Further, if F(T) = 1/2T(T+ 1), G(T) = 1/6T(T+ 1)(T+2) etc. I
confine myself to this hint. One must expect that N' will cease to be a model for a more
12
TH. SKOLEM
extended arithmetical theory than T. Let us consider the theory T I which arises when a function b defined by the equations bO=O, b(a+ 1)=a
is added to T. Simultaneously of course the U in the induction scheme then shall be understood to denote any propositional function that can be built when the function b is also taken into account. It is easy to see that N' is not a model for T I . Indeed one proves in T I the theorem (a=O) V (b(a)+I=a).
This means that every element except 0 has a predecessor, but this is obviously not true for N'. However, we get a model Nil of T I by omitting for 0;;;; r < n the requirement a. ~ 0 for the polynomials f(t), retaining an>O and if n=O also a o ~ O. Let T 2 be the theory obtained by adding to T I the recursive definition of the function a . .:. . b, namely a ...:.... O=a, a .z: (b+ 1)= b(a ...:.... b).
Then it is seen that Nil is still a model for T 2 • Just as the function a...:.... b in the case of N means a-b when a z-b and 0 when a ~ b, this is also true in T~ concerning Nil when the true propositions of T~ are those that can be derived from the provable propositions of T 2 by use of the definitions connecting T~ with T 2 • However, Nil again ceases to be a model if such a function as [a/2] is added to T 2 • I assert that a model N fII for the theory T 3 arising from T 2 by adding the function [alb], defined for b>O, can be chosen as the set of all polynomials where all a. are rational (eventually fractional) numbers, an (the highest coefficient) > 0 and a o non-negative integer when n= O. Instead of setting up the recursive definition of [iX/If] for N fII one can show that for arbitrary f(t) and g(t) there are uniquely determined polynomials g(t) and r(t) such that identically in t f(t) = g(t)q(t) + r(t) and 0
Hence after the definition of < and
=
~
r(t) < g(t).
between the elements of
PEANO'S AXIOMS AND MODELS OF ARITHMETIC
13
N'" we have that for the arbitrary elements iX and (3 of N'" two elements x and e exist so that
Then x is [iX/{3]' If instead of these very limited parts of number theory we take for example the primitive recursive arithmetic, the situation is as far as I can see today not simpler than in the general case I first treated above. Let F be the set of all primitive recursive functions, F I the set of all such functions of one variable. Then a recursive function !(x'Y)=!lI(x) exists which for y=O,I, ... yields all elements !lI(X) of Fl. According to my lemma a function g(t) exists such that if !i(X) and !i(x) are any two elements of Fl> a permanent relation <, = or > takes place between fM(x)) and fi(g(x)) when x>max (i, j). Correspondingly we put t.r t; !i=!i' !i>!i- Then every provable proposition in primitive recursive arithmetic will remain valid for the elements of it when it is the set of equivalence classes in F I defined by the relation = between the elements of Fl. Sometimes a model M for an arithmetical theory T remains a model also for a theory obtained by adjunction of a further recursive definition. This happens if and only if a function is definable in M satisfying the equations constituting the new recursive definition. I dare not say to what extent this remark may be utilized, but perhaps one will be able to set up models in an essentially simpler way then I have done here, for example for primitive recursive number theory. I attempted to prove that the set of all polynomials iX"tn + ...+ iXO' where iX, for r> 0 is = a, + b, V2, a, and b, non-negative integers, a"+b,, V2>0, iXo=ao and, if n=O, a non-negative integer, is also a model for the theory T above containing the recursive definitions of addition, multiplication and the relation <. If this could be proved, we would know with certainty that the theorem (10) is not deducible in the theory T. However, I did not succeed. I could only prove that this set of polynomials is a model for a theory containing addition, multiplication, the recursive definition of < and further all of the ordinary algebraic laws concerning addition,
14
TH. SKOLEM
multiplication and < as axioms, and finally containing the induction scheme with only one variable or in other words without parameters. But I think all that is of very limited interest. Therefore I will not on this occasion enter more closely into this matter. Another proof of the existence of different models of arithmetic can be found in the recently published book by Kleene [3]. I have not had the opportunity to study his proof thoroughly but my impression is that he uses a method similar to that used in the proof of Lowenheim's theorem. He obtains a realization of the predicates of arithmetic so that the axioms are fulfilled, but the realization is not isomorphic to the ordinary one. REFERENCES [1] D. HILBERT und P. BERNAYS, Grundlagen der Mathematik II, Berlin (1938). [2] TH. SKOLEM, Fund. Math., 23 (1934), 157-159. [3] S. C. KLEENE, Introduction to metamathematics, ArosterdamfGroningen (1952). Matematisk Institut, Universitet, Oslo, Norway.
G. HASENJAEGER
ON DEFINABILITY AND DERIVABILITY
I wish to discuss a kind of problem which is closely connected with the non-derivability proofs given by Th. Skolem using certain non-standard models of arithmetical axiom systems. Models of arithmetical systems may be non-standard in the sense that there are "too many" elements called numbers. In that case there must exist sets of individuals which are not (represented) in the model. This (weaker) form of being non-standard does not depend on a structure in the domain of individuals, and is related to similar derivability problems for the pure predicate calculi. Since there are too many complications in the case of higher order calculi, most of the following remarks are restricted to the first order calculi. § 1.
The Concept of Model
The domain of sets of individuals (and eventually sets of higher order) must be given by the model in the case of formal systems which contain bound variables for sets (or attributes i.e. propositional functions). In the first order case we can avoid this by considering defining formulas in place of the corresponding functions. To get a homogeneous treatment, however, I propose a homogeneous terminology as follows. A model S (frame, realisation) is a function giving a suitable designatum S(c) for each constant c and giving a suitable range S(v) for each variable v of a given formal system. 1 Some variables vI> V 2 may be characterized by the formation rules as being of the 1 This formulation presupposes that there is no typographical distinction between free and bound variables.
16
G.
HASENJAEGER
same type. Then "suitable" is to include S(v1 ) =S(v2 ) . (Thus the domain of individuals may be cited as S(x) for an arbitrary individual variable x.) Some constant c may be characterized as being of the same type as some variable v. Then "suitable" is to include S(c) EO S(v). Some constants or variables may be characterized as designating functions (from -- to -). Then "suitable" is to be understood as requiring the designata (in the case of constants) to be such functions, and the ranges (in the case of variables) to consist of such functions. The logical constants are used here in the sense of the twovalued logic (but this could be adapted to other purposes). A model is standard (Henkin) if and only if all ranges for variables for functions are extensionally maximal. 2 Given a model S and an assignment 1> of objects to the variables and constants 3 of a system, satisfying the conditions 1>(v) EO S(v) and 1>(c)=S(c) 3, a designatum s4>(m) is attached to each wff m of the system (see for instance [5], p. 83, 84). 1> may be called an assignment over S. For wff's m of the propositional type S4>(~O is T (truth) or F (falsehood), as technical counterpart for the concept of satisfaction. m is called S-valid if S4>(m)=T for every 1> over S. 4
§ 2.
The Concept of Universal Consequence
~( being a wff of the propositional type and r being a set of such wff's, m is called a universal consequence of (short: rUl-m) if and only if
Dl: for every standard S such that each element of misS-valid too.
r is S-valid,
2 The asswnption that such domains exist seems to be the only justification for considering impredicative comprehension principles, but these set-theoretical principles do not fully describe such domains. 3 This is only to simplify the notation of the definition of satisfaction in Tarski's sense. 4 "For every " is used here in the extensionally maximal sense. Certain restrictions of this may be described by modifications of S, and others are not needed here.
ON DEFINABILITY AND DERIVABILITY
17
To distinguish from a later definition the relation U~ is called standard universal consequence. The definition of universal consequence is a generalization of the following "classical" concept of consequence I~: r I~ ~l if and only if D2: for every standard S and every ep over S: if S ¢($B) = T for every $B E r, then S.p(m)=T. U~ seems to be a better description of the actual intentions of mathematicians in making deductions than I~ . The difference between U~ and I~ is this. I~ makes no distinction between free variables and constants; for under I~ each free variable is treated like a constant, in the sense that, for any interpretation a fixed (but arbitrary) denotation is given to the symbol. Since this is the treatment of the constants by U~, U~ may be considered as a generalization of I~, and UI- and l~ restricted to closed wff's coincide. On the other hand, all effective variables are implicitly generalized at the beginning of each formula by the definition of U~. Thus Ur gives the interpretation of the customary use of free variables in the universal sense, and so a kind of "legitimization" of such formulations of number theory as in HilbertBernays' Grundlagen der Mathematik I, § 6, § 7, where variables not generalizable in the language are used in the universal sense: number variables in the recursive number theory, and predicate variables for the formulation of complete induction in the wider number theory. Such variables let us call essentially tree variables. Thus U~ includes the intended concept of consequence for languages of several sorts: (i) all free symbols are regarded as potential names for the basic concepts of a theory (this case of U~ is reduced to I~), (ii) the general case (there are both extralogical constants and essential free variables, as in the above mentioned systems of number theory), (iii) there are essentially free variables but no extralogical constants.
18
G. HASENJAEGER
Our aim here is to show that case (iii) yields problems analogous to those of case (ii) arising from the incompleteness of the usual formalized number theories. In spite of the above generality of definitions from now on (until a final remark) we restrict our considerations to systems without bound function variables. These offer all problems except those arising from impredicativity. The three cases are given by the corresponding classification of the predicate symbols of a given first order language into variables and constants.
§ 3.
Incompletability Theorems for the relation of Universal Consequence
In the case (i) (where UI- reduces to 11-) a complete syntactical characterization of UI- is given by a general form of Godela completeness theorem. For case (ii) this is excluded by the fact that the class of universal consequences of the finite number-theoretical axiom system Z (Hilbert-Bernays I, p. 371) is not even arithmetically definable in the sense of Kleene-Mostowski 5, hence not axiomatizable in the usual syntactical sense. Similarly, as we shall show below, we get non-axiomatizability in case (iii). Consider, then, the special case {&} UI- 5B of universal consequence, where &,5B are wff's of the first order calculus, all predicate symbols of which (excepting at most the identity sign) are effective variables. This gives via arithmetization a relation which is not in P~2), hence not axiomatizable. Combining this with a similar result we prove
Thm. A: The restricted relation UI- is not in
P~2)
U Q~2).
Proof: It suffices to consider formulas not containing the identity sign. Let lY be a fixed formula valid exactly in the finite domains, & a fixed generally valid one, and To the class of all generally valid formulas. The well known syntactical definition of To shows that To E P~l), and by the theorems of Church and Post we have To ¢ Q~l). 6
A. Mostowski's terminology is used here; see [7].
19
ON DEFINABILITY AND DERIVABILITY
(1) Suppose U~ E P~2). Then from the equivalence (for the language without identity) ~! E
Fo if and only if
not
{2!} U~
~
we would get F o E Q~l). (2) Suppose U~ EQ~2). Then from the equivalence
2! E F o if and only if {&} U~ 2! we would get F o E Q~l). We remark that in the case (iii) the restricted UH is in the Boolean algebra generated by the elements of P~2). Thus in case (iii), as in case (ii), the question always arises whether a given concept of formal derivability yields a given universal consequence from given premises; and in both cases (when restricted to first order calculi) a proof of non-derivability may be arranged by the elementary method of re-interpreting all essentially free variables in the premises as syntactical variables, so that premises containing those variables change into schemata. But I think that the use of the following concept also in the first order case also helps us to understand the general case somewhat better. § 4.
The Concept of Universal K-Consequence
A less elementary arrangement for the intended non-derivability proofs which, however, is more instructive in the above mentioned sense can be given by adapting D1 to the given concept of formal derivability. We introduce into D1 a parameter K, denoting a class of models not necessarily standard, i.e. we define: 2! is a universal K -consequence of F (short: F U~ K 2!) if and only if D3: for every S E K such that each element of F is S-valid, 2! is S-valid too. Obviously D3 contains D1 as a special case. On the other hand D3 can be adapted by a suitable choice of K to various concepts of formal derivability in the following sense. If "F f- 2!" denotes
20
G. HASENJAEGER
that m is derivable from I', then adequacy of K for pressed by
r f- m if
and only if
r u~K m,
for all rand
f-
IS
ex-
m.
That a given K is adequate for a given f- is a completeness theorem in the sense of Henkin's completeness in the theory of types [5]. For the concept of derivability for a first order calculus (short: f-p) whose predicate symbols are characterized as variables by a substitution rule, the class K o of all models S satisfying the following condition can be taken for K. S",(A.!m(!))
E
S(Pl'I) for each
(! short for
Xl' •.• ,
ep over S. 6 xl'I)
Thus we have the
Thm. B: If rf-p
m,
then
rU~K.
m,
which justifies the underivability proofs by models E K o. For the other part of the adequacy, we get by a modification of the completeness proof of Godel (or Henkin) the
Thm. 0: If r
U~K.
m,
then
rf-p m,
7
which asserts that there are enough models for non-derivability proofs. On the other hand theorem A yields that there is no universal method to find such models. Thus instead of a detailed proof of 0 I shall discuss some special cases in which models have been found.
§ 5.
Special Cases of Non-derivability
In the case (iii) (no extra-logical constants) for standard S S-validity depends only on the cardinal number of S(x). If the • I.e.: S(P") is closed under all functions, defined by terms A!m:(!), of arguments given by 4>(v) for all variables v free in ).!m:(~). - If, as usual, ).·terms are not in the calculus, they may be adjoined in order to get simpler descriptions of this condition and of the substitution rule for predicate variables. 7 The idea of the application of that proof method is contained in § 5, (B).
21
ON DEFINABILITY AND DERIVABILITY
class of cardinals which yield models of the wff m: is called the spectrum of m: (short: Sp(m:)), we have {2t} Ub ~ if and only if Sp(m:)
c Sp(~).
Thus the questions formulated in Hilbert-Bernays I p. 124, footnote 1, and answered in [2], are examples of the problem here considered. Now I shall discuss the derivability between two formulas of the predicate calculus with identity, which is perhaps of some number-theoretical interest. Let ~(A) 8 be a single axiom for the theory of Boolean algebras and let %(B) 8 be a single axiom for the theory of fields of characteristic 2. To develop these theories the symbols A and Bare naturally to be considered as constants in the sense of § 2. It is well known that each of the formulas ~(A) and %(B) is satisfiable in precisely those finite domains whose cardinal is a power of 2, and in all infinite domains. Consider now the wff's ~(A) and %(B), where the constants A, B have been replaced by variables A, B. ~(A) and %(B) are valid for exactly the same standard models, namely those models whose domain of individuals is finite and is not a power of 2. So we have I'o.J
I'o.J
I'o.J
(1)
I'o.J
I'o.J
%(B) U~ ~ ~(A)
and (2)
~ ~(A)
Ur ~ %(B),
and the question arises if ~ ~(A) and ~ %(B) are derivable from each other. I shall treat only the case (1). For (2) see the final remarks. In order to show that ~ ~(A) is not derivable from ~ %(B) it would suffice to give a model S (E K o) for which ~ %(B) is valid but not ~ ~(A). But the lemma which would be used to construct the model 8 For simplicity I assume that each of these wff's contains exactly one predicate symbol other than identity, and that no individual constant appears among the basic concepts. In >8(A) A may be a symbol for the partial order or the Sheffer function. A general method is suggested by the following equivalence
Axyzu _ x
+y
=
z /\ x'y
=
u.
22
G. HASENJAEGER
by a general method (see below) gives an immediate proof by the methods used in [2]. Namely it suffices to show the Lemma ("), For each substitution Bjk£9l(!, a, A) there is a model 8 and an assignment cPo such that 8
a, A))) =8B(A))=T.
Proof of (*). Let k be the length of the sequence a= (aI' ... , a k) of the "parameters" of the substitution and p a finite Boolean algebra containing 2k+1 + 1 atoms. There is a model and an assignment cPo, partially defined by the following conditions
the domain of the algebra p, 8(9"): the class of all n-adic relations over 8(x) which are invariant under the group of automorphisms of (P, x) 9 for at least one k-tuple x, cPo(A): that basic concept of p suitable for >B(A). cPo(A) E8(A) and 8B(A))=T are obvious.
8(x):
Since 8
•.• ,
cP(ak )
,
consider the assignments cP such that cP(A) = cPo(A). For these cP 8
Now we show that each element of 8(Rn), in particular of 8(B), admits a non-commutative group 1: of automorphisms. The constituents of a k-tuple divide the class of atoms of p into at most 2k classes one of which must contain at least 3 elements, and the group 1: is the direct product of the permutation groups in these classes. Hence 1: is not commutative. Since finite fields admit only cyclic groups of automorphisms, no element of 8(B) is a possible basic concept for a finite field, and hence 8
23
ON DEFINABILITY AND DERIVABILITY
Since this includes all possible values of 4>(a), we get S",.(Va ,...,,%(J,rm:(r, a, A)))=T.
This completes the proof. From the lemma (*) the asserted underivability can be shown in the following ways, (A) or (B). (A). The method of the "Ruokverlegung der Einsetzungen" (see Hilbert-Bernays I, p. 225) shows that each proof from %(B) can be transformed into a proof from a finite class of formulas ,..." %(A-rm:1(r, ... )), ... , ,..." %(A-rm:,,(r, ... )) without the use of further substitutions. The implication i--.
permits us to derive the above substitutions in ,..." %(B) from a single substitution ,..." %(A-rm:(r, a, A)), and a proof of ,..." ~(A) from this can be transformed into a proof of
which is excluded by the lemma (*). Note that the S in the proof of (*) is not in K o, since generally S",(A-rm:(r, a, A)) is not in S(B) for 4>(A)*4>o(A). For, 4>(A) may require fixpoints other than 4>(0). In order to get a model in K o we have to modify the preceding proof nearly in the following way. (B). A contradiction (without use of substitutions) in the following set M consisting of (a) the wff ~(A), (b) the wff's ,..." %(BfP) for all possible variables P, (c) all wff's Vr(Pr +---+ (£:(r, ... )), where P is suitable chosen to avoid circularities in the dependence of P from the parameters in (£:(r, ... ), reduces to a contradiction in some finite subset E of M. By a procedure analogous to the "Ruckverlegung" we remove all predicate variables but A in such a way that all wff's (c) become 10
For the proof see [2], p. 274. Moreover the equivalence is valid.
24
G. HASENJAEGER
logical theorems. The finitely many cases of (b) are reduced as above to a single substitution such that a contradiction in
would arise, contrary to the lemma ("). Thus M is consistent. Each known completeness proof gives a "model" (S', ep') where S'(x) is the set of the ep'(z) for all individual variables z, whereas the domains S'(R") need only to contain the ep'(R"). If we define S'(P") to be the set of the ep'(Rn) for all R",
then by the truth of the wff's (c) the S'-validity is closed under the general substitution rule (see Godel [I], Satz X, Henkin [4]; also Henkin [6]). § 6.
Final Remarks
By the construction of the model S' in (B) we learn almost nothing about S'. Thus there may be some interest in the following model So, though one essential question concerning So remains open. So is constructed from the Boolean algebra (J0 consisting of the finite sets of natural numbers and their complements in the following way. SO(x): the domain of (J0, SO(R"): the class of all n-adic relations over SO(x) invariant under the group of automorphisms of ({J0, a) for at least one finite sequence a of elements of {J0.
lt can be shown that SO-validity is closed under substitutions of formulas even containing bound predicate variables (see [3]). is not SO-valid. But the question whether Obviously ,...., ~(A) ,...., '1:(B) is SO-valid requires a better knowledge of the infinite fields of characteristic 2. 11 A positive answer in connection with a which, formalization of the intuitive prooffor {,...., '1:( B)} U~ ,...., ~(A) I suppose, needs the axiom of choice, would give another independence proof for this axiom. Knowledge of the automorphisms 11 In the symposion I asserted too much. I neglected the difference between the model 8 in (*) and the model 80.
ON DEFINABILITY AND DERIVABILITY
25
of a suitable infinite field of characteristic 2 is needed also for the problem whether v-c 58(A) h ,...., ~(B), for there are not enough automorphisms in the finite fields to prove a lemma analogous to (*). REFERENCES [1] K. GODEL, Die Vollstandigkeit der Axiome des Iogisohen Funktionenkalkuls, Monatshefte fur Mathematik und Physik, 34 (1930), 349-360. [2] G. IIASENJAEGER, Uber eine Art von Unvollatandigkeit des Pradikatenkalkills der ersten Stufe. Journal of Symbolic Logic, 15 (1950),273-276. [3] , Some non-standard models of impredicative comprehension axioms, in preparation. [4] L. HENKIN, The completeness of the first- order functional calculus, Journal of Symbolic Logic, 14 (1949), 159-166. [5] , Completeness in the theory of types, Journal of Symbolic Logic, 15 (1950), 81-91. [6] , Banishing the rule of substitution for functional variables, Journal of Symbolic Logic, 18 (1953), 201-208. [7] A. MOSTOWSKI, On definable sets of positive integers, Fundamenta Mathematicae, 34 (1946), 81-112. Institut fur mathematiache Logik und Grundlagenforschung der Universitab, MUnster i W., Germany.
