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lr.
by y,
y new
4.2.2
pi,
38 v
INFINITARY PROPOSITIONAL LANGUAGES
< y.
I<&)I # I($:)[ 8 = (B
A
(ii).
v # v'.
p
v)
y. La
B 8
=
8 bo # 0 bo 1.
80
8 by
La
E = B.
no
h on
I' = ( A : A
L, by 4.3.4, I',6 < 6, 0 < 6 < a , t * [ A A o . . . A t . . .] a
i!*A 2 bo}. r Ao, . . ., A t , . . . bo f 0.
s
80
2
I',
= s*A
80 =
4.3.6
bo [A
1.
h
A'] E P,
s*A = s*A'.
t)
a
:
(i) no
p p PROOF: 4.3.5 10.4.3
4.2.2. (a,
CHAPTER 5
I N F I N I T A R Y P R O P O S I T I O N A L LOGIC
1.2)
1,
A
“I-A”)
A a
A,
formal proof),
kA complete
IkA FAA
provable from d A complete strongly A
A
A
“I-AA”) A
A A. A
all IkdA
all
A. no
L,
A l I - ~ l ! $--+ $1
‘230
A La,
l@
--+ $1
A
a
A’
a
l I - ~ * l ! $+$1, no Ly+, y
I - ~ t l ! $
--+
$1.
A’. y
y
4.1.3.
40
I N F I N I T A R Y P R O P O S I T I O N A L LOGIC
L,+
y‘
2
y’
< y,
L,+.
5.5.4. Leu, La,
no
w,
10.2,
10,
5.1 Description of the Formal Systems for a-Propositional Languages
Ag, 5
<
A,,,
,u
<
v
<
A “ A t ” , “A,,”
L, SFd,
V
V
f
La.
“A [A1 + A o ] ] ” Ao]]”.
[A0 +
“A
[A0 + [A1 +
Ao, A1 La.
5