G. KREISEL
MODELS, TRANSLATIONS AND INTERPRETATIONS
I propose to discuss these three syntactic relations between formal systems from the following points of view: their relation to the consistency problem, their relation to each other, and some of their uses in understanding informal mathematics. I shall deal not only with axiomatizable systems, i.e. those whose axioms form a (general) recursive set, but also with certain non-axiomatizable systems since they are better formalizations of such branches of mathematics as arithmetic and set theory. The systems in which these syntactic relations are established, will, in general, be mentioned explicitly. (Both the use of non-axiomatizable systems and explicit mention of metamathematical methods of proof may be regarded as a natural reaction to Godel's incompleteness theorems; for, these show (i) that axiomatizable systems are not satisfactory approximations to certain branches of mathematics, (ii) that various formalizations of these branches of mathematics are of different "strength", so that one may expect to learn essentially more from the particular proofs of a theorem than from its assertion). It turns out that most of our work is finitist lor, more precisely, is formulated in quantifier-free systems with decidable predicates and computable functions. This comes about as follows: syntactic relations can be arithmetized in elementary arithmetic, and, as is explained at length in [1], p. 113, quantifier-free proofs are particularly appropriate in arithmetic; further, proofs in the elementary 1 I do not need a definition of this word since I never try to show that some particular theorem cannot be proved by finitist means; the reader may give his own definition and see that it fits our work; if it doesn't he may wish to revise his definition.
27
MODELS, TRANSLATIONS AND INTERPRETATIONS
quantification theory of arithmetic may be replaced by quantifierfree ones, [1], pp. 122-123. It is worth noting that the treatment of non-axiomatizable systems is quantifier-free, too. The main conclusions are summarized at the end of each section. The reader is warned that current usage of the words "model", "translation", "interpretation" is not uniform. Notation
§ 1.
h h, (Z-f-), (Nw-f-), (N" -f-), (II-f-) mean in this order: can be proved in (8), (8 i ), Z (quantification theory of + and . with induction), N w (primitive recursive arithmetic), N" (ordinal recursive arithmetic of order iX), II (predicate calculus). Con 8: a formulation of "(8) is consistent" which satisfies Godel's second undecidability theorem. Prov, (m, n): m is (the number of) a proof in (8.) of the formula (with number) n. General recursive functions are denoted by Greek letters: v.(n) is the negation of n in (the numbering of) (8.); the value of a(n, m) is the term obtained by substituting the numeral Olffl) in the term n. 1\: for all functions f; /
V: there exists a function /
f.
N.R A system N" consists of the elementary calculus with free variables, defining equations for a particular primitive recursive ordering of the integers which has been established to be a wellordering with ordinal iX, and schemata for definitions and proofs by transfinite induction based on this ordering of the integers, cf. [l], p. 113. The system is determined by the particular ordering used, and not by the ordinal iX. For our purpose the non-axiomatizable (class of) systems consisting of all N", for a given iX, is unsuitable: in such a system every formula (x)(Ey)A(x, y) with recursive A can be decided since there is an effective method of constructing to every such A a primitive recursive ordering < A which is a well-ordering (of order w 2 ) if and only if (x)(Ey)A(x, y) [Markwald, Math. Annalen, 127 (1954) p. 148]. Kleene has announced (personal communication) that every arithmetic proposi-
28
G. KREISEL
tion has a similar equivalent. In other words, this class of systems is of very high degree of undecidability. § 2.
Models
The familiar consistency proofs of various geometries, the algebra of complex numbers, or-to take a modern case-of general set theory (G), [2], are obtained by means of models. This notion, which Tarski, [3], p. 20, calls "interpretation", may be defined for systems of the first order predicate calculus as follows: A system (8 1 ) has a model in (8 2) if the non-logical constants of (8 1) , i.e. its predicate symbols and function symbols, can be replaced by .expressions of (8 2) in such a way that the axioms of (81 ) go into theorems of (8 2 ) . The discussion of models is best subdivided into three groups: (8 1 ) has a finite set of axioms, (8 1 ) has an infinite, but recursive set of axioms, (81 ) is not axiomatizable. In the first case the model can be exhibited in full, in the other cases we need a syntactic proof to show that a given replacement of the non-logical constants of (8 1) constitutes a model. FROM MODEL TO CONSISTENCY
(i) If (8 1 ) is finitely axiomatizable and has a model in (8 2 ) then (N.,-H(Con8 2 --+ Con 8 1 ) , [4]. For certain systems (8 2) , if a finitely axiomatizable (8 1 ) has a model in (8 2 ) then h Con 8 1 ; e.g. if (8 2 ) is the system Z, or, more generally, if there is a normal truth definition in (8 2) for quantifierfree formulae of (8 2 ) , This is not possible for all (8 2 ) : consider a finitely axiomatizable system (82 ) which satisfies the conditions of Godel's second undecidability theorem, when h Con 8 2 is false though every system has a model in itself. (ii) If (8 1) has infinitely many axioms, which are mapped into the formulae <X2(n) of (8 2 ) and f- (Ey) Prov, [y, <X2(n)] or t- Prov, [n(n) , <x2 (n )] then t- (Con 8 2 --+ Con 8 1 ) . It should be noted that, e.g., one can prove in Z the existence of a model of (G) in Z and hence (Z-H (Con Z --+ Con G), but not (Z-H Con G.
MODELS, TRANSLATIONS AND INTERPRETATIONS
29
If (8 1 ) is not axiomatizable, we require a partially recursive function fl(n) such that if n is an axiom of (81) , fl(n) is the number of a proof in (8 2 ) of the model of n. In practice there is an important case where (8 2 ) is a subsystem of (8 1 ) and (8 1 ) and (82 ) are obtained by adding the same axioms to the axiomatizable systems (8~), (8~): in this case fl(n)=n. Below we shall consider extensions of (8') which are obtained from (8') by adding verifiable primitive recursive formulae; these non-axiomatizable systems may be handled by quite elementary means. (iii)
FROM CONSISTENCY TO MODEL
If (81 ) has a finite set of axioms and h Con 8 1 where (82 ) contains the system Z then a model of (8 1 ) may be defined in (82 ) : this is the famous Loewenheim-Skolem-Godel-Bernays completeness theorem. It is known that under the present conditions (81 ) does not generally have a recursively enumerable model, see, e.g., [6] or [7]. (The proof that the model is not necessarily recursive is given in [10].) The situation is not very different with higher order predicate calculi, [8]: if h Con 8 1 and (82 ) contains Z then a model of (81 ) may be defined in (8 2 ) where the models for the sets of (8 1 ) range over a class of arithmetic sets definable in P2 or 22 (in Mostowski's notation), [9]. MODELS OF ARITHMETIC AND SET THEORY
The construction of models has, perhaps, contributed more towards an understanding of mathematics, in particular of the impossibility of characterizing by axiomatic means the notions of "integer" and "set", than any other single work of mathematical logic. In picturesque language: every consistent system of arithmetic has models which are too large; i.e. while these models contain (representatives of) the numerals they also contain other terms like Skolem's functions; on the other hand every consistent system of set theory has models which are too small; i.e. they possess models by classes of sets, as in [9], which satisfy all the
30
G. KREISEL
closure conditions imposed by the axioms and rules of proof, yet these classes do not exhaust all "arbitrary" sets. SUMMARY
The notion of a model allows one to study relations between (axiomatizable and non-axiomatizable) systems which contain the same logical apparatus 2 (classical quantification theory), but the notion is not defined for systems with different logical character; in particular, it is not suitable for comparing essentially undecidable and decidable systems, or classical and intuitionistic systems. § 3.
Translations
Two well known consistency proofs, which do not use models, relate systems of different logical character to each other; one is the short consistency proof for the predicate calculus which shows that every theorem in this system is true in a universe consisting of a single individual, and, since the latter statement can be expressed in the propositional calculus, we have a relation between the predicate calculus and the propositional calculus; the other is G6del's proof, in primitive recursive arithmetic, which establishes that a formula is provable in Z if and only if a certain associated formula is provable in intuitionist arithmetic. Wang [11] has generalized these relations in his definition of "translation" which we reproduce with certain modifications: 3 A recursive function r(n) is an (8)-translation of the system (81 ) into (8 2 ) if
f- (Ey) Prov, (y,
n)
-'>-
(Ey) Prov, [y, r(n)]
2 Tarski [3], p. 22, footnote 17, mentions the interpretation of quantifiers but gives no details: presumably he means some such definition as
(x) +--> -, (Ex) -, .
3 (i) If A 1 of (S1) is associated with As of (Ss)' Wang requires that [-1 A 1 be equivalent to [-2 As in Def. 2 of p. 284 of [11]; he does not use this condition, which is not satisfied by the "translation" of the predicate calculus into the propositional calculus. (ii) He considers only Z-translations, which seems an artificial restriction.
MODELS, TRANSLATIONS AND INTERPRETATIONS
and
~ T[v1(n)]=v 2[T(n)]
31
(if n is a formula of (B1)).
(The definition assumes that the systems (B 1 ) and (B 2 ) contain symbols for negation). If (B) is externally consistent ([12]) with respect to recursive functions, the existence symbol may be eliminated: ~ Prov. (p, n) ~ Prov, [n(p), T(n)]. TRANSLATION TO CONSISTENCY
If (B 1 ) has an (B)-translation into (B2 ) then
~
(Con B 2 ~ Con B 1 ) .
CONSISTENCY TO TRANSLATION (LIMITATION)
There are systems (B), (B 1 ) , (B 2 ) such that ~ (Con B 2 ~ Con B 1 ) , but (B 1 ) has no (B)-translation into (B 2 ) . Let (B) be the system Z, (B1 ) the system (Q) of [3], p. 51, (B2 ) the propositional calculus IIo. Then, by [5], (Z-H Con Q and hence (Z-H (Con IIo ~ Con Q), but (Q) cannot be Z-translated into any decidable system, such as IIo : if T(n) is the proposed translation, let T1(n)=O if (the formula with number) T(n) is provable in IIo' T1(n) = 1 if T(n) is not so provable; then there is a term q of (Q) such that T1(q) = 1 has the number q, and by the diagonal argument we get a contradiction. (See the remark below.) MODEL TO TRANSLATION
If (B 1 ) has a model in (B 2 ) and if this model is established in (B) then (B 1 ) has an (B)-translation into (B 2 ) . TRANSLATION TO MODELS (LIMITATION)
There are finitely axiomatizable systems (B 1 ) and (B 2 ) which can be N w-translated into each other, but have no models in each other. Take for (B1 ) the elementary quantification theory of addition, for (B 2 ) that of multiplication (or of some suitable monadic predicate). Then, since these systems are decidable (in N w) we can map Ai of (Bi) into 0 = 0 if h Ai and into 0 i= 0 if h Ai is false. But it is known that the systems mentioned above have no models in each other.
32
G. KREISEL
Remark 1. The examples above, which establish limitations, are artificial because the comparison system (8) is stronger than either (8 1 ) or (82)' It would be interesting to decide whether Bernays' set theory with class variables but without the axiom of infinity (cf. appendix of [7]) can be Z-translated into Z: for, if it can we should have a finitely axiomatizable system containing Z which can be Z-translated into Z without having a model in Z (see [5]); if it cannot be Z-translated into Z we should be able to prove in Z the consistency of a system from the consistency of Z without there being a Z-translation into Z. Remark 2. If (82) is a quantifier-free system, the notion of "negation" presents a difficulty: either, if Al (of (81 ) ) is related to A 2(a ) we relate ,AI to ,A2(O(
Apart from the consistency of (8 1 ) relative to (82)' a translation of (81 ) into (82) seems to provide, in general, little information of any interest about (81) ; this is particularly true of the translation of the predicate calculus into the propositional calculus. To make further progress it is desirable to analyze this translation more closely: Speaking informally, the translation weakens the formula which is translated: "A is identically true" is replaced by "A is true in a universe with a single individual" (This notion of a weakening can be defined for any systems (8 1 ) , (82) with respect to a common extension (8): f- (AI -+ A 2 ) but not f- (A 2 -+ AI)') In general, a translation replaces a proposition Al of (81 ) by a weaker one, A 2 , of (82): the formula A 2 does not "express" the "full content" of AI'
MODELS, TRANSLATIONS AND INTERPRETATIONS
33
These informal criticisms will be made clearer in the next section and will there be formulated syntactically. Oonclusion. The notion of "translation" applies to a wider class of pairs of systems than that of a "model". However, two important topics in the foundations of arithmetic cannot be treated by means of this notion: there are no translations of elementary arithmetic into quantifier-free systems although these systems have a special claim to attention. A translation into (8 2 ) does not generally extract from a formula Al of (8 1 ) all that-naively speaking - one can learn from a proof of Al in (81 ) about the system (8 2 ) , § 4.
Interpretations
GENTZEN'S Hauptsatz or HERBRAND'S theorem establish an interesting connection between the predicate calculus II and the propositional calculus Ilo ; the contrast between this work and the translation discussed in the last section will make our criticisms much clearer. I consider a formula (x)(Ey)(z)A(x, y, z) (or m:) where A(x, y, z) is quantifier-free. (i) m: can be proved in III precisely if there are quantifier-free terms yo(a), ... y..(a, a l ... a,,) of III such that A[a, yo(a), Ut] VA[a, Yl (a; Ut), a 2 ] V .. VA[a, y,,(a; Ut· .a,,), a..+ 1 ] (i)
is an identity (or: a theorem of IIo)' Read: either the function yo(a) satisfies A [a, yo(a), Ut] for all Ut, or it does not when a l =~; then Yl(a;~) satisfies A [a, Yl(a; al ) , a 2 ] for all a2 etc. From each formula of the Herbrand disjunction (i) one can derive m: by means of classical logic: the disjunction is stronger than m: (in classical logic) in contrast to the translation of III into IIo. (ii) Since the analysis above of III by Ilo cannot be extended to other systems (see [13], p. 114) it is desirable to give another formulation of Herbrand's theorem which is intuitively obvious: Suppose m: were false in the sense that if x=xo and Z= j(y) then -, A [x o, y, j(y)] for all y. To prove m: in classical logic, it is sufficient to show that this is impossible, i.e. to show A (Ey)A[x, y, j(y)]; I
3
34
G. KREISEL
one does this by constructing a functional Y(f, x), i.e. a term containing the symbols f and x, such that for all x and f we have A{x, Y(f, x), f[Y(f, x)]}.
5
These considerations can be sharpened in two directions: (a) the functionals Y which are needed for our purpose, can be enumerated in advance and depend, of course, on the particular system, (b) the functionals are such that in order to prove ~ we need not assume A{x, Y(f, x), f[Y(f, x)]} for all functions, i.e. one does not need the notion of an arbitrary function, but (e.g. in arithmetic) (x) (n) A {x, Y(f", x), f"[Y(f,,, x)]} -+~,
where I" ranges over a suitable sequence of functions (a base, [1], p. 120). 6 We can thus list the functionals Y v Y 2 , •• , and instead of the sequence of disjunctions in (i) we associate with ~ a sequence .. A{a, Y.(f, a), f[Y.U, am, ••
In general, each such formula is stronger than ~ (classically), since ~ can be derived from A{a, Y.U, a), I[Y.U, am by classical logic, while one cannot read off from ~ which particular functional is needed. This discussion of Herbrand's theorem suggests three general types of relations between systems (8 1 ) and (8 2 ) : Definition: A recursive function 1(a, n) is a (i) disjunctive, (ii) complete disjunctive, (iii) strong disjunctive interpretation if the value of 1(a, n) is the number of a formula A~") of (8 2 ) when a is a formula Al of (81 ) ,
¥
I In classical logic, 2( f-> ~ (x)(Ey) A[x, y. f(y)] f-> (x)(z) A [x, f(x), z], the two equivalences being dual to each other. As explained below, the equivalence ~(x)(Ey)A[x,y,f(Y)] f-> ¥(x)(z)A[x,f(x),z] does not hold constructively, if "function" is interpreted differently in existential and universal propositions. In particular, if 2( holds classically, t\(x)(Ey) A [x, y, f(x)] 1 holds constructively (intuitionistically), but not always ¥(x)(z) A [x, f(x), z]. • The values of these functionals are determined when the values of f are given for a suitably large, but finite set of arguments of t. e.g, if Y(f, x) is f{fU(x)]}. For details, see the appendix.
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MODELS, TRANSLATIONS AND INTERPRETATIONS
and (i) if h Al then, for some n, h A~~) if ,1 L I A then for each n A (,,) is "false'" l ' , 2 ' (ii): (i) and: if, for some n, f-2 A~") then hAl; (iii): (i) and: if (B2 ) is a subsystem of (B1 ) then Al can be proved from A~") in (B1 ) (for each n) (A~") is "stronger" than AI)' If (B2 ) is quantifier-free "A~") is false" means that there is a substitution for the free variables of A~") which reduces this formula to a false numerical formula. In symbols (p, a, n are free variables):
f- Prov, (p, a) -+ Prov, {n(p), 1[a, n'(p)]} f- Prov. [P, 'Jf1(a )] -+ Prov, {nl(p), V2{0'2[1(a, n), n~(p, where
0'2
n)]}},
is the Godel substitution function for (B2 ) .
TRANSLATION TO INTERPRETATION
An (B)-translation is an (B)-disjunctive interpretation (of course, not necessarily complete). INTERPRETATION TO TRANSLATION (LIMITATION)
There is a (complete) interpretation of the extension of Z by verifiable primitive recursive formula in the same extension of N., but there is no translation in this direction. INTERPRETATION TO CONSISTENCY
From an (B)-interpretation of (B1 ) in (B2 ) we get f- (ConB2-+ ConB1). CONSISTENCY TO INTERPRETATION
Every axiomatizable consistent system has a complete, 7 strong disjunctive interpretation in numerical arithmetic: associate with Al the sequence Prov, (n, a), i.e, "AI can be proved in (B1 ) " : there7 Note that the trivial interpretation of JSL 16, p. 248, § 16, is not complete: with Al is associated the formula I PrOVI [p,1I1(a)] with the free variable p; if this can be proved in (8 (for one a) Con 8 1 , Now if (x) I PrOVI (x, q) (or 0) has the number q, (Z - f-) (Con 8 1 -+- 0) and, hence, if (8,) contains Z, also f-t 0: 0. of (8t) is associated. in the trivial interpretation with -, 0 of (81 ) ; this formula cannot be proved in (81 ) , but ,0 can be proved in (8a).
t)
I-t
36
G. KREISEL
fore the interpretation of non-axiomatizable systems as above offers the most interesting problems. USES OF INTERPRETATIONS
It seems best to confine the discussion of the uses of interpretations to those of the particular interpretation described above, the so called no-counterexample-interpretation. The notion itself is so general that its main use seems to consist in the general orientation which it affords and in negative results, e.g, [13] remark to theorem 1 and theorem 3; further it allows one to formulate Mostowski's question whether there exists an analogue to Herbrand's theorem for intuitionist logic, as follows: does there exist a disjunctive interpretation of the intuitionist predicate calculus in the intuitionist propositional calculus which remains invariant for every extension of these two systems by the same consistent set of quantifier-free axioms? The no-counter example-interpretation has been useful both in logic and in mathematics. !J-consistency. A proof of the co-consistency of Z in a sharp form can be given by means of our interpretation. Since the proof requires a certain amount of formal detail it is relegated to the appendix. Computable Functionals are brought into prominence by the nocounter example-interpretation, and are handled by constructive means. The notion of a computable functional seems to deserve attention because it presents difficulties which do not touch computable functions. Intuitionism. For me at least, the no-counterexample-interpretation seems to provide a better approach to Brouwer's ideas than the formal systems given by Heyting; in particular, the passage from arithmetic to intuitionistic analysis with its free choices becomes very natural. - Brouwer means by "function" a computable function; therefore when he wishes to use (something like) the concept of an arbitrary function, he needs a new word, and chooses "freie Wahlfolge". He does not speak of the existence of a "Wahlfolge", nor of propositions about all "functions". - An
MODELS, TRANSLATIONS AND INTERPRETATIONS
37
intuitionistic proof of (x)(Ey)(z)A(x, y, z) or V(x)(z) A [x, !(x), z) t requires the existence of a computable function !; a refutation on the other hand, i.e. a proof of 1\ (Ex) (Ez) , A [x, !(x), z) requires t for every Wahl!olge ! the existence of numbers x t and Zt such that ,A[xt , !(x t ), Zt]. One cannot expect these two possibilities to be exhaustive (yet another formulation of the failure of the law of excluded middle). 8 Remark. I believe that this asymmetry in the use of "function" in existential and universal propositions is characteristic of Brouwer's writings. But there is also an independent justification from the point of view of computing techniques. "Existence of a function" in connection with a computation can refer only to some method of computation having been made available; on the other hand a universal proposition about functions may enter instructions as follows: we may have a random element in the machine, and the instruction may state that at some particular stage the machine should print 0 if the random sequence has a property and I if it the sequence does not have the property: to show that this instruction is not circular (for any random sequence) requires a proof about all freie Wahlfolgen. (Examples of such proofs are given in the appendix). Arithmetic and Analysis. Especially convergence theory and compactness theory, where non-constructive proofs occur, is greatly unified if the results are expressed in quantifier-free formulae and proved by quantifier-free methods. The elimination of non-constructive methods is explained in various publications, but it is worth while giving a simple example. It is known that (even) for (computable) reals iX, f3 one cannot generally decide whether iX = f3 or iX < f3 or f3 < iX; but suppose that the ordering of 8 The no-counter example-interpretation retains the law of excluded middle in the following form: it replaces (x)(Ey)(z)A(x, y, z) by 1'(x)(Ey) A [x, y, I(y)] and its negation by 1\(Ex')(Ez') ,A[x', g(x'), z"]. Now, if 1/
for x = X o and I = 10 we have, A [x o, y, lo(y)] for all y then, for any g, take x' =xo'z' =fo[g(xo)]: so, if ~(x)(Ey)A[x,y,/(Y)] can be refuted, 1\(Ex')(Ez') , A[x', g(x'), z'] can be proved. 1/
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G. KREISEL
the continuum has been used in the proof of a quantifier free formula B[n, A.(n)] V B[n, ~(n)] V B[n, ~(n)] (whose variable n ranges over the integers) as follows: (X={3 --+ B[n, A.(n)], (X<{3 --+ B[n,
(3<(X --+ B[n,
~(n)],
~(n)].
Consider any n; B[n, A.(n)] is decidable; if it is false we have a proof of (X =1= {3: by calculating (X and {3 to a sufficient degree of accuracy (depending on the proof of (X={3 --+ B[n, A.(n)]) we can decide (X < {3 or {3 < (x, and then proceed. (The argument assumes that the methods of proof used are externally consistent.) I learnt recently that some 30 years ago Skolem recommended the use of quantifier-free methods in arithmetic: though he did not mention function variables, I believe that the no-counterexample-interpretation is in the spirit of Skolem's recommendation. CONCLUSION
One may take one of two views of the no-counter exampleinterpretation: since any classical theorem ~ can be proved by finitist methods in a version which, classically speaking, is at least as strong as ~, a preference for finitist methods is confirmed by the fact that classical methods are no stronger than finitist ones; and a preference for classical methods is confirmed by the fact that they do not lead to (quantifier-free) formulae to which a finitist could object. My own view on such matters is: one man's meat is another man's poison.
APPENDIX We shall prove the w-consistency of Z if a formula (Ey)(x1)(EYl)· .. (x,.)(Ey,.)A(y;
9
in the following form:
XI • • X,.,
Yl .. y,.)
can be proved in Z there is an ordinal recursive functional tP(p) of 9 R. Gandy has informed me that hr. possesses a proof of the w-consistency of Z which is based on Gentzen's work.
MODELS, TRANSLATIONS AND INTERPRETATIONS
39
order <e2 such that p[l.P(p)] is not (the number of) a proof in Z of the formula --, (xl)(EYl)' . (x,,)(EY,,)A [l.P(p);
Xl' .
X", Yl' .y,,].
In other words, given a function p whose arguments and values are integers, it is not true that the value of p(m) is the number of a proof of --, (Xt)(EYl)' . (x,,)(Ey,,)A(o(m>; Xl' . X", Yl' .Y,,) for all m. The reason for this somewhat lengthy definition of "w-consistency" is given on p. 41 of [7]. Incidentally, the proof below gives a solution to problem 2 of JSL 17 (1952), p. 160. An analogous but simpler argument establishes the w-consistency of the elementary quantification theory of addition and multiplication (without induction). We show that this argument cannot be formalized in Z while it is well known that the consistency of this quantification theory can be proved in Z. Since the proof uses the notion of a computable functional it seems desirable to begin with a section on this notion. § 5.
Computable Functionals
Informal Idea. (i) By a "computable function" with integral arguments and values is meant a method which provides for any integer 0("> an integer O(m,,> (its value). It should be noted that this definition applies to representations of integers by numerals only: e.g. the function lX which satisfies the relations lX(O) = 1, lX(n+ 1) = 0 is computable; yet, if in some particular system (Ex)A(x) is undecidable, these defining conditions do not decide the value of lX[,u.,A(X)] in the system concerned; in other words, representations of integers by ,u-symbol expressions are (properly) ignored in the definition of computability. (ii) By a "computable functional" (whose arguments range over integral valued functions of the integers) with integral values is meant a method which provides for any such function f an integer o(m,>; the method should be such that once a sufficiently large, but finite number of values of f has been computed the value of m, should be determined; it is not assumed that f is computable,
40
G. KREISEL
i.e, that a method of computing t has been given in advance. The notion of a computable functional is not as definite as that of a computable function since there is no analogue to the representation of integers by numerals. However, we shall see below that in practice this difference is less serious than appears. Standardization. Kleene has given standard forms for computable functions and functionals: (i) primitive recursive functions ~(l; m, n), i(n) such that, if tP is a computable function there is an integer l~ with the properties
(a) for all n there is an m such
~(l.p;
o(n), o(m))=o
(b) for all n, tP(o(n))=i{,u.,[f)(l.p; o(n), x)=O]}; (ii) a primitive recursive predicate T(l; m), such that if (/J is a computable functional there is an integer l.p with the properties
(a') For all t there is an m such that T[l.p, t(m)) where t(m) denotes
.II p:(i)+\ Pi being the i th prime (Po=2).
,<m
(b
/)
For all [, (/J(f) = j{,ullT [l .p, t(y)]}.
Observe that, if these conditions are satisfied, for any given [, one can verify this fact a posteriori by means of a finite procedure: if there is an n satisfying (a') one need only calculate the values of t for arguments ~ n, and then one can choose the smallest n satisfying condition (a /). The values of t for arguments exceeding n do not enter into the calculation of the functional. The indefiniteness of the notion derives from the fact that the (crucial) condition (a') may be satisfied for all t of some large class (e.g. all arithmetically definable functions), but not for certain impredicative functions. In practice the analogy with computable functions is to some extent restored by the indefiniteness of the notion of a "correct proof" for condition (a): e.g. most of the proofs of condition (a) proceed by ordinal induction with respect to some ordering -<: to establish the well-ordering character of <, one has to show that for all functions t there is an n such that.., [f(n+ 1) -< t(n)), i.e.
MODELS, TRANSLATIONS AND INTERPRETATIONS
41
a proposition of the same form as condition (a'). 10 Below we shall be concerned with two kinds of classes of functionals: the first are so simple that their computability is apparent, the second are so defined that they are computable for all functions if they are computable for arithmetically definable functions (provided the proofs of computability employ the standard substitution rules): thus the vagueness in some contexts of the concept of "arbitrary function" does not spoil the work below. (If the reader feels that there can be no "constructive" treatment of propositions about arbitrary functions since the class of arbitrary functions includes highly non-constructive entities, he should either consider simple examples from the elementary calculus with free variables or should compare the situation with the propositional calculus; an identity of the latter is a schema: provided the truth values of the constituent propositions have been decided the truth values of the compound proposition is decided, and it is true whatever truth values have been assigned to the constituent propositions; the propositions about arbitrary functions which we consider have a similar property: they are decided provided the values of the functions concerned have been assigned at a suitable finite number of arguments: only now the (number of these) arguments which are used in deciding a given proposition may depend on the particular function used; the proposition is an identity if it is true for an arbitrary choice of values of the function. The fact that we often use "definitions" of functions which do not provide an immediate method of calculation or, on inspection, turn out not to be unambiguous definitions at all, has a perfect analogue with "definitions" of properties (propositional calculus).) 10 More precisely thus: analysis of such an argwnent often shows that the argwnent does not need well-ordering with respect to arbitrary (des. cending) sequences, but only with respect to sequences of a restricted type; e.g, in the consistency proof of [15] only primitive recursive sequences appear. But the proof that the particular orderings < used in [15] are well 1> orderings, makes no use of this restriction; and the fact that the orderings are well orderings with respect to arbitrary sequences is needed, e.g., in the full statement of the no-counterexample-interpretation of arithmetic.
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G. KREISEL
Ordinal Becursioe Functionals. The following simple class of funotionals suffices for our w-consistency proof. Let I, denote an ordinal recursive function of order < <X • 4>(/; m, n) is the maximum of the numbers m and I(x) for x ~ n. w(/; m, n) is a kind ofiteration functional which may be computed by means of the following relations: w(f; m, 0)=(/>(/; m, 0), w(f; m, n+ 1)=4>[/; m, w(f; m, n)]. The notion of an ordinal recursive functional of order <X whose arguments are the function variable I and the individual variable m is defined inductively as follows: The variable m and a numeral 0("1 are such functionals; if t:P, lJI are such functionals so are rp(f; t:P, lJI), cot]; t:P, lJI}, 1(t:P), li(t:P) , and ,u",[x ~ t:P & A(f, x)] where A is a formula of the predicate calculus whose non-logical constants are symbols for ordinal recursive functionals of order <x, and whose quantifiers are bounded. It should be noted that the definition is easily modified so that instead of restricting / i to a class of ordinal recursive functionals we can take them to be general recursive or even arithmetically definable. The modifications necessary for "representing" these functionals in arithmetic (see below) are obvious. N.B. To distinguish those functionals which are obtained by letting t, range over the class of general recursive functions, from Kleene's uniformly general recursive functionals, we shall call them general recursive I-functionals ("I" for: iteration). The I-functionals are a proper subclass of Kleene's class. Arithmetic Functionals (cf. [14], p. 3). The informal notion is this: if (x1)(EYl)' . (x),,(Ey,,) A(~ . .x"' Yl' . Y,,), say m, is "true" then A (EYl)" (Ey,,)A[xl YZ(Yl)' ·Y,,(Yl· ·Y,,-l), Yl' .y,,],
and
q... u"
is a computable functional, where Y is the number of the n-tuple in some linear ordering. Plainly, the functional is computable for "all" functions if it is computable for arithmetically
43
MODELS, TRANSLATIONS AND INTERPRETATIONS
definable functions; in particular if the expression above is computable for m=p,.,1 -, (EYI)(x 2)(EY2)' . (xn)(EY,,)A(xl
•• X"YI'
.Y,,),
=m
g2(a) = p,.,I -, (EY2)(xa)(EYa)' . (x,,)(EY,,)A(mx 2.. x" a Y2' .y,,), etc.,
then it is computable for all functions. The informal notion is defective because it depends on the vague notion of a "true" arithmetic formula. When this condition is sharpened in the form that 2l is provable in a system (8) of arithmetic, the problem arises what conditions (8) must satisfy (in addition to being a consistent extention of recursive arithmetic). For instance (8) may be externally consistent with respect to the class of general recursive functions (see [14], Theorem IV), without being w-consistent. In this case, if I- 2l, then our functional is computable for all general recursive functions, but not necessarily computable in the naive sense set out at the beginning of this section. This explains the apparent paradox of [14]: there it is shown (para. 23) that any externally consistent system (8) has a no-counterexample-interpretation by means of functionals of the system (8) itself, while (para. 17) there are externally consistent extensions of Z which do not have such an interpretation by means of computable functionals; in particular, these extensions do not have such an interpretation by means of general recursive 1-functionals. The notion of an arithmetic functional allows one to construct a computable functional which is not a general recursive 1functional. Outline of the Proof. Observe that the class of general recursive functions can be enumerated in elementary arithmetic by means of a 2-quantifier formula, and similarly the class of general recursive 1-functionals can be so enumerated. Observe next that there is a primitive recursive enumeration 17,,(a) of functions which are ultimately zero. By means of a Godel substitution function we construct a formula (x)(Ey)(z)A(x, Y, z) which expresses the following proposition: for each integer x, if x is the number of a general recursive I-functional XU) then there is a function 17" and a
44
G.
KREISEL
proof III Peano's arithmetic of the formula ",A{O"',X(17011)' 17011[X(17011)]}". The formula (x)(Ey)(z)A(x, y, z) is true; otherwise there would be a general recursive I-functional X(f) with number x such that, for all y, there is no proof in Peano's arithmetic of "-, A {O"', X(t]01I), 17011[X (t]01I) ]}": since general recursive I -functionals are computable, we should have A{O"', X(t]I/)' 171/[X(17I/)]}' and hence (x)(Ey)(z)A(x, y, z). Evidently the argument just given cannot be formalized in any extension of Z by defining relations for recursive functions, since to any such system the method of [15] can be applied, which yields general recursive I-functionals. The reason is that though (it can be proved in Z that) the formula (Ey)(z)A{O"', y, z}
can be proved in Z for each numeral 0"', one cannot infer in the extensions of Z mentioned above, that (x)(Ey)(z)A(x, y, z). As a corollary it follows that in these systems one cannot prove (all instances of) the principle (x)(Ey) Provo [y, £x(x)] --* (x)~(x)
where £x(x) is the number of the formula "~(O"')"; not even if ~(n) is restricted to 2-quantifier formulae. Details. Consider the class of I-functionals where t, ranges over all the functions j,ul/[l)(l, n, y) = 0] l = 0, 1, 2, .. i.e. not only over those l which satisfy (x)(Ey)[l)(l, x, y)= 0]. For any givenintegernone can determine systematically (i) whether n is the number of such a functional, (ii) the parameters ll' .. lk
MODELS, TRANSLATIONS AND INTERPRETATIONS
45
Here 'fJm is the m th function in some linear ordering of functions of the integers which are ultimately zero. There is a primitive recursive function a(n, a, b) with the following property: if a is the number of a formula (Ey)(z)9l(y, z) then a(n, a, b) is the number of the formula 9l{rp(O(1I>, O(b)), 'fJb["P(O(1Il, O(bl)]}, where 0(11) denotes the nth numeral (0', 0", Oil', .. ). sia, a) is Godel's substitution function. Let < YI' Y2> be the yth pair of integers in some ordering of pairs of integers, and, for each numeral O(1Il, let p(n) be the number of the formula (Ey)(z) {, 0(0(11), y, z) V Prov {Yl' VC1[0(1I), s(a, a), Y2]}}'
with the free variable a. Let q(O(1Il) denote the expression s[p(O(1Il), p(O(1Il)].
Then (the value of) q(O(1I)) is the number of (Ey)(z){,O(O(1Il, y, z) V Prov {Yl' Ya[O<1Il, q(O<1Il), Y2]}}
(i)
For each numeral 0(11) (i) can be proved (in Z) as follows: If (i) were false, n would be the number of a computable functional since (y)(Ez)O(O(1Il, Y, z). Hence "P(O'"l, m) is computable. Also , Prov {bI> vo [0(11), q(O'"l), b2 ]} with free variable b. But, for each b2, Ya[O(1II, q(O(1II), b2] is the number of the formula , { , 0[0(11), "P(O(1Il, O(",,), 'fJb, "P(0(1Il, O(b,))] V Prov {"Pl(O(1Il, O(b,»), ya[O(1Il, q(O(1II), "P2(0(1I), O(b,»)]}}
(ii)
But, by a straightforward extension of [12], pp. 312-324, since "P is computable, we can prove in Z: for each b2 if (ii) is not provable in Z then ,0[0(11), "P(O(1I), b2), n: "P(o(n>, b2)] V Prov {"Pl(o(nl, b2), ya[o(nl, q(o(nl), "P2(0(111, b2)]}
(iii).
If n is the number of the functional N(f), we have, rewriting (iii), ,own), N('fJb,),
n; N('fJb,)]
V Prov {NI('fJ",) , va[o(n), q(o(nl), N 2 ('fJb,m
(iv).
46
G. KREISEL
But, as in the discussion of interpretations above, we get from (iv) (Ey)(z){, O(O
It follows that the functional
flll{' C(n, Y, f(y)] V Prov {Yl' va[n, q(n), Y2]} is computable, but not a general recursive I-functional. Representation of Functionals in Arithmetic. Let Zk be the extension of Z by defining relations for the general recursive functions i: I ~ i ~ k, I consider the representation of a general recursive functional in which the functions t, occur. By the representation of such a functional 8 we mean a term XU) with the following properties: (i) XU) is a term of the predicate calculus made up of the function variable [, the symbols t. and the nonlogical constants of Z, (ii) for each term t(a) of Z,. the defining relations for 8 (as given in the section on ordinal recursive funotionals) can be proved in Zk when t(a) is substituted for f(a) and X(t) for 8(1). Note that this requirement is metamathematical, and is to be verified by a syntactic argument. The representation has the following two properties which are the formal equivalent in arithmetic of the informal requirement of computability. (i) Given any term t1(a) of Zk there is a term t of Zk such that for any term !:a(a) of Zk (x)[x ~ t ~ t1(x)= !:a(x)] ~ X(t1)= X(!:a)
can be proved in Zk (by means of a proof whose rank depends on the rank of t 1 ) . Informally speaking this means that if the values of t 1 , !:a agree over the range x ~ t the values 8(t1 ) , 8(!:a) are equal: only a finite number of values of t 1 are needed to calculate the value of 8(t1 ) . (ii) Given terms t1(a) and t of Zk there is a term t' of Zk such that x ~ t ~ t1(x) = 1'Jt'(x)
can be proved in Zk' Informally speaking this means that to any
MODELS, TRANSLATIONS AND INTERPRETATIONS
47
function t 1 of Zk there is a function 11t' which is ultimately zero, and has the property that E(11t,)=E(t1 ) . The representation of funotionals is established in detail in [14], &-7. It should be observed that this representation applies, in particular, to ordinal recursive functionals of order 8 (or higher orders too), since the substitution method of [15] applies to any extension of Z by verifiable quantifier-free formulae. The representation solves Problem 2 of JSL 17 (1952), p. 160. (When I stated the problem I did not realise that the functionals considered could be represented in a quantifier-free extension of Z, but thought that one would need a principle of ordinal induction of order 8 for quantified formulae of Z: no substitution method is known for this extension of Z).
§ 6.
(a)-Consistency
Without loss of generality we may confine ourselves to formulae with three quantifiers: (Ex)(y)(Ez)A(x, y, z). Lemma 1. There is an ordinal recursive functional EU, m) of order 8 such that if -, (y)(Ez)A(n, y, z) is proved in Z by a proof with number m then -, A {o
can be proved in Z. to which the function symbol p has been added. The substitution method applies to this extension of the system Z., and thus we get a functional c[)(p) (actually of order <8 2), and a functional lJI(p, a) such that if c[)(p) is substituted for n, p[c[)(p)] for m and lJI(p, a) for t(a) the formula of lemma 1 is false. This
48
G. KREISEL
means that p[(l)(p)] is not the number of a proof of the formula -, (y)(Ez)A[(l)(p), y, z].
This proves the w-consistency of Z in the sense explained at the beginning of the appendix. Remark 1. Observe that the functional (l) is of order ~ e while, if (x)(Ey)(z)A(x, y, z) has been proved in Z there is a functional BU, a) of order <e such that A{a, BU, a), f[BU, a)]}. The reason is clear: "w-consistency" refers to proofs with numbers p(m) whose rank (number of quantifiers) is unbounded; therefore if the substitution method were applied to each of these proofs p(m), the resulting functionals would be of order exceeding any ordinal < e. Remark 2. It is interesting to consider the analogous proof of the w-consistency of the elementary quantification theory of addition and multiplication without induction. Since all functionals of the predicate calculus may be enumerated by means of a functional of order w 2 one might expect to be able to prove the w-consistency of arithmetic without induction in Z itself. This is not true. What is true is that for each k one may prove in Z the w-consistency of formulae with ::( k quantifiers of the elementary quantification of addition and multiplication, e.g. by means of a truth definition as in [5]: this depends on the fact that a proposed example p(m) of an w-inconsistency, i.e. a sequence of proofs p(m) of -, S2!(o(m)) together with a proof of (Ex)S2!(x), may be restricted in advance; if 2{(Olm)) is of rank k and if it can be proved in the predicate calculus at all, it can be proved by a sequence of formulae none of which contains more than k quantifiers. When one applies the analogue to lemma 2 one gets functionals of order Wk (where WI = w and wn+! =w;n) because in the representation of the functional E in Z one applies induction to formulae with k quantifiers (i.e, s-formulae of the second kind of rank k are used, if the proof is formalized with the help of the s-symbol). Thus one cannot expect to prove the w-consistency of arithmetic without induction in Z itself. This result may be established by means of a ve.. y elegant observation due to Gentzen: Gentzen's Lemma. A proof in Z of a formula A may b ; replaced
MODELS, TRANSLATIONS AND INTERPRETATIONS
49
by a proof of A in which there is only a single application of the rule of induction. If induction has been applied to the formulae A1(x), .. , Ak(x) in this order, take a variable a not occurring in the proof, and apply the principle of induction to the formula 2£(x), namely [a=1 ~Al(X)] & ... & [a=k~Ak(x)], i.e. use: 2£(0) & (x)[2£(x)
~
2£ (x + 1)] . ~ (x)2£(x).
(i)
Form this formula, A1(x), . . Ak(x) can be derived in elementary arithmetic without induction: A1(0), A1(x) ~ A1(x+ 1) can be so derived; we substitute 1 for a in (i) and hence we get (x)A1(x); A 2(O) can be derived in elementary arithmetic from (x)A1(x) and so can A 2(x) ~ A 2(x+ 1), hence (x)A 2(x) can be derived from (i), and so forth. This argument can evidently be formalized in Z itself. And from it we prove, again in Z, that Con Z follows from the w-consistency of arithmetic without induction. For, if the formula 0 = 1 could be proved in Z it could be proved by means of a single application of induction (i). Thus, since 0 -:f' 1 can be proved in arithmetic without induction we should have a proof in this system of a formula 2£(0) & (x)[2£(x) -+ 2£(x+ 1)] & (Ey) --, 2£(y), or: a proof of (Ey) --, 2£(y) together with proofs of 2£(0), 2£(1), 2£(2), .. Thus arithmetic without induction would be eo-inconsistent. Since Con Z cannot be proved in Z, the w-consistency of arithmetic without induction cannot be proved in Z, as was suggested by the analysis of our w-consistency proof above. Remark 3. As has been pointed out, the externally consistent, but w-inconsistent system of para. 24 in [14] has a no-counterexample-interpretation by means of functionals of the system itself. The functionals are not, e.g., general recursive, and the present w-consistency proof cannot be applied to that system. REFERENCES [1] G. KREISEL, A variant to Hilbert's theory of the foundations of arithmetic, The British Journal for the Philosophy of Science, 4, 14 (1953), 107-129.
50
G. KREISEL
[2] W. ACKERMANN, Die Widerspruchsfreiheit der allgemeinen Mengenlehre, Mathematische Annalen, 114 (1937). [3] A. TARsKI, A. MOSTOWSKI, and R. M. ROBINSON, Undecidable Theories, Studies in Logic and the Foundations of Mathematics, Amsterdam (1953). [4] A. MOSTOWSKI, On models of axiomatic systems, Fundamenta Mathematicae, 39 (1952), 133-158. [5] G. KREISEL and H. WANG, Some applications of formalized consistency proofs, Fundamenta Mathematicae. [6] A. MOSTOWSKI, On a system of axioms which have no recursively enumerable arithmetic model, Fundamenta Mathematicae, 40 (1953), 56-61. [7] G. KREISEL, Note on arithmetic models for consistent formulae of the predicate calculus, II, Proceedings of the XIth international congress of Philosophy, XIV, 39-49. Amsterdam (1952). [8] L. HENKIN, Completeness in the theory of types, Journal of Symbolic Logic, 15 (1950), 81-91. [9] G. KREISEL, Remark on complete interpretations by models, Archiv fur mathematische Logik und Grundlagenforschung. [10] , Note on arithmetic models for consistent formulae of the predicate calculus, Fundamenta Mathematicae, 37 (1950), 265-285. [11] H. WANG, Arithmetic translations of axiom systems, Transactions of the American Mathematical Society, 71 (1951), 283-293. [12] D. HILBERT and P. BERNAYS, Grundlagen der Mathematik, Vol. II (1939), Berlin. [13] G. KREISEL, On the concepts of completeness and interpretation of formal systems, Fundamenta Mathematicae, 39 (1952), 103-127. [14] , Some concepts concerning formal systems of number theory Mathematische Zeitschrift, 57 (1952), 1-12. [15] W. ACKERMANN, Zur Widerspruchsfreiheit der reinen Zahlentheorie, Mathematische Annalen, 117 (1940), 162-194. Department of Mathematics, University of Reading, England.
ABRAHAM ROBINSON ORDERED STRUCTURES AND RELATED CONCEPTS
§ 1.
Introduction
The work which is described in the present report owes its existence to the following circumstances. As is well-known, A. Tarski, in a famous paper [1], gave a method by which it is possible to decide whether or not a statement X, formulated in the restricted calculus of predicates in terms of the relations of equality, order, addition and multiplication, can be deduced from a set of axioms X for the concept of a real-closed ordered field. It turns out that for any such X, either X or X is deducible from X, that is to say, the set of axioms X is complete, or, as we shall also say in the sequel, the concept of a real-closed ordered field is complete. Tarski makes use of a generalization of Sturm's theorem and of a method of elimination of quantifiers, and states that a similar procedure leads to the result that the concept of an algebraically closed field of given characteristic also is complete in the sense mentioned above. The same fact was proved independently by the present author (cf. [2], [3]) by an entirely different procedure which is based on some simple results of Steinitz' field theory. The methods which will be discussed in the present paper arose from an attempt to evolve a general metamathematical principle on the basis of which the completeness of certain concepts, including that of a real-closed ordered field, can be deduced without recourse to a detailed procedure of elimination. t'.J
§ 2.
Model-Completeness
Let X be a consistent and non-empty set of propositions which is formulated in the restricted predicate calculus in terms of certain relations and (individual) constants- the latter set may be empty-
52
ABRAHAM: ROBISON
and let M be a model of K. To simplify the analysis without limiting the generality of our considerations, we shall assume that the elements of M actually belong to the formal language and denote themselves. Thus, M consists of a set of (individual) constants, and of a set of relations, such that for each relation R( , ... , ) of order n which belongs to M, and for each ordered n-uple 1Zt, .. " all of constants of M, R(IZt, ... , all) either subsists or does not subsist in M. We now define the set of statements N by the condition that in the former case, the "atomic" statement R(IZt, ... , all) shall belong to N while in the latter case, the negation '" R(IZt, ... , all) belongs to N. N is called the complete diagram of M (compare [2]). Then every model of the set K u N is a model of K which is an extension of M (or, with a wider definition of the notion of a model, which is isomorphic to an extension of M) and conversely, every extension of M which is a model of K is also a model of K U N. The set K will be called model-complete if for every model M of K, the set K U N is complete, N being the complete diagram of M. That is to say, for every statement X of the restricted predicate calculus which contains only relations and constants of M, either X or '" X is deducible from K U N. We note that since M is a model of K, all the relations and constants of K are contained also in M. It can be shown by means of suitable examples that the concepts of model-completeness and of ordinary completeness are not comparable, that is to say, neither includes the other. A set K which is not model-complete will be called model-incomplete. We now state the following: Main Theorem: In order that the (non-empty and consistent) set of statements K be model-incomplete it is necessary and sufficient that there exist models M, M ' of K, and a statement X, which contains only relations and constants of M, such that M ' is an extension of M, and such that X is satisfied by M' although it is not satisfied by M. Moreover, X has the form X
= (3Yl) ...
(3Yk) V(Yv .. " Yk)
ORDERED STRUCTURES AND RELATED CONCEPTS
53
where V does not contain any quantifiers and is a conjunction of atomic statements and negations of such statements (or of either alone). It is not difficult to see that the conditions of the theorem is sufficient. Indeed, M is a model of K u N, where N is the complete diagram of N. It follows that X is not deducible from K U N. On the other hand, M' also is a model of K U Nand M' satisfies X, so that "-' X is not deducible from K U N either. This shows that K is model-incomplete. The condition is also necessary. For given K, supposed nonempty and consistent, consider the set of all ordered pairs {M; Y} where M is a model of K and Y is a statement whose relations and constants are contained in M. We shall say that {M; Y} is decidable if either Yor "-' Y is deducible from the set of statements K U N, N being the complete diagram of M; otherwise the pair is said to be undecidable. For any such pair let n {M; Y} be the number of quantifiers in Y. According to our present assumption there exist undecidable pairs (for the given K). Moreover, for every statement Y, there exists an equivalent statement in prenex normal form, y', which contains the same relations and constants. Accordingly, there exist undecidable pairs {M; Y} in which Y is of prenex normal form. Among all such pairs, there must be at least one, {M; Z} say, for which n{M; Z} is a minimum,
n{M;Z}
~
n{M; Y}
for all undecidable pairs {M; Y} in which Y is of prenex normal form. It can be shown that for all undecidable pairs, n {M; Y} > 0, the reason being, briefly, that any Y which does not contain any quantifiers, and which holds in a given structure M, also holds in its extensions. Accordingly, n{M; Z} > O. Moreover, we may assume that Z begins with an existential quantifier. Indeed, let {M; Z} be any undecidable pair in which Y has prenex normal form and begins with a universal quantifier. Then {M; "-' Y} also is an undecidable pair, and if we express
54
ABRAHAM ROBINSON
___ Y in prenex normal form in the usual way as a statement Y', then {M; Y'} is an undecidable pair in which Y' begins with an existential quantifier. Suppose then, that in the above pair {M; Z}, Z begins with an existential quantifier, so, Z=(3y)W(y)
where W may contain further quantifiers. According to our assumption, ---Z is not deducible from K u N so that K u N U {Z} must be consistent and hence, must possess a model M'. Then M' contains a constant c such that W(c) is satisfied in M'. It then follows from the minimal property of {M; Z} that {M'; W(c)} is a decidable pair. That is to say either W(c) or --- W(c) is deducible from K u N' where N' is the complete diagram of M'. But M' is a model of K u N' which does not satisfy --- W(c) and so W(c) must be deducible from K u N'. It follows that there exists a finite number of elements of N', VI' ... , Vm , say, such that VI/'- ... r-; Vm:> W(c)
is deducible from K. Then i.e. VI/'- ... r-: Vm:>Z
also in deducible from K. We write VI/'- ..• r-; Vm= V(a I, ... , ak)
where we have distinguished all the constants a v ... , a k which belong to M' but which do not belong to M. Accordingly, these constants are contained neither in X nor in any of the elements of K. It then follows from the rules of the calculus of predicates that the statement [(3YI)'" (3Yk)V(YI' ... , Yk)]:>Z
also is deducible from K. We propose to show that the statement X
=
(3YI) ... (3Yk) V(YI' ... , Yk)
satisfies the conditions of the theorem.
ORDERED STRUCTURES AND RELATED CONCEPTS
55
Indeed, V(Yv ... , Yk) has the required form since V(~, ... , a k) is a conjunction of elements of N'. Also, X holds in M' since V(a 1 , •.• , a k ) holds in that structure. On the other hand it is not possible that X hold also in M. For in that case there would exist constants bv ... , bk in M such that V(b v ... , bk ) holds in M. Since V does not contain any quantifiers it would then follow further that V(b v ... , bk ) and hence X = (3Yl) ... (3Yk) V(Yl' ... , Yk) holds also in all extensions of M. X would accordingly be deducible from N. But X ~ Z is deducible from X, and so the statements X, X ~ Z and hence Z would all be deducible from K u N. This is contrary to the assumption that the pair {M; Z} is undecidable. Thus X does not hold in M; the theorem is proved. As stated above, model-completeness does not in general entail completeness in the ordinary sense. However, it can be shown that if the model-complete set of statements X possesses a primemodel then K is also complete in the ordinary sense. The model M of the set of statements K is said to be a prime model of K if every model of K is isomorphic to an extension of M. § 3.
Applications Let K be a set of axioms for the concept of an algebraically closed field formulated in terms of the relations of equality, addition, and multiplication. We wish to show that X is model-complete. For this case, we may interpret the main theorem as follows. Consider any finite system of equations and inequalities which belong to the following types: 0<. =
(3,
0<.
=1= (3,
0<.
+ (3 =
y,
0<.
+ (3 =1= y,
0<.(3 =
y,
0<.(3 =1=
y.
where the Greek letters stand either for unknowns Yi or for certain constants in an algebraically closed field M. In order to prove that K is model-complete it is then sufficient to show that if such a system has a solution in an extension of M then it has a solution already in M. This is a special case of the following. Let
P.(Yl'.",Yk)=O q.(yv ... , Yk)=I=O
i=l, i= 1,
,n ,m
56
ABRAHAM ROBINSON
be any system of polynomial equations and inequalities with coefficients in an algebraically closed field M. If the system possesses So solution in some extension of M then it possesses a solution already in M. To prove this assertion one may use either purely algebraic means (theorem of Hilbert-Netto, elimination theory, specialization of transcendental parameters) or, again, a metamathematical argument which reduces the algebra required still further. The concept of an algebraically closed field is not complete in the ordinary sense. However, the concept of an algebraically closed field of given characteristic is complete also in the ordinary sense, by the result quoted at the end of the preceding section. Similar applications of our main theorem lead to the conclusion that the following concepts also are model-complete. - The concept of a real closed ordered field. The remark at the end of section 2 then shows that this concept is also complete, which is Tarski's result. The concept of a completely divisible ordered abelian group. This concept also is complete. - The concept of an algebraically closed field which is valued non-trivially in an ordered abelian group. (The group does not reduce to the neutral element). It is not difficult to formulate concrete mathematical theorems, e.g. on the solubility of systems of equations and in-equalities in structures of one of the above types, which can be derived directly from our results. REFERENCES
A. TARSKI and J. C. C. McKINSEY, A decision method for elementary algebra and geometry (First ed., 1948), Second ed., Berkeley and Los Angeles (1951). [2] A. ROBINSON, On the meta-mathematics of algebra, Studies in logic and the foundations of mathematics, Amsterdam (1951). [3] , On the application of Symbolic Logic to algebra, Proceedings of the International Congress of Mathematicians, 1950 (Pub. 1952), 686-694. [1)
Department of Mathematics, University of Toronto, Canada.
HAO WANG
ON DENUMERABLE BASES OF FORMAL SYSTEMS § 1.
Introduction
I feel greatly honoured by Professor Heyting's invitation to speak on such an important occasion. I sincerely hope that you will find some of the things which I have to say of interest. The question of denumerable bases is especially interesting when we are concerned with formal systems of set theory. A collection K of sets forms a base of a formal set theory, if a proposition "for all sets x, Fx" is true if and only if, for every set a in the collection K, "Fa" is true. The collection K forms a denumerable base, if we can enumerate its members. There are, of course, many different methods of enumeration. For example, there are effective ones and ones which are not effective. Given any formal system, there are methods of enumeration which cannot be formalized in the system. A number of powerful tools, including the Skolem functions, the axiom of choice, Hilbert's selection operator, and the Godel numbers, have been applied in the study of denumerable bases of formal systems. The object of this paper is to examine and reflect on the applicability of these tools to the investigation of several questions closely related to that of denumerable bases: (1) the essence of quantifiers; (2) the limitations of formal systems; (3) unintended interpretations of formal systems; (4) the concept of an arbitrary set or function. The purpose of this paper is primarily a review and synthesis of known results. § 2.
The Skolem Functions A formal system can usually be formalized within the framework
58
HAO WANG
of quantification theory (first-order predicate calculus). Let us consider arbitrarily such a system. For brevity, let us assume that the system contains a single axiom. If there are more but altogether finitely many axioms, we can of course consider the conjunction of all these axioms. On the other hand, when there are infinitely many axioms, some of the results to follow can be proved analogously while others will be affected. Each time it will be clear as to which is the case. Suppose that the axiom is brought to the prenex normal form. For definiteness, suppose further that the axiom p is of the form p: (x)(3y)(3z)(w)(u)(3v) H(x, y, z, w, u, v)
which contains no other variables besides those which are explicitly shown. Of course, the actual cases of interest use longer and more complex formulas, but the considerations would be entirely similar. Following Godel [3], let us call "Skolem functions for p and M" any functions !(x), g(x), h(x, w, u) in M such that for any elements x, w, u of M the following is true: p*: H[x, !(x), g(x),
W,
u, h(x, w, u)].
By the axiom of choice, there exist such functions [, g, and h. These functions can be applied fruitfully in many considerations. Skolem uses these functions to prove Lowenheim's theorem that if a quantificational formula p is satisfiable at all, it is satisfiable in a denumerable domain. Thus, if 0 is an arbitrary object, the domain is the union of all sets M., such that M o = {a}, and Mi+l contains all and only those objects which either belong to M i or are j(a), g(a) or hia, b,c) for some a, b, and c that belong to M j • lt is known that a generalization can be similarly proved for an infinite set of formulas and that the use of the axiom of choice can be dispensed with. (Cf. Skolem [21], Church [1], pp. 83-89). In later sections we shall have occasion to refer to formulas related to each other in the manner of p and p*. We shall call p* the de quantified form of p, and p the quantified form of p*.
ON DENUMERABLE BASES OF FORMAL SYSTEMS
§ 3.
59
Enumeration in a Formal System
An interesting question is to examine the possibility of enumerating in a system L all the elements of a model of L. This would clearly be a fertile ground for the operation of the diagonal argument. For instance, if the axiom of the system L is just the formula p given in the preceding section, then an enumeration of a model of L would be provided by (say) a function F such that: F(I)=A (A being the empty set), F(2 a)= f[F(a}], F(3 b)=g[F(b)], F(2
a
.
b
3 .5 C)=h[F(a), F(b), F(c)].
In this way, each element of a denumerable model of p is correlated with a positive integer, although many integers are not used. Such functions can be defined in many formal systems. In most cases, however, no such function enumerating the model of a given formal system could be proved to exist in the system itself. In short, if such a function could be constructed, we would be able to get by the diagonal argument a set not already in the model, and a contradiction would ensue. Let us consider first a fairly strong system B with a finite number of axioms. For example, the Bernays-Godel set theory (slightly reformulated by using only one kind of variable) or the QuineHailperin theory (ordinarily known as the "New Foundations") is certainly strong enough for the purpose. Indeed, much weaker systems would do. It should be clear from the considerations to follow that a wide class of systems can be treated in a uniform manner. Let Be be the system obtained from B by adjoining Hilbert's s-operator and the accompanying s-rule and s-terms, If the axiom of B is p, then we can derive a dequantified p* in Be by putting: f(x} = ell(3z)(w)(u)(3v) H(x, y, z, w, u, v), g(x)=ez(w}(u)(3v} H[x, f(x), z, w, u, V], h(x, w, u}=e"H[x, f(x}, g(x), w, u, V].
60
HAO WANG
Now it is possible to define in S. a formula which amounts to an enumeration of all members of the Skolem model of p*, as follows:
.
definition 1.8 of Wang [24] which, however, is more easily adapted to cases where different axioms and functions are employed. Consider now the following axiom of limitation: AL. (x)(3j)[E(j)=x]. Accordingly to this, there are nothing other than the objects enumerated by E(j). Intuitively it appears plausible that if S. is consistent, then the addition of AL does not destroy consistency. More exactly, if S: is S. plus AL, then, by Skolem's method (cf. Wang [26], pp. 59-60), we can, assuming the consistency of S., get a model of S. which satisfies also AL. It is, however, not clear whether S: is translatable into S •. For instance, if we replace, in formulas of S:,
S:
it is not clear whether or not all theorems of will be turned into theorems of S •. Since S. is consistent if S is, we have: Theorem I. If S is consistent, then S: is consistent. A second application of the enumeration is the construction of an assertion of set existence which is independent of the axioms of S.:
DE.
(3y)(m)[m
E
Y
=
I
m
E
E(m)].
ON DENUMERABLE BASES OF FORMAL SYSTEMS
61
Thus, by the familiar diagonal argument, the negation of DE is Hence, by Theorem I, we have: provable in Theorem II. The formula DE is not provable in B.; moreover, if it is possible to find functions t, g, h, in S for which p* is consistent with p, then DE (containing these functions [, g, h) is not provable in B. Incidentally, when a system has an infinite number of axioms, it is no longer possible to formulate something like "E(j) = x" in the system itself, because we would need an infinitely long expression. The effect can, however, be obtained by using variables on a higher level which enable us to make induction on the length of the defining formulas for sets.
B:.
§ 4.
Reduction to Finitely Many Axioms
A closely related question is that of reducing infinitely many axioms to a finite number. Thus, let T be the system obtained from B. by adding an axiom schema to the effect that every number set (i.e., set of positive integers) expressible in S. exists: Given any formula Fm of B., (3y)(m)[m E y Fm]. Either T is inconsistent, or else the axiom schema is not derivable from a consistent finite set of formulas in the system. Indeed, given any consistent finite set, we can construct a formula of the form DE (see preceding section) which is not derivable. Zermelo's set theory is, for example, a system in which every number set, if expressible in the system, can be proved to exist in the system. Many subsystems of Zermelo's set theory, as well as certain systems constructed by Quine and the speaker (some of which equivalent to the theory of real numbers) possess the same property. Let Q be any such system, then it would appear that Q., obtained from Qby adding Hilbert's s-operator and its associated apparatus, is not finitely axiomatizable without introducing further new symbols. There is, however, a slight complication. While we assume that every number set, if expressible in Q, can be proved to exist in Q, it is not true that every number set, if expressible in Q., can be proved to exist in Qe- Thus, there are formulas Fm in Q. which
=
62
HAO WANG
contain the Hilbert s-symbol such that "(3y)(m)(m E y - Fm)" need not be a theorem of Q€" Let us define Q; as the system obtained from Q. by strengthening the axiom schema of Q to read: if Fm is a formula of Q., then (3y)(m)(m E y _ Fm). Then the argument sketched above does establish the following conclusion: Theorem III. If Q; is consistent, then no consistent finite set of formulas in the system can yield all the cases of the strengthened axiom schema. It should, however, be remarked that, while the s-theorems establish the relative consistency of Q. to Q, the relative consistency of Q; to Q is in general not known. The reason for switching from Q to Q., it will be recalled, is to make sure that, for example, functions t, g, h can be obtained so that the de quantified p* is derivable from p. If it happens that Q is a system in which such inference can be carried out, then there is no need to introduce Hilbert's s-operator, and we have the stronger theorem: Theorem IV. If Q is consistent, then no finite consistent set of formulas of Q can yield all cases of the general axiom schema asserting the existence of number sets. The problem now is, what are the systems Q in which we can obtain theorems of the de quantified form p* either from or in place of p? In the first place, Theorem IV is true of Q if either Q contains a suitable principle of choice or well-ordering, or can be consistently extended to include such a principle. This method of treatment is employed in Wang [26] and amended in McNaughton's review thereof. A different method is used in Mostowski [15]. Instead of trying to derive the dequantified p* from p, Mostowski begins with axioms which are of the form p*, containing function symbols but no bound variables. The addition of function symbols amounts to the same thing as switching from a system Q to a related system Q;. In any case let Q be a system to which the conditions for Theorem IV apply and L be a subsystem of Q containing only a finite number of axioms. We can define a formula of the form DE (see preceding section) and get rid of the quantifiers in favor of functions (i.e.,
ON DENUMERABLE BASES OF FORMAL SYSTEMS
63
replacing p by its de quantified form p*). The result, call it DEL, is independent of the axioms of L. Thus we get a number set which cannot be proved to exist in L. If we call L I the system obtained from L by adding DEL, then we can again construct similarly a formula DELI which is not provable in L I. Continuing thus, we have: Theorem V. If Q satisfies the conditions described above and L o is a subsystem of Q with finitely many axioms, then there is a sequence of subsystems L o, L I, L z, ... and a sequence of formulas DELo, DELl> DELz, .•. asserting existence of number sets such that DELi is provable in L, if and only if i :S j. In particular, we know that L o may be a system such that all predicative number sets expressible in Q are provable in L o. When such is the case, the formulas DELo, DELI' DELz, '" define a sequence of impredicative number sets which are in a sense of monotone increasing complexity. Let Q be the same and L be an arbitrary subsystem with finitely many axioms. Mostowski proves in [15] the further theorem: Theorem IV. If Con (L) is the arithmetic formula expressing the consistency of L, then Con (L) is a theorem of Q. A more detailed proof of the same theorem which depends more closely on the constructions of Wang [26] is given in McNaughton [14]. In his proof, McNaughton uses the enumeration of sets of L, as described above, to provide in Q a "transformed" normal truth definition of L (terminology of p. 270, Wang [28]). This is accomplished by saying, instead of formulas being satisfied by finite sequences of sets of L, that formulas are satisfied by finite sequences of their corresponding integers in the enumeration.
§ 5.
Certain Special Methods of Enumeration
In a system of set theory, there are axioms which assert unconditionally that such and such sets exist, and also axioms which assert that if such and such sets exist, then certain other sets related to them in certain definite manners also exist. For example, the axiom of infinity in the Zermelo set theory belongs to the first kind, while the axiom of power set, the axiom of sum set, etc. are
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of the second kind (given a set, its power set exists; given a set, its sum set exists). There are also axioms which do not seem to generate new sets, e.g., the axiom of extensionality. The general method of enumeration discussed in § 3 amounts to treating all the axioms formalistically and in a uniform manner. There is a strong temptation to approach the matter on a more intuitive basis. For example, we are inclined to disregard the axiom of extensionality and just enumerate the sets required to exist by the axioms which assert set existence conditionally or unconditionally. When we enumerate such a model, a special proof is usually necessary to establish that the axiom of extensionality is also satisfied. In certain special cases, on account of peculiarities of the system under consideration, we can even disregard many of the axioms which assert set existence and construct a model by enumerating only those sets which are required to exist by a few of the axioms of the system. Whenever we face such a situation, we have to make special efforts, on the one hand, to establish that the other axioms of set existence are also satisfied, but, on the other hand, since fewer axioms are directly involved in constructing the model, the model thus obtained is more transparent and often more informative. An interesting example is the system G presented in Godel [4]. Godel enumerates a model for G by taking into account merely the axioms A4, BI-B8, and using an additional clause which takes care of the union of all sets enumerated up to a given stage. On account of the last clause, Godel's enumeration uses transfinite ordinals and proves deep results about the system G. If we omit his last clause, we get a model which is denumerable (by using just positive integers) and rather simple. This simpler model is described in Wang [24] and Myhill [16]. It obviously satisfies A4 and BI-B8. This fact is employed in Rosser-Wang [18] (p. 128) to prove the consistency of G relative to the Zermelo-Fraenkel set theory. That the model should satisfy also all other axioms of Gis, though plausible, not at all obvious. Indeed, this is true only
ON DENUMERABLE BASES OF FORMAL SYSTEMS
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because of many peculiar properties of the system G. This truth is established in Suszko [23J and Myhill [16J, independently of each other. Suszko and Myhill have proved that the consistency of G is not impaired if we add as an additional axiom the assertion that all sets belong to the simple model which satisfies A4 and BI-B8. This result is similar to the general Theorem I proved above. But it is, on account of special properties of G, stronger in that a simpler enumeration function is used and there is no need to add Hilbert's s-operator. Incidentally, Suszko's proof uses auxiliary metalinguistic concepts which can be dispensed with by using the arithmetization of syntax. Myhill's presentation of his proof is rather condensed. There are places where the speaker is not able to follow his steps of argument. It does not appear justified to consider the major significance of this result about G as proving that there are no nameless sets. In the first place, there are other known methods of giving every set in G a name, e.g., by using Hilbert's s-operator and then taking the denumerable totality of all the s-terms of Ge- In the second place, those who believe in the existence of nameless sets would assert, truly I think, that there are in any case sets which are not available in the system G, and that, therefore, although there are no nameless sets in G, there might still be nameless sets. We note incidentally that the more direct method of enumeration considered in this section is not satisfactory when impredicative definitions are involved in the introduction of new sets. For example, suppose we have an axiom (3x)(y)[y
EX
=
(3z) --, Y
E
zJ.
Superficially this says merely that there exists a set of members y defined by the condition (3z) --, y E z. Yet on account of the fact that the set x thus introduced also falls under the range of values of the bound variable z, no model consisting of a single set could satisfy the axiom. - When the defining condition contains additional free variables, the situation gets more complex of course. Therefore, if we apply the method of the present section to cases 5
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where the axioms contain impredicative defining conditions, it is not necessarily true that the sets enumerated are enough to form a model. The method of § 3 is free from this difficulty, because it begins by bringing all quantifiers in the defining conditions to the beginning of the axioms and then proceed in a uniform manner.
§ 6.
Principlesfof Choicerand Hilbert's e-Operator
Clearly there is some connection between the axiom of choice and rules governing Hilbert's s-operator. Occasionally, the s-rule is referred to as a generalized principle of choice. This is misleading. For example, if the axiom of choice were a special case of the s-rule, why does the consistency of the axiom of choice not follow from the s-theorems according to which application of the s-rules can be dispensed with if the s-operator occurs in neither the axioms nor the conclusions? Indeed, if the axiom of choice were derivable from the s-rule, we would, by the s-theorems, be able to derive the axiom of choice from the other axioms of set theory. But, as we know, the independence of the axiom of choice is an unsolved problem. Of course, there is at least one difference between the s-rule and the axiom of choice. The former makes a single selection, while the latter requires that a simultaneous choice from each member of a given set be made and that all these selected items be put together to generate a new set. Hence, there is no reason to suppose that, in general, the axiom of choice follows from the s-rule. If the following (x)(3y)(z){z
E
y
=
(3w)[w
EX
& z=e..(u
E
w)]}
happens to be a theorem in a certain system of set theory, then the axiom of choice does follow from the s-rule in that system. But then it would be highly unlikely that the s-operator did not appear in the axioms of the system. There are also cases where, although the s-rule would yield the desired result, the axiom of choice would not. For example, in the Zermelo theory we can infer "(x)R(x, e,;Rxy)" from "(x}(3y)Rxy" by the s-rule, but we cannot infer "there exists I, (x)R(x, Ix)" from
ON DENUMERABLE BASES OF FOBMAL SYSTEMS
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"(x)(3y)Rxy" by the axiom of choice, on account of the absence of a universal set in Zermelo's theory In recent years it has become rather customary to use in quantification theory systems of natural deduction, more or less along the line of Gentzen and JaSkowski. To get definiteness, let us consider the system of Quine [17]. In this system, similarly as in several other systems, an attemp is made to formalize the following common mode of mathematical reasoning: "There exist y such that Fy. Let x be one such and consider whether this x has such and such properties." In Quine [17]. the step is translated simply into "From (3y)Fy, infer Ez", Since the x chosen is not an arbitrary object but rather an arbitrary object which happens to have the property F, the variable "x" in the above formalization cannot be expected to behave like ordinary variables in mathematical logic, As a result, Quine is led to the introduction and use of several concepts and restrictions: flagging a variable, no variable is to be flagged twice, alphabetical order of variables, finished deductions. It would seem more natural to introduce the s-operator and dispense with the various restrictions. Thus, let us modify the system of Quine [17] in the following manner: strengthen UI and EG by permitting s-terms as substitution instances of the bound variables, and replace EI by: "From (3x)Fx, infer F(e.cFx)". In the resulting system, there is no longer need to flag variables, every line of a proof is valid, and there is no longer need to consider the alphabetical order of variables. Completeness of the system can easily be verified similarly as in Quine [17], and it becomes an easy affair to prove soundness. The system thus obtained is quite similar to systems in. HilbertBernays [9]. The second s-theorem can be proved for this system in a manner which is more direct than the method used in HilbertBernays [9] and is along the line of Gentzen's argument. For this purpose, we have to assume as an additional primitive rule the derivable rule "From Fx::> p, infer (3x)Fx::> p if x is not free in p". This rule is no longer derivable if we omit EI. Thus, suppose we have a derivation in the above system of q from
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PI' ... , Pm' and none of q, PI"" Pm contains Hilbert's s-symbol.
The problem is to show that we can find in the above system another derivation of q from PI' ... Pm such that Hilbert's e-symbol does not occur in the proof. The rules to be applied are A, TF, Cd, the strengthened UI and EG, the revised EI, and VG (cf, Quine [17]). Suppose that e"Flx, ... , e"F"x
are all the s-terms introduced by EI, arranged in the order in which they appear in the proof. It follows that if instead of using EI, we just write down each of FI(e"Flx), , F"(e"F,,x), we get again a proof of q, but this time from PI' , Pm' together with FI(e"Flx), ... , F,.(e"F"x). The problem now is to find another proof in which the additional premises are dispensed with. Consider the last of these: F"(e"F,,x). By the rule Cd, we can prove "Ff&(e"Ff&x):> q" from PI> ... , Pm' FI(e"Flx) , ... , F"_I(e"F"_IX) without using EI. Throughout the proof, we can replace e"F"x everywhere by a new variable 'v' (say) and obtain a proof of "F"v:> q". By the additional rule assumed especially for this purpose, we can then derive "(3v)F"v:> q" without using EI. But "F"(e,,Ff&x)" was originally introduced by EI. Its premise "(3x)Ff&x" is, therefore, derivable from PI>'" Pm' FI(e"Flx), ... , F"_I(e,,Ff&_lx) already. Hence, since "(3v)F"v:> q" is also derivable from the same, q is already derivable from PI' ... , Pm' FI(e"Flx), ... , Ff&_I(e"Ff&_lx) without using EI. Hence, we have succeeded in getting rid of the premise Ff&(e"Ff&x), Repeating the same process, we can get rid of the other premises Ff&-I(e",Ff&_lx), ... , FI(e"Flx) one by one and obtain a proof of q from PI' ... Pm in which EI is not applied. In the resulting proof, there may still appear s-terms introduced by A, T F, and V I. Suppose that e"Hlx, ... , e~,;.c
are all the s-terms in the resulting proof, arranged in the order in throughout which their last occurrences appear, Replace e~,;.c the proof by a new free variable, and we get again a proof of q
ON DENUMERABLE BASES OF FORMAL SYSTEMS
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from Pt' ... , Pm' in which EI is not used. Repeat the same with eJ]k-tX, and so on. We finally get a proof of q from P» ... , Pm in which no s-terms appear. This completes the proof of the second s-theorem. § 7.
Unintended Interpretations of Formal Systems
The attempt to find a comprehensive formal logic in which we can develop all ordinary mathematics has led to several interesting axiom systems for set theory. Usually a system is said to be categorical if any two models for the system must be isomorphic. It is well-known that none of these systems of set theory is categorical, nor could any of them be made categorical by adding new axioms. In the first place, since each of these systems contains indenumerable sets, there is by the Lowenheim-Skolem theorem, always some model for each system which is denumerable and therefore different from the intended model. Secondly, by Godel's incompletability theorem, each such system must necessarily contain undecidable sentences; therefore, there exist at least two non-isomorphic models for each system such that the undecidable propositions are true in one and false in the other. There remain the questions of relative categoricity. Vaguely the concept is merely to consider whether two interpretations which agree in certain special aspects are isomorphic. A more exact definition can be given in two parts. Definition. (a) A system 8 is categorical relative to a subsystem 8 t of 8 if and only if any two models of 8 which contain isomorphic submodels of 8 t are isomorphic. (b) If P v ... , P k are obtainable in a system 8, 8 is categorical relative to the predicates P v ... , P k , if and only if, any two models of 8 in which the interpretations of Pt, ... , P k are respectively isomorphic are isomorphic. For example, 8 may be a set theory containing only the membership primitive predicate and PtX, P'J1j, Paxy may be respectively the property of being a natural number, the property of being zero, the successor relation; the system 8 is said to be categorical relative to its natural numbers, if and only if, in any two models of 8, isomorphic interpretations of P v P 2 , P a yield isomorphic interpretations of
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the membership predicate. Similarly, a set theory may also be categorical relative to its ordinal numbers. Some simple questions of relative categoricity are studied in Wang [29]. These questions are simple because they are concerned with rather weak systems of set theory. It is not obvious that stronger systems can be treated entirely similarly. There are, however, questions which do not appear too difficult. Is the system G described in Godel [4], when extended by adding "V =L" as a new axiom, categorical relative to the ordinal numbers 1 Is the system G, when extended by adding an axiom of limitation along the line of § 3 above (resp. along the line of § 5 above), categorical relative to the natural numbers1 Probably the most frequently discussed unintended models are the models of set theory associated with the Lowenheim theorem or the Skolem paradox. A few comments on this topic may not be completely out of place here. Of course, the paradox of Skolem is pretty closely related to Cantor's notion of the indenumerable. The classical diagonal argument was given and accepted in the absolute sense, i.e., without reference to any formal system as logicians now understand it. There is absolutely no law which would correlate all sets of positive integers one-by-one with all positive integers. For, as you will recall, every enumeration of sets of positive integers always leaves out a set K which differs from the n-th set in that n belongs to the n-th set if and only if it does not belong to K. Let S be, for instance, Zermelo's set theory. On the one hand, we seem to be able to define in S the set of all sets of positive integers and prove in Sits indenumerability. On the other hand, the Lowenheim-Skolem theorem tells us that every set definable in S must be denumerable, if S is consistent. Hence, the paradox. The paradox can be dissolved either by admitting the inadequacy of formalism to intuition or by refusing to countenance any (absolutely) indenumerable sets. The Lowenheim-Skolem model is an unexpected interpretation which disagrees with our intention. The system S is not categorical because it admits both the intended and the Lowenheim-Skolem
ON DENUlIlERABLE BASES OF FORMAL SYSTEMS
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interpretations. What is more, the system S is not faithful to Cantor's notion which does not admit the Lowenheim-Skolem interpretation. The remedy is to keep the intuitive interpretation firmly before our mind as we work with the formalism S. It is, of course, not necessary that we always restrict the means of communication to formal systems. The fact that we do in a sense understand the informal diagonal argument proves the possibility of communication by other means. According to this line of reasoning, the Skolem paradox is one among the many arguments to show that intuition cannot be completely formalized. A more drastic solution is to reject absolute indenumerability. The diagonal argument merely shows that given any enumeration of sets of positive integer, there is a set of positive integers not in it. Unless we assume that there is a universal set which contains as members all sets of positive integers, we cannot get an absolutely indenumerable set out of the argument. True, once we give a. denumerable set and claim it to be including the totality of all sets of positive integers, we can be soundly refuted by the diagonal argument. But if we admit this and still contend that all sets are denumerable, the argument is quite powerless to refute us. When such a position is adopted, the Skolem paradox automatically disappears. The resulting theory of mathematics would become something like the ramified theory of types, except that it is not necessary to restrict the type or order indices to finite numbers. Vaguely we feel that each formal system is constructed with a unique intended model, which may be called the standard model, in mind. The speaker shares with many the discomfort over the unqualified notion of a standard model. The notion of standard model relative to certain preassigned interpretations of certain specific notions is easier. There remains, nevertheless, the question of specifying the preassigned interpretations. For example, one might feel that there is obviously only one standard model of set theory because the membership predicate is preassigned on interpretation. If, however, we specify the preassigned interpretation by stating explicitly certain conditions to be fulfilled, it is quite
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possible that if we get one model which satisfies these preassigned conditions, we can get more. Since we cannot formalize entirely the membership relation, the explicit specification of the preassigned interpretation may also involve insurmountable difficulties. Even if we leave the interpretation to our intuition, there is no assurance that our intuition is sharp enough to be capable of singling out a unique model that is standard. In this connection, the situation with number theory is much better than the situation with set theory. The standard interpretation of positive integers can be specified, for example, by emphasizing that every positive integer is either 1 or obtainable from 1 by applying the operation of adding 1 a finite number of times. Indeed, the Hilbert-Bernays truth definition already provides a standard model of number theory. For set theories such as Zermelo's, we know of neither standard models nor any models which could supply a comparable amount of information. § 8.
Completeness of Quantification Theory
This section contains a brief review of the history of the Lowenheim theorem and a proof of the completeness of quantification theory which appears to be more straightforward than the standard proofs. The following result is proved in Lowenheim [12]: (i) Every quantificational formula, if satisfiable in any (nonempty) domain at all, is satisfiable in a denumerable domain. In Skolem (19], Lowenheim's proof is simplified and an extension is obtained: (ii) H a denumerable number of quantificational formulas are
simultaneously satisfiable in any domain at all, they are simultaneously satisfiable in a denumerable domain. Skolem's proofs of (i) and (ii), as well as L6wenheim's original proof of (i), all make use of the axiom of choice. A proof of (i) is given in Skolem (20] in which the axiom of choice is not applied. Finally, in Skolem (21], improved proofs of (i) and (ii) are given. These proofs are very interesting.
ON DENUMERABLE BASES OF FORMAL SYSTEMS
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Skolem merely assumes that the quantificational formulas be given in prenex normal form. But, for brevity, let us consider a single formula in the Skolem normal form:
Skolem takes H(I, ... , 1; 2, ... , n+ 1) as HI' and then considers all the m-tuples of the positive integers no greater than n+ 1, ordering them in an arbitrary manner with the m-tuple (1, ... , 1) as the first. If (k il , ... kim) is the i-th m-tuple, then Hi is Hi-I> & H(k il
•••
kim; n(i-l)+2, ... , ni+ 1).
He then considers all the m-tuples of the positive integers used so far, and again couples the i-th m-tuple with the n-tuple consisting of the n consecutive integers starting with n(i - 1) + 2. In this way he defines a sequence of quantifier-free conjunctions HI> H"" ... all gotten from the formula A. (Cf. Skolem [21], p. 23 if and Skolem [22], p. 28 if). His proof of (i) contains two parts: (iii) If a quantificational formula A is satisfiable at all, then none of the formulas -, HI> -, Hz, ... is a tautology. (iv) If none of the formulas -, HI> -, H 2 , •• , is a tautology, then the formula A itself is satisfiable in the domain of positive integers. A little while later, Herbrand proved the following theorem (Herbrand [7], and Herbrand [8], p. 112): (v) If -, A is not a theorem of quantification theory (in other words, if the system determined by A is consistent), then none of -'-l HI> -, Hz, ... is a tautology. From (iv) and (v), the completeness of quantification theory follows as a corollary: (vi) If A is consistent, then A is satisfiable in the domain of positive integers; therefore, if -, B is not satisfiable in any domain (i.e., B is valid), then -, -, B (and therewith B) is a theorem of quantification theory. In his considerations, Herbrand also uses prenex rather than
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Skolem normal forms. Actually (v) is only a rather unimportant part of Herbrand's important theorem which he considers to be a finitary interpretation of the Lowerheim theorem: (vii) A formula A is consistent (i.e., .. A is not a theorem of quantification theorem) if and only if, none of -. HI> .. H s, ... is a tautology; moreover every proof yields effectively a proof in the "Herbrand normal form". Soon after, Gadel independently proved (iv)-(vi) in a peculiarly clear and exact manner (Godel [2]). His proof of (iv) which is widely known through Hilbert and Ackermann is rather different from Skolem's (for Skolem's argument, cf. Hilbert-Bernays [9], pp. 186188 and Wang [25]). From (vi), it follows that given any axiom system S with a finite number of special axioms and formulated in quantification theory, S contains no contradictions if and only if it has a model in the domain of positive integers. Gadel also extends this to apply to cases where there are a denumerable infinity of special axioms: (viii) If a system formulated in quantification theory is consistent, then it has a model in the domain of positive integers. Another extension of (vi) given in Godel [3] is the completeness of the extended quantification theory obtained by including identity and axioms for identity. In Henkin [5], an interesting alternative proof of (viii) is given in which Henkin introduces the method of constructing maximum consistent extensions of sets of formulas, presumably inspired by similar constructions in modern algebra. The method may be described as follows. All the quantificational formulas are enumerated in a definite manner so that each is correlated with a unique positive integer in the standard ordering: qI> qs, etc. If a consistent set So of quantificational formulas is given, its maximum consistent extension is constructed in the following manner. If So plus q1 is consistent, then S1 is So plus q1; otherwise, S1 is the same as So' In general, if S" plus q,,+1 is consistent, then S,,+1 is S" plus q,,+1; Otherwise, S,,+1 is the same as S". The maximum consistent extension is the union of the sets S•.
ON DENUlIIERABLE BASES OF FORMAL SYSTEMS
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Henkin succeeds in proving (viii) by application of this method and a device of introducing denumerably many suitable new symbols. If we include Hilbert's s-symbol and its associated apparatus, we do not need these new symbols because certain s-terms would serve their purpose. Thus, we get a rather simple proof of (viii) and therewith of (vi) in the following manner. Let So be a consistent set of closed quantificational formulas. Add the s-expressions and the s-rule. Then So remains consistent by the second s-theorem, Construct the maximum consistent extension (call it S) of So within the enlarged framework. Call a closed formula true or false according as whether it belongs to S or not. We assert that hereby we get a model of So in the denumerable domain D of all constant s-terms (i.e., those containing no free variables). In the first place, if P is an arbitrary predicate with n arguments and all , an are n constant s-terms, then either P(~, ... , an) or -, P(~, , an), but not both, is true, because S is maximal consistent. In the second place, "neither p nor q" is true if and only if neither p is true nor q is true, again because S is maximal consistent. In the third place, (3x)F(x) is true if and only if there is some constant e-term a such that F(a) is true, because "(3x)F(x) = F(e",Fx)" belongs to S. Moreover, since So is a subset of S, all formulas in So are true. Hence, we get (viii). In particular, So may contain a single formula, and (vi) follows immediately. It may be noted in passing that we could, if we wish, also think of the s-expressions as expressions for Skolem functions. Thus, for example, given a formula (3y)F(x, y), the s-expression e"F(x, y) of course corresponds to the Skolem function f(x) such that F(x, f(x». Corresponding to constant s-terms, we introduce Skolem functions with zero arguments. If we look at the matter in this manner, we may describe the above proof as an extension of Skolem's proof for (ii). A further sharpening of (viii) and the Lowenheim theorem is given in Hilbert-Bernays [9} and extended in Wang [25]. Thus,
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given Zermelo's set theory or some other system 8, we can as usual use an arithmetic proposition Con (8) to express the consistency of 8. For example, we can assume that Con (8) says in effect that the proposition 0= 1 is not a theorem of 8. Let N. be obtained from elementary number theory (Peano axioms and recursive definitions of addition and multiplication) by adding Con (8) as a new axiom. The result is then as follows. We can exhibit an arithmetic predicate P in N. such that if we replace the membership predicate of 8 throughout by P, then all theorems of 8 turn into theorems of N." This is sharper than the usual forms because it answers clearly the questions of expressibility and provability of the models. Moreover, if 8 is consistent, then the proposition Con (8) is not only true but obtainable from numerically true (verifiable) propositions by using merely one quantifier. Another extension of the various theorems given above is to the many-sorted quantification theories. As is indicated in Wang [27], such extension is entirely straightforward. The contrast between one-sorted and many-sorted quantification theories is more basic than that between first-order and higherorder predicate calculi. While higher-order predicate calculi and theories formulated therein all are many-sorted systems, there are many-sorted theories such as geometry dealing with points, lines, and planes which are not formulated in higher-order predicate calculi. While many-sorted quantification theories treat all predicates on an equal basis, higher-order predicate calculi reserve a special pedestal for the membership relation. Indeed, an n-th order predicate calculus is nothing but a n-sorted theory which contains a single axiom schema (the axiom of comprehension) asserting that every meaningful formula in the system defines a set. An immediate corollary of (vi) and (viii), useful for the study of completeness, is: (ix) Every sentence of a system 8 formulated in quantification theory, if true in every model of 8, is a theorem of 8. Thus, for simplicity, let us assume that 8 has a single special axiom A. If p is true in every model of 8, then "A :J p" is valid
ON DENUMERABLE BASES OF FORMAL SYSTEMS
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or true in every model (of quantification theory). For, if a model is not one of S, A is false and therefore "A :J p" is true; if a model is one of S, then p is also true in it by hypothesis. Hence, by (vi), "A :J p" is a theorem of quantification theory and, therefore, p is a theorem of S. A similar proof is available if S has infinitely many axioms. A similar result holds for many-sorted theories. Such results are of interest if we wish to make a given system complete by introducing a suitable definition of completeness. Thus, if we wish to make a system S complete, all we have to do is to define completeness to mean provability of every formula which is true in all models of S. By (ix), it follows that S is complete by this definition. This somewhat trivial trick seems to provide a clue for methods of manufacturing nontrivial concepts of completeness. Thus, given a system S, (ix) leads us to look for significant properties common and peculiar to all models of S. If, for a given system S, it happens that there are such properties, then the system S is complete in the interesting sense that every formula which is true in all models possessing these properties is provable in S. For each particular system S, special considerations are required to find suitable properties and prove the completeness of S relative to them. The concept of completeness of predicate calculi, introduced in Henkin [6], seems to be a good example. § 9.
Arbitrary Functions
What is an arbitrary set? What is an arbitrary function? There appear to be two extreme answers. On the one end, there are those who would like to deal only with computable or recursive functions. On the other end, there are those who do not hesitate to accept all the higher infinities in Cantor's theory. In the speaker's opinion, not enough attention is paid by logicians to the middle position presented and defended, perhaps not very clearly, by the semiintuitionists such as Borel, Lebesque, and Lusin. For most purposes, computable functions are sufficient in the theory of positive integers. Indeed, Ackermann and Kreisel have succeeded in interpreting quantifiers in the arithmetic of positive
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integers by computable functions. Yet it seems misguided to attempt to build up analysis or set theory solely on the basis of computable functions. The desire to see exactly how much function theory we can get by using solely computable functions is apparently innocent. It could, nonetheless, be harmful if, through prolonged exclusive occupation with computable functions, one were led to refuse to countenance other mathematical functions. In this connection, it is of interest to note that, although Brouwer does not seem willing to admit any function which is not computable, he does permit the use of free choice sequences and spreads which are not generated by computable functions (see e.g., Kleene [11]). On the other hand, there are those who would rather keep the whole Cantorian theory of the transfinite, either for its beauty or for its usefulness. It seems, however, indisputable, that the human mind is at present quite incapable of forming a clear and distinct idea of the indenumerable or the impredicative. A beauty which everybody fails to comprehend is probably a luxury which we can go without, at least insofar as mathematics is concerned. As for the application in "useful" mathematics, the speaker is of the opinion that there is no need to go beyond the denumerable. When we seem to need the indenumerable in function theory and measure theory, indenumerability relative to certain restricted means is usually sufficient. At one time or another, Weyl, Russell, Chwistek all attempted to develop mathematics, as far as possible, on a basis that does not appeal to the indenumerable or the impredicative. In recent years Lorenzen has successfully carried the program further, using nothing which resembles a formal system. In Wang [30], the speaker discusses a theory which is an extension of the ramified theory of types to the transfinite so that for any acceptable ordinal number iX of Cantor's second number class, there are variables and sets of order or type iX. In the speaker's opinion, the theory is essentially equivalent to the body of methods which Lorenzen admits and applies. The theory has on the lowest order (the O-th order) a denumerable
ON DENUMERABLE BASES OF FORMAL SYSTEMS
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totality consisting of (say) all the positive integers or all the finite sets built up out of the empty set. On the first order are these same sets plus sets of them which can be defined by properties referring at most only to the totality of all sets of the O-th order (or, in other words, by formulas which contain no bound variables of the first or a higher order). Similarly, for every positive integer n, the sets of order n + 1 includes all sets of order n together with sets of them defined by properties referring at most only to the totality of all sets of the n-th order. The sets of order t» includes all and only sets of the finite orders. For any ordinal number (X + 1, the sets of order (X + 1 are related to those of order (X in the same way as the sets of order n+ 1 are to those of order n (n a nonnegative integer). For any ordinal number (X which is the limit number of a monotome increasing sequence PI> P2' ... of ordinals, the sets of order (X are related to the sets of order PI> P2' ... in the same way as the sets of order ware related to those of finite orders. An important question is the nature of the ordinal numbers employed in describing the theory. For example, as Godel remarks in his discussion of Russell's mathematical logic, if we permit sufficiently many ordinals introduced by impredicative definitions, then we have more or less the ordinary impredicative axiomatic set theory. Since we wish to use exclusively predicative sets, we should use all and only ordinals definable by predicative definitions. It seems difficult to get a more exact characterization of these ordinals. For instance, the following method, though reasonable, is inadequate because we can again get new constructive ordinals by the diagonal procedure. Thus, let R be that part of the theory which contains only the sets and variables of finite orders. This system R, as we know, can be described exactly. Assume that an arithmetization of the syntax of R is given. Consider all the functions of positive integers which can be proved to exist in R. Each function has a defining formula which has a Godel number. The Godel number of the defining formula of a function will be called simply the Godel number of the function. We can then define ordinals in Rand represent them by positive integers in the following manner.
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(1) 1 represents the ordinal number O. (2) If m represents IX, 2m represents IX + 1. (3) If {IX..} is a monotone increasing sequence of
ordinal numbers, for each n, m.. represents IX.. , and k is the Godel number of the function of R which enumerates the sequence {ml' m 2, ... } then 3.5 k represents the limit ordinal of {IX..}. In this way, we get a totality of ordinal numbers N l . Let us expand the system R by permitting all numbers in N l to serve as indices of sets and variables. Call the resulting system R l and arithmetize its syntax. We can then define ordinals in R l and represent them by positive integers, as with R. In this way, we get a new totality N 2 of ordinal numbers from which we can construct a system R2 • And so on. Since the initial system R is well-defined, and since there is a definite method of extending a given system to get a new one, we have defined both a definite totality of ordinal numbers and also a comprehensive theory which is the union of all the partial theories. It is not easy to say much that is positive about these ordinals. They all belong to Cantor's second number class, and they are all constructive in the sense that no impredicative definitions are employed in their definitions. Moreover, there are more such ordinals than the constructive ordinals of Church and Kleene (see Kleene [l0]). Indeed, there are more ordinals in N l (the set of ordinals defined from R) than their constructive ordinals. Thus, the representation of ordinals of R is the same as theirs except that under condition (3) they use only recursive functions, while we use functions of R. But it is well-known that all recursive functions are available in R while many functions which can be proved to exist in R are not recursive. Hence, there are more constructive ordinals by the definition given above than the Church-Kleene ordinals. Instead of or in addition to attempting to build up mathematics on a predicative basis, one may try to prove the consistency of systems of impredicative set theory. It is well-known that the problem is very difficult. Let us examine a little closely the nature of the difficulties. For example, if we take the system described in Gadel [4] and
ON DENUMERABLE BASES OJ' J'ORMAL SYSTEMS
81
replace the axiom of infinity by an axiom giving the empty set 0, the resulting system (call it S) has a simple denumerable model, as is well-known. Suppose now we add a simple axiom of the form (1)
(3Y)(x)[x
E
Y ;:; (3X)FXx],
where FXx contains no other free variables besides X and x, no other class variables besides X. Let us try to get a model for the extended system by enlarging the known model for the original system. It can happen that (1) is true already in the given model. This is of course not true for most of the interesting cases. So let us assume that (1) is not true in the given model. The natural thing to do would appear to be the addition of a new class X to the model of the original system S. For a given set x, if there is a class X in the original model, such that FXx is true, x E X is of course true. But it can happen that F Xx is false for every X in the original model, then the truth of x E X would depend on the truth of F K». Since F K» can contain x E X as a part, it can happen that F K» is true if and only if x E X is not true. As a result, in order to get a model for the system S plus (1), we must introduce other classes in addition to the class X. In certain simple cases, it is not difficult to introduce a new class X and get a model of S plus (1). This is in general true for cases where the new class X does not affect the truth of (3X)FXx: in other words, either F K» is false, or else there is already a class X in the original model for S such that F Xx is true. In such a case the truth of x E X is determined univocally by the original model for S and by introducing X we get easily a model for the extension. Similarly, such simple extensions are also possible when several bound class variables occur in the defining condition but the truth of the defining condition is not affected by the new class introduced by the condition. For example, consistency proofs of the number theory Z of HilbertBernay cannot be formalized in S but can be formalized in an extension of S with two new axioms, one of the form (1) and one with two general class quantifies in the defining condition. These 6
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two axioms do satisfy the requirement of the preceding paragraph. As a result, we can extend the model of S by adding two new classes and obtain a model for the extension of S which contains two apparently impredicative axioms of class existence. This is not surprising since, as we know, the same consistency proof can also be formalized in other systems which contain no such axioms but include suitable induction axioms or predicative classes of one level higher. But the example may be of interest in so far as it illustrates that many impredicative definitions can be replaced by predicative conditions. ADDITIONAL NOTE
Mr. Richard Montague has read the paper and offered useful criticisms which have induced several changes in the text. In addition, he has called my attention to an error with regard to the proofs of Theorem IV, Theorem V, and the second half of Theorem II. In these theorems I speak of "finding the functions [, g, h" in the system S or in the system Q. I failed to distinguish two senses in which the functions [, g, h can be found in a system. The proofs are valid only if the functions t, g, b are part of the original system and thereby characterized by the original axioms of the system. If, as is often the case, the functions are introduced as descriptions by contextual definitions, then the proofs are no longer valid because these contextual definitions amount to new axioms while the proofs only take care of the original axioms. The reader is asked to bear in mind this distinction and to note that the reference to Wang [26] in the paragraph immediately following the statement of Theorem IV is incorrect, because in the treatment of Wang [26], the functions i, g, h are gotten by using contextual definitions and Mr. Montague's criticism applies. (All these proofs can be rectified by a supplementary argument because in the cases concerned the relative consistency of to Q can be proved.) I should like to stress two points here. In the first place, the criticism does not affect the proofs of Theorem I, Theorem III, and the first half of Theorem II, because the use of e-expressions
Q:
ON DENUMERABLE BASES OF FORMAL SYSTEMS
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does make the functions i, g, h. part of the original system. Hence, the main conclusions which are affected by Mr. Montague's criticism are correctly established if we use e-expressions. In the second place, even if we do not use the e-expressions, the questionable conclusions of Wang [26] and the present paper can all be justified by the extension of Wang [26] in McNaughton [14], independently of how t, g, h are obtained. Thus, as is stated in the text, a proof of Theorem VI is given in McNaughton [14]. Theorem IV is an immediate corollary. Theorem V also follows from Theorem VI. Thus, Con (L o) is a theorem of Q and only a finite number of axioms of Q are used in the proof of Con (Lo). Let L 1 be L o plus these axioms. Similarly, Con (L 1 ) is by Theorem VI a theorem of Q, and let L 2 be L 1 plus the finitely many axioms employed in the proof of Con (L 1 ) . And so on. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]
ALONZO CHURCH, Introduction to Mathematical Logic, Princeton (1944). KURT GODEL, Monatsh. Math. Phys., 37 (1930), 349. , Proc. Nat. Acad. Sci. U.S.A., 25 (1939), 220. , Consistency of Continuum Hypothesis, Princeton (1940). LEON HENKIN, J. Symbolic Logic, 14 (1949), 159. , J. Symbolic Logic, IS (1950), 81. JACQUES HERBRAND, Comptes rendus Acad. Sci. (Paris), 188 (1929), 1076. , Recherches aur la theorie de la demonstration. (1930). DAVID HILBERT and PAUL BERNAYS, Grundlager der Mathematik, vol. 2, (1939). S. C. KEENE, J. Symbolic Logic, 3 (1938), 150. , Proc. Int. Congo Math., 1950, (1952). LEOPOLD LOWENHEIM, Math. Annalen, 76 (1915), 447. ROBERT McNAUGHTON, J. Symbolic Logic, 18 (1953), 265 (review of Wang [26]). , "A Non-Standard Truth Definition", Proc. Am. Math. Soc., 5 (1954), 505. A. MOSTOWSKI, Fund. Math., 39 (1952), 133. JOHN MYHILL, Proc. Nat. Acad. Sci. U.S.A., 38 (1952), 979. W. V. QUINE, J. Symbolic Logic, IS (1950), 93. BARKLEY ROSSER and HAO WANG, J. Symbolic Logic, IS (1950), ll3. TH. SKOLEM, Vidensk. Skrifter I, Mat. naturw. Klasse, Oslo, No.4, (1920).
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[20] TH. SKOLEM, Proc. 5th Scand. Congo Math., 1922, (1923), 217. [21] , Vidensk. Skrifter I, Mat. naturw. Klasse, Oslo, No.4, (1929). [22] , Lea entretiens de Zurich, 1938, (1941), 25. [23] ROMAN SUSZKO, Studia philosophica, 4 (1951), 301. [24] llAo WANG, Proc. Nat. Aead, Sci. U.S.A., 36 (1950), 479. [25] , Methodos, 3 (1951), 217. [26] , Math. Annalen, 125 (1952), 56. [27] , J. Symbolic Logic, 17 (1952), 105. [28] , Trans. Am. Math. Soe., 73 (1952), 243. [29] , Math. Annalen, 126 (1953), 385. [30] , "The formalization of mathematics", J. Symbolic Logic, (1954), 241. Department of Philosophy, Emerson Hall, Harvard University, Cambridge, Mass. U.S.A.
L. HENKIN
THE REPRESENTATION THEOREM FOR CYLINDRICAL ALGEBRAS In this paper we shall deal with certain algebraic structures introduced and studied by Alfred Tarski and his student F. B. Thompson. Unfortunately these concepts are not widely known, being presently available in the literature only in abstract form [1]. We shall, therefore, precede a description of our own work by an account of the basic ideas of Tarski and Thompson, mentioning only those concepts and results necessary to render intelligible the later sections of this paper.
§ 1. Basic Concepts Consider first a first-order formalism g:. Such a formalism is comprised of certain primitive symbols together with rules for forming these symbols into terms and well-formed formulas. The symbols include quantifiers, truth-functional connectives such as a negation sign and an implication sign, an equality sign, parentheses, a list of individual variables Xv x 2' X a, ... , and finally a list of constants divided into three sorts: individual constants, predicate constants, and operation symbols. We presume that the reader is familiar with the way in which formulas are constructed from these symbols [2]. In connection with such a formalism g: we often have occasion to consider a model 9)(. Such a model consists of a domain of individuals 1, X, together with a function which assigns to each constant g of the formalism a value g. If g is an individual constant then g is an element of X; if g is an n-ary predicate constant then 1
This may be an arbitrary non-empty set of elements.
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L. HENKIN
{j is a set of ordered n-tuples of elements of X; if g is an n-ary operation symbol then {j is an n-ary operation which associates
an element of X with each ordered n-tuple of elements of X. Given a formalism g: and a model 9R there is defined the important relation of satisfaction which holds between certain wellformed formulas G and elements IX of X'" (i.e, infinite sequences IX each term of which is an element of X); and we presume that the reader is familiar with this definition 2. There is thus associated with each wff G a certain subset G* of XO> defined by the condition that IX E G* if and only if IX satisfies G. Let A be the class of all sets G* (for all wffs G of the formalism The class A is closed under the operation U of union, since it results from the definition of satisfaction that (0 V H)* =G* u H* for any wffs G and H. Similarly A is closed under the operation 1 of complementation (with respect to X"'), since (,....., G)* (G*). From these observations it follows that the system 2{= (A, u, n, I) is a Boolean algebra of sets in which the zero element is the empty set (i.e., (,....., (:11. =xI ))*), and the unit element is XO> (i.e., (Xl =xl )*). Among the elements of A are the diagonal elements do;, defined by the law di j = (x.=x i)*, i, j = 1,2,3, .... Furthermore, in addition to the Boolean operations on A we distinguish certain unary operations, the (outer) cylindrifications C i , defined by the law 3 C, (G*) = ((3x.)G)*, t.: 1,2,3, ... , for any wff G. The system ~=(A, u, n, I , dij, C.) is an example of a highly proper (wdimensional) cylindrical algebra (with diagonal elements). More generally, let 'YJ be any ordinal number and X any set. By X'I we mean the set of all functions 4 whose domain is 'YJ (we
m.
=,
2 A precise definition of satisfaction was first given in Tarski [3]. A short account of this definition appears in Mostowski [4]. In terms of the relation of satisfaction other important semantical notions such as truth and consequence may be defined. 3 This definition is justified by the fact that if G and H are two different wffs such that G* = H*, then ((3xilGl* = ((3xilHl*, as easily follows from the definition of satisfaction. 4 These functions may be identified with sequences of type TJ whose terms are in X.
THE REPRESENTATION THEOREM FOR CYLINDRICAL ALGEBRAS
87
identify 'YJ with the set of ordinals less than 1]), and whose values are in X. By the diagonal element d il of X'7, i, j = 1,2, ... <1], we mean the subset of X'I consisting of all those elements IX of X'I whose ith term IXi is the same element of X as its jth term IXi' By the (outer) cylindrification C, of X'I we mean the operation on the subsets of X'I such that, for any subset Y of X'I the set 0iY consists of all elements IX of X'I which differ from some element {3 of Y at most in the ith term 6. Now we define a highly proper 1]-dimensional cylindrical algebra (with diagonal elements) to be a system A=
°
6 The diagonal elements and the cylindrification operators have a simple geometric interpretation. If the elements of X be thought of as points in a linear space, then the elements of X'I may be thought of as points in a product space of 1] dimensions. The elements of dii are then seen to form a hyper-plane perpendicular to the two-dimensional plane containing the ith and jth axes, and passing through the diagonal line of this plane consisting of all points whose ith and jth coordinates are equal. The operation 0i' acting on a point set Y of the product space X'I, is seen to have as value a cylinder generated by translating Y parallel to the ith axis. For the case where 1J is 2 or 3 these elements and operations are easily visualized.
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and model IDl, so that each element of 58 is a set G* for some wff G of l1, then 58 is certainly locally finite-dimensional; for whenever Xi is an individual variable which does not occur freely in the wff G, we have C.(G*)=G*. On the other hand there certainly exist h.p.c.a.'s which are not locally finite-dimensional, as we see by considering the class of all subsets of some XW. Thus not all h.p.c.a.'s arise from a formalism and model. On the other hand, if we restrict ourselves to locally finitedimensional h.p.c.a.'s of dimension w, then we obtain a simple positive answer to our question. For let 58=
THE REPRESENTATION THEOREM FOR CYLINDRICAL ALGEBRAS
89
operators. Among these are, for example, the following: YnC.Y = Y, and for i and j both different from k, di;=C,,(d'k 1'1 dn,)' What Tarski and Thompson have done is to choose a select list of these laws and employ them as axioms to define a new, abstract, algebraic concept of cylindrical algebra, in the same way as the concept of Boolean algebra is abstracted from the notion of an algebra of sets. Specifically, for each ordinal number 1] we define an 1]-dimensional (abstract) cylindrical algebra (with diagonal elements) to be any system 5B=(A, +, " -, d.i, C.) such that (A, +, " -) is a Boolean algebra, for each i, j= 1,2, ... <1] dij is an element of A and C. is a unary operation on A, and the following axioms are satisfied. PI. C.O=O (where 0 is the zero element of the Boolean algebra). P2. y. C.Y = Y for each YEA. P3. C.( Y . C.Z) = C.Y . C.Z. P4. CoOfY =Cp.Y. P5. d.. =1 (where I is the unit element of the Boolean algebra). P6. For i and j both different from k, d.i=C"(d.,, . .di/',). P7. For i different from j, C.(dij,y),Cddij'-Y)=O. With regard to such an axiom system one of the most basic questions which can be asked is whether it is complete; i.e., whether every law holding for all h.p.c.a.'s follows in a purely algebraic way from those laws selected as axioms. A closely related question is whether every cylindrical algebra satisfying the given axioms is isomorphic with an h.p.c.a.. In the present instance the second question has a negative answer, as the following considerations show. In any finite-dimensional h.p.c.a. it is easily seen that the only elements Y such that CiY = Y for every i are Y = 0 and Y =1. On the other hand we can construct a finite-dimensional cylindrical algebra 5B containing elements Y of this kind other than 0 and 1. It will follow that cannot be isomorphic to any highly proper c.a.. To construct m, we start with any two finite-dimensional h.p.c.a.'s of the same dimension N, say 5B1 = U, 1'1, -', di:), C~l» and 5Bz = ( A z, U, f1, -', d~:), Ci2» , whose unit elements X~ and X: are disjoint. Let A be the class whose elements are all unions Y U Z such that Y E At> Z E A z. Let dii=d~:) U di~), and let O, be the operation on A such that Gi(Y U Z)=C~l)y U G~2)Z for any Y EAt> Z E A z. Then it can easily be shown that the system 5B = (A, u,
m
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L. HENKIN
n, I , d t;, Ct ) satisfies all of the axioms PI-P7 for cylindrical
algebras, is finite-dimensional (in fact has dimension N), and possesses the element X~ (different from its unit element X~ V X:) with the property that CiX~ =X~ for all i. An analysis of the construction of >8 shows that it is essentially the direct product of the algebras >81 and >8 2 , Indeed, any direct product of cylindrical algebras is again a e.a., since the axioms for c.a.'s are universal equations. On the other hand an algebraic study shows that every finite-dimensional h.p.c.a. is simple and directly indecomposable. These observations explain why we cannot expect every c.a. to be isomorphic to an h.p.o.a.. And at the same time they suggest a natural way to broaden the concept of an h.p.o.a. so as to obtain a new family of structures which will be closed under the formation of direct products. These new structures are called proper cylindrical algebras (p.c.a.'s). The proper c.a.'s are defined in the same way as the h.p.c.a.'s, except that the unit element is not restricted to be a set of the form X''', but may be a union of disjoint sets of this form (all with the same value for TJ). The diagonal element d i ; is the set of all sequences (of type TJ) of this unit element whose ith and jth terms are identical. The cylindrification operator Ci' applied to any subset Y of this unit element, yields as value the set of all sequences of the unit element which differ from a sequence of Y at most in the ith term. An TJ-dimensional p.o.a. is thus a system >8 = 8 will be called dimension-complemented, if the union of the dimension indices of any finite number of
THE REPRESENTATION THEOREM FOR CYLINDRICAL ALGEBRAS
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elements of >8 has an infinite complement (with respect to 'YJ). Clearly every Lf.d, cylindrical algebra is dimension-complemented. Now we can show: Every dimension-complemented o.a. is isomorphic to a proper c.a.. This constitutes the representation theorem for cylindrical algebras mentioned in the title of the paper. [See added note at end of paper.] Before discussing the proof of the representation theorem, however, we wish to consider briefly the question of whether there arise in a natural way other models of the axioms PI-P7 than the p.c.a.'s; for if this is not the case then the process of abstracting the algebraic concept of a cylindrical algebra loses much of its point. It happens, however, that the general theory of formal systems furnishes certain very important cylindrical algebras other than the p.c.a.'s, and we proceed to give a brief description of these. Consider again a first order formalism ~. We may obtain a formal calculus, ~+, from ~, by adding to the list of primitive symbols and rules of formula formation a specification of certain formulas as formal axioms and prescribing formal rules of inference. We shall assume that these include a standard set of axioms and rules for the pure first-order functional calculus 7. Now we may associate with each wff 0 of ~ the set [0] of all wffs H such that 0 - H is a formal theorem of ~+. It can be shown that two different sets of this kind are disjoint. Let A be the class Then we may define an operation of all sets [OJ (for all wffs 0 of + on A by the rule that [G]+ [H]= [G V H], for it can be shown that whenever G GI and H _ HI are formal theorems of ~+, so is the wff (G V H) = (GI V HI)' In the same way we define operations " -, and G., i = 1, 2, 3, ... , by the following rules: [G]· [H] = [G A H], - [G] = [,....., G], Gi[G] = [(3x i)G]. Finally, we define dii , i, j = 1, 2, 3, ... , to be [x.=xil Then the system (A, +, ., -, d ii , G.) can be shown to be a c.a. satisfying axioms PI-P7. This last observation gives to the representation theorem for c.a.'s important metamathematical consequences.
m.
,
See, for example, Church [5].
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Still another way of forming c.a.'s, although in a sense trivial, is worthy of note. Namely, we start with an arbitrary Boolean algebra 9!= 01) is a I-dimensional c.a. satisfying all of the axioms PI-P7. Since a proper I-dimensional c.a, is in particular an algebra of sets, the representation theorem for I-dimensional c.a.'s implies the representation theorem for Boolean algebras. A Boolean algebra can also be used to form an 1]-dimensional c.a., for arbitrary 1], by choosing dij=I for all i, j <1], and defining 0iY=Y for all i<1] and any element Y of the Boolean algebra 8.
§ 2. Some Remarks on a Certain Feature of the Proof of the Representation Theorem We have observed above that although cylindrical algebras, both proper ones and others, are closely connected with formalisms, models, and formal calculi, the basic definitions can be given without any reference to these metamathematical concepts, so that the representation theorem for c.a.'s may be regarded purely as a theorem of algebra and set theory. Indeed, we have pointed out that this theorem may be regarded as a generalization of the representation theorem for Boolean algebras. It might be expected, therefore, that a proof could be devised along purely algebraic lines, generalizing in some manner the familiar proof of Stone [6] in the case of Boolean algebras. However, we have not succeeded in finding such a proof. Instead, our proof generalizes another proof of the representation theorem for Boolean algebras which is of a metamathematical nature [7]. That is, when an arbitrary c.a. is given, and it is required to construct a proper c.a. isomorphic to it, we introduce certain formal systems and their models as elements of the construction process. Such a procedure may be criticized by some who feel that algebraic theorems should be given purely algebraic proofs, or 8 We emphasize again that many of the ideas described in this section of the paper were originally developed by Tarski and Thompson.
THE REPRESENTATION THEOREM FOR ,CYLINDRICAL ALGEBRAS
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that in proving theorems about a given class of structures all constructions should remain in the same domain. We have no real defense against such criticism, other than to repeat that in the present case the metamathematical proof is the only one we have been able to find, and to report that it seems to us of just as much interest and validity that metamathematical concepts should be applicable to the solution of algebraic problems as that geometrical methods should find application in algebra, or vice-versa. There is, however, another feature of our metamathematical construction which perhaps requires more discussion. Namely, we shall extend the ordinary notion of a formula and speak of formulas of infinite length. More specifically we shall consider formalisms in which the primitive symbols are the same as in the usual first order formalisms, but in which the rules of formula formation provide that a basic formula may be constructed by placing a sequence of individual symbols of type r; after a predicate symbol - where r; may be either a finite or transfinite ordinal number. Further formulas are built up from the basic ones by means of truth-functional connectives and quantifiers in the usual way, so that only a finite number of these appear in anyone formula. Each formula thus has the form of a well-ordered sequence of symbols. It may be objected that formulas are physical objects, or at least classes of physical objects, so that infinitely long formulas do not really exist. In response to such objections we take the position that from the mathematical point of view it does not matter what the true nature of a symbol may be, but it is only important that the rules governing the use of symbols be clearly stated. If we allow ourselves a broad form of set theory as a metalanguage within which to carry out metamathematical discussions, then it is quite clear that we can develop a theory of symbols in a precise way so as to admit infinitely long formulas. One way to do this is simply to identify symbols with certain sets - say the ordinal numbers - so that the whole theory of expressions simply becomes a part of the theory of transfinite sequences of ordinal numbers. Alternatively we could axiomatize our theory, taking symbols and expressions as undefined objects
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and providing suitable axioms governing their use. In this case we would need, in addition to the binary operation of concatenation, a new primitive operation, infinitary in character, which would permit us to construct infinitely long formulas. We shall not here choose between these two methods of setting up foundations for a theory of signs, nor will we enter further into details of either of these methods. But we shall allow ourselves to speak of and operate with infinite formulas in an intuitive fashion, contenting ourselves with this remark concerning the possibility of justifying these activities. A formalism of the kind described above, admitting formulas of infinite length, can be transformed into a first-order calculus by adopting one of the standard sets of axiom schemata and formal rules of inference which have been considered in the literature for formalisms containing only formulas of finite length. It will then be found that the metatheory of the new calculus can be developed in very much the usual way: standard formal theorem schemata can be derived, various metatheorems concerning normal forms, duality principles, the so-called regularity theorem, all can be established for the new calculus without essential changes from the original derivations. There is one simple respect, however, in which the new calculus differs from standard ones. Namely, given a wff of an ordinary calculus it is always possible to construct a closure of it, by prefixing universal quantifiers, so as to obtain a wff without free variables which is a formal theorem if and only if the original wff is a formal theorem. In the new calculus this is not always possible. For a formula such as gX1 x 2 xa ... contains infinitely many distinct free variables, while any formula containing it can have only finitely many quantifiers and so cannot have all of its variables bound. This distinction between new and old calculi requires certain complications in the proof of the representation theorem. In connection with our new type of formalism it is also possible to consider models. These again consist of a domain of individuals, together with an assignment of values to each constant of the formalism. In the case of predicate constants g which are followed
THE REPRESENTATION THEOREM FOR CYLINDRICAL ALGEBRAS
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by sequences (of individual symbols) of type 'fJ, in constructing wffs, the value g assigned by the model will naturally be a class of sequences (of individuals) of type 'fJ. Given a formalism and model of this kind, the relation of satisfaction can be defined without essential change from that employed in the case of ordinary formalisms. It may happen that the formalism contains among its primitive symbols individual variables which are given in a list having ordinal type ~' In that case the relation of satisfaction will be so defined as to hold between well-formed formulas and sequences (of individuals) of type t. In terms of the notion of satisfaction other semantical concepts, such as truth and consequence, can be defined in the ordinary way for the new type of formalism, and the theory of these concepts can be developed largely along familiar lines. However, we wish to note one respect in which the new theory differs from the old. Namely, for an ordinary first-order functional formalism it is true that if is a set of formulas such that for each finite subset F' of r there exists a model M' satisfying every wff of F', then there exists a model M which satisfies every formula of For formalisms containing wffs of infinite length this is not in general true, as one easily sees by considering the following set r of wffs:
r
r.
This fact also entails complications in the proof of the representation theorem. § 3.
The Representation Theorem for Cylindrical Algebras
Lei n be any initial transfinite ordinal number
9.
Let Sll= (A, +, "
9 Our proof does not hold for finite-dimensional c.a, 's, and indeed at present writing it is not known whether the representation theorem holds for such algebras. (But see note at end of paper.) On the other hand a ;-dimensional c.a. for which ; is transfinite but of the same cardinality as some smaller ordinal, 'f), can always be regarded as an 'f)-dimensional c.a. simply by renumbering the cylindrification operators 0i and diagonal elements dij' Similarly a proper c.a. of ; dimensions can be expressed as a p.c.a. of 'f) dimensions simply by systematically rearranging the terms of the sequences which make up the elements of the algebra. In view of these
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HENKIN
- , Co, d'i) be any n-dimensional cylindrical algebra which is dimensionaUy complemented. Then there exist an q-dimensional proper
m=
m.
Note added November, 1954. As originally announced at the Symposium, the representation theorem stated that all infinite-dimensional c.a, 's were isomorphic to proper c.a, 's, I have since discovered a flaw in the original proof, so that the additional hypothesis restricting the theorem to dimensionally complemented algebras must be added for that proof to be valid. Of course the condition that a c.a, be dimensionally complemented, while sufficient, is certainly not necessary for the c.a, to be representable as a p.c.a. However, it has proved possible to strengthen somewhat the representation theorem given here, so as to obtain an algebraic condition both necessary and sufficient for a given c.a, to be representable. This form of the theorem is applicable uniformly to c.a.'s of any dimension, finite or infinite. Having obtained an algebraic condition both necessary and sufficient for a c.a, to be representable, the question still remains whether or not aU cylindrical algebras are representable. For the finite-dimensional case we have been able to find examplee which show that the answer is negative, and there is some reason to believe (though it is not yet proved) that the same is true for the infinite-dimensional case. Finally we add two notes of an elementary nature concerning the axioms Pl~P7. (i) Axiom PI can be derived from the other axioms P2-P7 except in the case of one dimensional c.a, 's, when it is independent. (ii) Although the law dii = d i, can be derived, in general, from axiom P6, in the case of a 2-dimensional c.a, the law d l l = d n is independent of the axioms and must be taken as an additional axiom.
REFERENCES [1] A. TARSKI and F. B. THOMPSON, Abstract 85: Some general properties of cylindrical algebras, and A. TABSKI, Abstract 86: A representation theorem for cylindrical algebras, Bull. Am. Math. Soc., 58 (1952), 65-66. facts the representation theorem for $-dimensional c.a, 's reduces to that for "I-dimensional c.a. 's, and therefore we restrict our consideration to the case where "I is an initial number.
THE REPRESENTATION THEOREM FOR CYLINDRICAL ALGEBRAS
97
[2] D. HILBERT and P. BERNAYS, Grundlagen der Mathematik, Berlin, (1934, 1939). [3] A. TARSKI, Der Wahrheitsbegriff in den formalisierten Sprachen, Studia Philosophica, 1 (1936), 261-405. [4] A. MOSTOWSKI, Sentences undecidable in formalized arithmetic, Amsterdam (1952). [5] A. CHURCH, Introduction to mathematical logic, Ann. of Math. Studies, No. 13 (1944). [6] MARSHALL STONE, The representation theorem for Boolean algebras, Transactions of the Am. Math. Soc., 40 (1936), 37-111. [7] LEON HENKIN, Boolean representation through propositional calculus, Fundamenta Mathematicae, 41 (1954), 89-96. Department of Mathematics, University of California, Berkeley Calif. U.S.A.
.JERZY LOS
QUELQUES REMARQUES, THEOREMES ET PROBLEMES SUR LES CLASSES DEFINISSABLES D' ALGEBRES Cette conference a pour but l'assemblement de certains resultats, concernant la theorie des algebres universelles, et plus partioulierement ceux qui s'occupent des classes d'algebres definissables par voie elementaire. La theorie des algebres universelles eut pour debut les resultats frappants de G. Birkhoff [1,2]. Nous trouvons deja chez lui la notion de la classe d'algebres definissable par voie elementaire d'un type special, a savoir les classes equationellement definissables, et leurs caracteristique algebrique, Neanmoins en meme temps la theorie des classes definissables trouve une base logique plus large dans les travaux de A. Tarski, devenant, grace aux recherches de Tarski et de son ecole, une discipline presque independante, plaoee sur les confins de I'algebre et de la logique. Je ne considere point comme rna tache d'aujourd'hui de vous presenter ici la problematique de la discipline envisages toute entiere et son developpement chronologique. Je me bornerai it quelques themes choisis qui ru'interessent speeialement, et en liaison avec lesquels je desire vous presenter quelques questions ouvertes.
1. Par une algebre on comprend un ensemble non-vide A et certaines operations 01' ... , On definies dans A. Sous une operation on comprend ici une fonction d'un certain nombre de variables choisi d'avance, definie sur A et it valeurs dans A. On designe les algebres par une seule lettre, p. ex. T', F'. On ... ,O~). Les algebres eorit alors r=
LES CLASSES DEFINISSABLES D'ALGEBRES
99
et les operations Oi et O~ sont respectivement du meme type, ce qui veut dire qu'elles sont des fonctions du meme nombre de variables. Dans ce qui suit, nous designerons par m: la classe de toutes les algebres d'un type fixe et nous ne parlerons que d'algebres, de cette classe. Nous supposerons que les operations O; de l'algebre p.: ... , On> E sont des fonctions de k; variables. A la classe mest adjointe une langue d'une theorie elementaire, c. ad. l'ensemble des formules formees par la voie habituelle des signes 1 , •.. , On' des variables individuelles XI' x 2' ••• et des connectives logiques: I'identite =, l'implication -+, la disjonction V, la conjonction 1\, I'equivalence - , la negation ' et les quantificateurs ll-general et. E-existentiel. Les signes o, sont regardes '" '" de ki variables. Nous designons l'encomme signes de fonctions semble de toutes les formules par E. Les signes °1 , ••. , On qui apparaissent dans une formule iX E E peuvent etre interpretes par les operations 01> ... , On de chaque Par cette interpretation iX devient vraie ou algebre donnee fausse. Nous disons que iX est vraie ou fausse dans r. Ce fait nous permet d'introduire deux operations, l'une sur les algebres l'autre sur les sous-ensembles X C E. La premiere operation assigne a chaque algebre rEm: l'ensemble E(r) compose de toutes les formules iX E E qui sont vraies dans r, la seconde assigne a chaque sous-ensemble X C E la classe m(X) de toutes les algebres rEm: dans lesquelles chaque formule de X est vraie. L'operation E(·) peut etre prolongee encore de telle faeon qu'elle soit applicable non seulement aux algebres mais aussi aux sousclasses m: o C m:. On pose dans ce but
m
°
rEm.
rEm,
E(m: o )
=
IT E( r).
rE'lr.
L'ensemble E(mo ) est alors compose de toutes les formules iX E E qui sont vraies dans chaque algebre E Pour XC E, m(X) est une sous-classe de m. Ce n'est certes pas chaque sous-claese o C m qui peut etre presentee sous la forme mo = m(X), XC E. Les sous-classes qui se laissent presenter sous cette forme sont dites classes E-definissables.
r mo'
m
100
JERZY Z.OB
Des fois il est plus commode d'exprimer cette definition de Ia faeon suivante: (1.1) La classe ~o C ~ est E-definissable (E-def.) si et seulement si ~o = ~(E(~o»' Une classe ~o C ~ qui peut etre presentee sous la forme ~o = ~(X), X etant un ensemble fini, est dite axiomatisable, ou (selon Tarski [23]) classe arithmetique. Les operations ~( . ) et E( . ) sont respectivement du type 211 --+ 2w et 2~ --+ 2E • Neanmoins depuis longtemps est connue une operation dans E, c.-a.-d. une operation du type 211 --+ 211 (Tarski [19]), nommee operation de consequence. Cette operation assigne it ohaque sousensemble XC E, l'ensemble Cn(X), de toutes les formules <X E E qui se laissent deduire elementairement (o.sa-d. uniquement sur Ia base du calcul fonctionnel d'ordre premier) des formules de X. La theorie generale de cette operation a ete etudiee par Tarski d'une faeon detaillee, c'est pourquoi je trouve superflu de m'en occuper. Je rappelle seulement qu'un ensemble XC E est dit non-contradictoire, si Cn(X) # E; un systeme, si Cn(X) = X; un systeme axiomatisable, si X = Cn(X) = Cn( Y), Y etant un ensemble fini. Enfin pour un systeme X, Ie systeme
,X
IT
=
C,,(X+Y)-E
Cn(Y)
est dit pseudo-complementaire de X. Je rappelle aussi les theoremes suivants: (1.2)
Cn(X)
2
XCY
Cn(Y)
[19].
Y 1InI;
(1.3)
Un systeme X est axiomatisable si et seulement si I
X = Cn(ep)
~(X)
= ~(Cn(X».
X· (1.4)
[21].
En posant ~l (~o) = ~(E(~o»' on introduit dans la olasse ~ une operation du type 2~ -+ 2~ qui ressemble a. celle de consequence, definie dans E. Une analogie entre les classes E-def. et les systemes est alors frappante. Ces deux types d'ensembles sont definis par
101
LES CLASSES DEFINISSABLES D'ALGEBRES
une condition de la forme F(A)=A. On obtient les classes E-def. en prenant pour F I'operation mI, tandis que pour obtenir lea systemes on prend pour F l' operation On. U est connu d'ailleurs que, aussi bien les systemes que les classes E-def. forment une structure (lattice) Brouwerienne. En tenant compte de (1.4) et de la definition de I'operation m(·) on peut facilement prouver que cette operation est un homomorphisme dualiste de ces structures. Le fait que cet homomorphisme eat un Isomorphisme, resulte du theoreme de GOdel, qui est habituellement exprime: (1.5)
Pour un ensemble X non-contradictoire,
m(X)~,p,
Car une consequence de ce theoreme est que
E(m(X»=x, pour les systemes X,
(1.6)
ce qui montre l'existence de I'operation inverse. Le theoreme sur l'isomorphisme de la structure des systemes et la structure des classes E-def. n'est, a vrai dire, qu'une formulation differente du theoreme de Godel. Avec I'aide de (1.2) et (1.3) il s'en suit que:
me,
Ilm, , = ,p, mI., ..., ml,.=,p.
(1. 7) Si t E Test une famille de classes E-def. et alors il exist un ensemble fini ~, ... , tn ET, tel que
m
(1.8) Pour qu'une classe E-def. o soit axiomatisable il faut et il suffit, que la difference m- o soit une classe E-def.
m
Le theorems (1.7) est nomme (Tarski [24]) theoreme de compacite des classes E-def. (compactness theorem). U est simplement une traduction dans la langue des classes E-def. du theorems (1.2). Le theoreme (1.8) caracterise les classes axiomatisables et il est une traduction analogue de (1.3). II est a remarquer que (1.7) (Rasiowa [14]) ainsi que (1.8) (Henkin [3], p. 419) furent demontres independamment de (1.2) et (1.3). L'algebre des systemes et la theorie des structures Brouweriennes ont ete si profondement examinees par Tarski et ses collaborateurs ([21], [22]), [11], [12]), que en vue de ce que nous avons deja mentionne sur ce sujet, on ne peut s'attendre a trouver des pro-
102
JERZY l.OS
blemes nouveaux dans l'algebre des classes E def.. c.-a.-d. dans la theorie qui s'occupe uniquement des classes E-def., sans faire usage d'autres notions. En realite, les problemes nouveaux lies aux classes E-def. se developpent principalement dans d'autres directions. J'ai !'intention de vous rendre compte en lignes generales, de trois de ces directions. La direction premiere s'occupe it caracteriser des classes E-def. par moyen de certaines operations sur les algebres, telles quelles p. ex. le produit direct. La seconde traite de conditions sous lesquelles certaines classes definies par voie non-elementaire sont des classes E-def. La troisieme, enfin, contient les problemes d'extension des algebres. Cette classification n'a pas de pretention d'etre complete, en plus, les directions mentionnees ne sont pas disjointes. Nous verrons dans la suite de proches connections qui les lient. Je voudrais encore remarquer que, tant dans les considerations preeedentes, que dans celles qui suivent, la supposition que nous nous occupons des algebres, dans la plupart des cas n'est pas essentielle. On pourrait aussi bien parler plus generalement des modeles de theories elementairea arbitraires. Les difficultes d'une telle generalisation apparaissent la. seulement, ou on parle de produits directs, d'homomorphismes et de congruences. Le sens de ees notions, quant aux algebres, est tres clair; leur generalisation aux modeles presente des diffioultes. 2. Designons par R les equations de E, c.va-d. les formules de la forme {}=T, ou {}, T sont des "polynomes" formes de signes 0l> ... ,0" et de variables xl> x 2 ' ••• (nous appelons de tels "polynomes" des termes), et par 0 designons ees formules de E qui sont formees sans quantificateurs (les formules ouvertes). Evidemment RCOCE. Posons R(SJio)=R.E(SJio) et O(SJio)=O.E(SJio)' La olasse SJio est dite equationellement definissable (R-def.), si SJio= SJi(R(SJio))' et definissable par formules ouvertes (O-def.), si SJio= SJi(O(SJio))' Les classes R-def. ont ete examinees par Birkhoff [2], qui a trouve les conditions neoessaires et suffisantes pour qu'une classe donnee SJio soit R-def. Comme c'est oonnu, il faut et il suffit pour
103
LES CLASSES DEFINISSABLES D'ALGll:BRES
ce but, que ~o soit close par rapport au produit direct, aux sousalgebres et a. la division par les congruences, c.-a.-d. que (2.1)
L'algebre produit
(2.2) a. ~o.
Chaque sous-algebre
~rt
des algebres T', E ~o,appartient
ro'
d'une algebre
r E ~o'
(2.3) Chaque algebre quotient Tl--: d'une algebre tient a. ~o' 1
r
a. ~o.l appartient E ~o'
appar-
Le theorems de Birkhoff donne l'exemple typique d'une solution complete d'un probleme qui appartient a. la premiere direction mentionnee. Les autres problemes, pour la plupart, ne se laissent point resoudre par un theoreme de forme si simple que celui de Birkhoff. P. ex. pour les classes O-def. nous avons le theoreme suivant (Los [6, 7]): (2.4) Pour qu'une classe E-def. ~o soit O-def. il faut et il suffit, que ~o soit close par rapport aux sous-algebres (c.-a.-d. que la condition (2.2) soit remplie). Neanmoins ce n'est pas une oaracteristique des classes O-def. analogue a celIe de Birkhoff, car elle permet seulement de distinguer les classes O-def. parmi les classes E-def. Pour une caraeteristique complete des classes O-def. il est neoessaire d'introduire la notion du champ logique. On appelle ainsi les triples P = (B, (J, T'», ou B est un corps de Boole 2, r= (A, Ov ... ,0..) une algebre de ~ et enfin e(x, y) est une fonction definie pour x, YEA, a. valeurs qui appartiennent a. B et qui remplit les conditions (x, Y, z, xv,." x k,' Yv ... , Yk, E A): e(x, y)=ep si et seulement si X=Y; e(x, Y)=e(Y, x); e(x, y)+e(y, z):J e(x, z), k;
e(0,(x1 ,
••• ,
x kj), O,(Yv ... , Yk;» C
!
i-I
e(x i, Yi)' i
=
1, 2, ... , n.
1 ~Tt denote le produit direct des algebres T t et Tj,..." I'algebre quotient (voir p. ex. G. Birkhoff, Lattice theory, New York, 1948, Foreword on algebra, vii-ix). 2 Selon l'usage nous denotons une algebre de Boole par une seule lettre - B. Pareillement un groupe par G.
104
JERZY LOB
Si B est un Corps a deux elements, Pest dit champ simple sur r. Pour un algebre donnee T, il n'y a evidemment qu'un seul champ simple sur r. Le champ Po=
eII(xl"""", YI"""")=e(x, Y)/1. 4 On dit que l'algebre rest obtenue par I'operation (P), effeotuee surles algebres I', si rest I'algebre du champ simple P, obtenu par Ie precede suivant:
I. sur chaque algebre r, on etend un champ simple P,; II. on construit Ie produit direct ~P, de tous les champs P t; III. dans le corps du champ ~P t on choisit un ideal premier I et on divise Ie champ produit ~P t par cet ideal: P = ~P til. On voit de suite que l'operation (P) n'est pas univoque; Ie resultat de son application depend du choix de I'ideal 1. Mais, ce qui est essentiel, elle est applicable aux algebres et comme resultat elle donne egalement des algebres.
$r"
3 f et g appartenant a elles sont alors des fonctions, qui pour t E T prennent une valeur dans l'ensemble d'algebre r,. h appartient a $B t , c'est une fonction dont Ill. valeur pour t E T appartient a B,. 4 xl-- et YI __ denotent les elements de I'algebre quotient rl--, e(x, Y)/l - un element du corps quotient BjL, parce que e(x, y) E B pour x, Y de r.
LES CLASSES DEFINISSABLES D'ALGEBRES
105
On peut demontrer, que:
m
(2.5) Pour qu'une classe o soit O-def. il faut et il suffit, que mo soit close par rapport aux sous-algebrea et a l'operation (P). Nous avons aussi le theorems suivant: (2.6) Une condition necessaire (mais pas suffisante) pour qu'une classe mo soit E-def., est que o soit close par rapport a I'operation (P).
m
Une consequence de (2.5) et (2.6) est (2.4). Tenant compte de (2.6) on peut demontrer que la classe des algebres de cloture bicompactes, n'est pas E-def. Nous verrons que ce resultat peut etre egalement obtenu par une methode differente, II est a remarquer que les champs logiques ont eM introduits sous une forme moins algebrique, dans [5]. Dans ce travail ils furent nommes "matrices algebriques". Par une voie analogue a. celIe qui donne une caraoteristique des classes O-def., on peut obtenir aussi une oaracteristique des classes E-def. Les demonstrations et les notions, qu'il faut introduire dans ce, sont bien plus difficilis et compliquees ; elles exigent l'usage de quelques theoremes et constructions de L. Rieger [15, 16]. Pour ne pas compliquer oet expose par un grand nombre de notions nouvelles, je me bornerai a. presenter quelques conditions neoessaires pour qu'une classe soit E-def. ... ,O~), ~
2
E<'"
r E=<
2
E>'"
AIEl,Ov ... , On), OU 0i(X1 ,
... ,
Xk.)=O~E)(xv
... , Xk)
pour un ~ arbitraire tel que Xv ... , x k ; E A E• L'operation de la sommation des algebres trouve une large application dans la theorie des groupes. II est connu que la somme de chaque suite croissante de groupes est un groupe. Ce theoreme est valable plus generalement pour les classes O-def.:
r m
m
(2.7) Si o est une classe O-def., o E o ' ~ < IX, une suite croissante d'algebres, I'algebre somme 2 r E appartient aussi a mo. E<",
106
JERZY l.OS
Le theorems analogue pour les classes E-def. est faux. En 1951 C. Ryll-Nardzewski 5 a demontre le theoreme suivant:
m
mo' ~ < une suite croissante mo' qui contient com me sous-
(2.8) o etant une classe E-def., F o E d'algebres, il existe une algebre FE algebre I'algebre somme ! r; 0<'"
IX,
m
Ce theorems exprime une condition necessaire pour que o soit E-def. II permet de deduire que la classe des algebres de cloture separables, ainsi que la classe des algebres finies, ne sont pas E-def. Le theoreme (2.8), applique plus generalement aux modeles, pour lesquels il est aussi valable, donne le resultat bien connu de Tarski et Kuratowski [21,4] d'apres lequel la classe des ensembles bien ordonnes n'est pas E-def. La premiere demonstration de Ryll-Nardzewski (pour Ie cas IX = w o ) s'appuya, d'une faeon inattendue, sur l'existence de la limite generalises, d'un type special, de Banach et Mazur. C'est une operation qui assigne a chaque suite bornee x, de nombres reels un nombre reel Lim Xi; cette operation etant additive, multiplicative, pour les suites convergentes elle est egale a la limite habituelle. Aujourd'hui nous savons deja deduire le theoreme de Ryll-Nardzewski des theoremes generaux sur l'extension des algebres (voir theorems (4.1)), neanmoins la methode de RyllNardzewski se trouve utile pour certaines demonstrations, comme nous le verrons par la suite (theoreme (3.4)). Les theoremes (2.7) et (2.8) ne paraissent point renfermer toute la problematique Me a la sommation des algebres, -Iusqu'a present nous nous sommes ocoupes de caracteristiques algebriques des classes definies par voie logique. En posant la question a rebours, on peut s'interesser a la earaoteristique logique des classes definies algebriquement. Le probleme de E. Marczewski qui demande quelle est la earaeteristique logique des classes closes par rapport a la division par congruence [10] 6, ainsi que la question de 6 -Iusqu'e present pas publiee, • Contrairement a l'opinion du compte rendu dans Math. Rev. 14 (1953), 347 ce problems fut publie pour la premiere fois par Marczewski [10J.
107
LES CLASSES DEFINISSABLES D'ALGEBRES
caraoteriser les classes closes par rapport a la sommation, appartient type de problemes. Le problems de Marczewski peut etre resolu (par affirmation de I'hypothese de Marczewski) en utilisant les champs logiques. Cela exige des recherches detaillees sur la structure de certains champs (dits champs libres) et n'est pas facile. Je ne connais, par contre, aucune caracteristique des classes closes par rapport it la sommation. Je suppose qu'une telle classe peut etre presentee sous la forme ~o=~(X), ou l'ensemble X consiste des propositions de la forme
a ce
IT···IT!·.. 2 ""
"'k
""'+1
1X
1XEO
"'k+l
(un seul changement de quantificateurs dans le prefixe) qui appartient it E(~o)' mais la demonstration m'est inconnue. En 1951 j'ai pose la question, si le theoreme suivant est vrai: (2.9)
Si le produit direct fini Ftx ... x I'; appartient, pour
chaque n, it une classe E-def. appartient aussi it la classe ~o'
~o'
le produit direct infini
00
~
F"
,,~1
Un theoreme qui donne une solution partielle de ce probleme fut demontre par A. Mostowski [13] (Ie cas de toutes les algebres F" identiques). Le theoreme (2.9) sans aucune restriction fut demontre par R. L. Vaught, a qui j'avais communique ce probleme par ecrit. 7 Le theoreme (2.9) exprime aussi une condition necessaire pour qu'une classe soit E-def. La classe des algebres de cloture bicompactes, p. ex., ne remplit pas cette condition, parce que les algebres de Boole finies munies de cloture discrete (a=a) sont bicompactes, mais Ie produit direct d'une suite infinie de telles algebres n'est pas bicompact. 3. Le groupe G est dit (Iineairement) ordonne, si pour les elements de G une relation d'ordre < est definie, telle que (3.1) 7
a-cb implique xay<xby, pour a, b, x, y R. L. Vaught me
1'110
E
G.
communique sans demonstration.
108
JERZY Z.Os
Si la relation < n'ordonne G que partiellement et remplit (3.1) pour a, b, x, y E G, Ie groupe G est dit partiellement ordonne. Nous savons qu'il y a des groupes qui ne se laissent pas ordonner. La classe .2 des groupes qui admettent un ordre, peut etre definie comme la olasse des algebres (du type approprie), qui verifient la proposition
!II .. , II <
""
(x,
"'k
(X etant la conjonction des axiomes des groupes (eorites a l'aide de la multiplication et l'inversion) et de la condition (3.1) eorite dans des variables convenablement choisies. Nous voyons que la classe .2 est definie par une proposition qui n'est pas elementaire ; de sa seule definition il ne resulte pas, qu'elle soit une olasse E-def. J'ai demontre dans [8) qu'elle est non seulement E-def., mais en plus une classe O-def., neanmoins pas axiomatisable. Dans mon travail [7) on trouve un resultat plus general:
(3.2) Chaque classe d'algebres definies par une proposition nonelementaire de la forme
! II ... II R
(x,
(X
EO,
:tot
Xl
est O-def. II parait interessant qu'un theoreme encore plus general se trouve vrai: (3.3) Chaque classe d'algebres definie par une proposition nonelementaire de la forme
II ... II ! .,. ! ... II ... ! ... II ... II r,
est O-def.
Pk
R,.
R~
S
Q
"'.
(x,
(X
EO
"'.
8
La demonstration de ce theorems, suivant la methode de [8), n'est pas difficile. II est essentiel qu'au commencement les quantificateurs se rapportent aux relations et non aux fonctions, et 8 Ce theorems a ete, a ce qu'il parait, demontre dernierement aussi par Tarski. La demonstration de Tarski ne m'est pas connue.
LES CLASSES m!:FINISSABLES n'ALGEBRES
109
qu'ensuite apparaisse une proposition elementaire de la forme
IT ... IT lX,
lX
E
0,
x,
~1
o.va-d. une formule ouverte preoedee de quantificateurs generaux, contenant toutes les variables Xi qui apparaissent dans lX. On peut facilement trouver une proposition de la forme
LIT.·· IT L ... L R
"'1
"'.
-'+1
lX,
lX
E
1II.+1
0
qui ne definit pas de classe E-def. Hormis la classe £, on envisage souvent la classe ~ des groupes G qui remplissent Ia condition: chaque ordre partiel dans G se laisse prolonger a. un ordre Iineaire. La classe ~ est definie par une proposition de la forme
IT LIT··· IT L .,. L lX, -<
<:lit
"'.
"'.+1
lX
EO
"'.+1
elle n'est dono pas assujettie au theorems (3.3). Le probleme de savoir si elle est E-def., reste ouvert. Je ne sais meme pas decider si chaque sous-groupe d'un groupe de ~ appartient a. ~. Neanmoins nous avons le theorems suivant: (3.4) 00
L
11-1
Si Gn
E ~
est une suite croissante de groupes, la somma
Gn appartient aussi
a.
~
J'ai l'intention de donner ici une esquisse de la demonstration de ce theoreme, pour mettre en evidence l'emploi de la methode de Ryll-Nardzewski mentionnee a. l'occasion du theorems (2.8). Posons G=
L G.. et soit -< un ordre partiel dans G. La relation < 00
.. -1
envisagee dans chacun des groupes G.. separement, est dans G.. un ordre partiel et en vertu de la supposition -< se laisse prolonguer dans G.. a. un ordre lineaire < ... Pour a, bEG posons:
!.. (a, b) =
~
1, si a, b E Gn et a < .. b; 0, dans Ie cas contraire,
110
JERZY l.OS
La relation: a-cb si et seulement si Lim n
In (a, b)= 1
est, ce qui est facile it verifier, un ordre Iineaire dans G et un prolongement de -<. 4.
Soit 2l, comme jusqu'a present, la classe des algebres
T= et 2l' soit la elasse des algebres T' = ' Les algebres de 2l et de 2l' ne sont pas semblables, on obtient une algebre T' E 2l' en ajoutant it une algebre T E 2l, m operations de types choisis d'avance. Pour une algebre donnee T' E 2l' designons par T'12l l'algebre . Nous avons alors T'I2l E 2l, pour T' E 2l'. Soient 2lo C 2l et 2l: C 2l' deux classes E-def. Le probleme d'existence pour une algebre donnee T E 2lo ' d'une algebre T' E 2l', telle que Test une sous-algebre de T'I2l, est nomme probleme d'extension. Dans le travail [7] j'ai demontre le theorems suivant, qui est une solution generale du probleme d'extension: (4.1) Pour l'existence d'une algebra T' E 2l:, telle que l'algebre donnee T E 2l est une sous-algebre de T'I2l il faut et il suffit, que 0(2l:12l) C O(T) (2l:I2l designe la classe des algebres T'12l, OU T' E 2lJ
La demonstration de ce theoreme se fait par l'emploi du theoreme de Godel, en faisant usage des ainsi dites "descriptions" des modeles (diagrams), qui ont eM introduites par A. Robinson ([17], p. 74). Remarquons que (4.1) permet de prouver (avec l'aide de (2.7)) Ie theoreme de Ryll-Nardzweski (2.8) et aussi les theoremes (2.4) et (3.2). Un autre probleme d'extension est Ie suivant: que deux algebres T 1 et 2 appartenantes it la olasse E-def. 2lo soient donnees. Quand existe-t-il une algebre T E 2lo ' telle que ainsi T 1 que T 2 sont des soua-algebres de T? Ce probleme est dir probleme de l'extension commune. R. Suszko et moi-meme, nous avons resolu ce probleme en demontrant que:
r
LES CLASSES DEFINISSABLES D'ALGEBRES
III
(4.2) Pour l'existence d'une extension commune r E mo des deux algebres r l , r 2 E mo il faut et il suffit, que pour chaque disjonction iX(XV ... , x k) V {J(xk+l' , xk+l) a variables separees qui appartient a o(mo), ou bien iX(XI, , x k) appartient a o(rl ) et a o(r2), ou bien {J(xk+V ... , xk+l) appartient a o(rl ) et a o(r2). Le systeme X est dit O-complet, si pour chaque disjonction a variables separees <x(xv ... , x k) V {J(xk+V ... , xk+l), qui appartient a x ·0, ou bien <x(xv ... , x k) appartient a X· 0, ou bien {J(xk+V ... , xk+l) appartient a X· O. Le theorems (4.2) permet d'inferer : (4.3) Pour qu'existe une extension commune r E mo de chaque famille d'algebres T', E mo , t E T, il faut et il suffit que E(mo) soit un systeme O-complet. Le theoreme (4.3) permet de prouver ce theorems, qui a eM demontre independamment dans [6]:
m
(4.4) Dans une classe E-def. o il existe une algebre fonotionellement O-libre, c.ad. une telle algebre r E mo' que O(mo)=O(r), si et seulement si, E(\l!) est un systeme O-complet. Les problemes de l'extension commune sont importants a l'egard des ainsi dits produits libres des algebres (R. Sikorski [18]). Speeialement importants sont ces problemes poses pour les classes R-def. Soit f}(x l) un terme dans lequel n'appartient qu'une seule variable Xl' et tel que I'equation f}(x l) = f}(x2) appartient a R(mo ) ' Dans ces suppositions nous disons que f} definit une constante dans mo' Un terme qui definit dans moune constante, distingue dans chaque algebre r E mo un element. L'ensemble de tous les elements ainsi distingues forme une sous-algebre de chaque algebre
rEmo'
Evidemment une condition neeessaire pour l'existence d'une extension commune de deux algebres est, que les sous-algebres des constantes soient isomorphes. La question de savoir si cette condition est aussi suffisante, dans Ie cas des classes R-def., est un probleme qui reste ouvert.
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JERZY Zoos
OUVRAGES CITES [1] G. BIRKHOFF, On the combination of subalgebras, Proc. Cambridge Phil. Soc., 29 (1933), 441-464. [2] , On the structure of abstract algebras, Proo. Cambridge Phil. Soc., 31 (1935), 433-454. [3] L. HENKIN, Some interconnections between modern algebra and mathematical logic, Trans. Amer. Math. Soc., 74 (1953), 410-427. [4] K. KURATOWSKI, Les types d'ordre definissables et les ensembles boreliens, Fund. Math., 29 (1937), 97-100. [5] J. Los, 0 matrycach logicznych, Prace Wroclawskiego Tow. Naukowego, Seria B. Nr 19, Wroclaw (1949). [6] , The algebraical treatment of the methodology of elementary deductive systems, Studia Logics 2, (1954), 151-212. [7] , On the extending of models I. Fund. Math., 42, (1955), 38-54. [8] , On the existence of linear order in a group, Bulletin de I'Academie Polonaise des Sciences, C1. III, 2 (1954), 21-23. [9] and R. SUSZKO, On the extending of models II, Fund. Math., sous presse. [10] E. MARCZEWSKI, Sur les congruences et les proprietes positives d'algebres abstraites, Coll. Math., 2 (1951), 220-228. [11] J. C. C. MCKINSEY and A. TARSKI, The algebra of topology, Annals of Mathematics, 45 (1944), 141-191. [12] and , On closed elements in closure algebras, Annals of Mathematics, 47 (1946), 122-162. [13] A. MOSTOWSKI, On direct powers of theories, The Journal of Symbolic Logic, 17 (1952), 1-31. [14] H. RASIOWA, A proof of the compactness theorem for arithmetical classes, Fund. Math., 39 (1952), 8-14. [15] L. RIEGER, On free totE,complete Boolean algebras, Fund. Math., 38 (1951), 35-52. [16] , On countable generalized c-algebraa, with a new proof of Godel's completeness theorem, Czechoslovak Math. Journ. 1, (76) (1951), 29-40. [17] A. ROBINSON, On the metamathematics of algebra, Amsterdam (1951). [18] R. SIKORSKI, Products of abstract algebras, Fund. Math., 39 (1952), 211-228. [19] A. TARSKI, Uber einige fundamental Begriffe der Meta-Mathematik, Comptes Rendus de la Societe des sciences et de Lettres de Varsovie Cl. III, 23 (1930), 22-29. [20] , Fundamentals Begriffe der Methodologie der deduktiven Wissenschaften I, Monatshefte f. Math. u. Phys., 37 (1930), 361-404. [21] , Grundziige des Systemenkalkills, Fund. Math., 25 (1935), 503-526 et 26 (1936), 283-301.
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[22] [23]
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, Aussagenkalkiil und die Topologie, Fund. Math., 31 (1938). , Arithmetical classes and types of mathematical systems, Bull. Amer. Math. Soc., 55 (1949), 63-64. [24] , Some notions and methods on the borderline of algebra and metamathematics, Proceedings of the International Congress of Mathematicians, Cambridge, Massachusetts, U.S.A., August 3()"" September 6, 1950, Amer. Math. Soc., Providence (1952), I, 705-720.
Universite, Toruri, Pologne.
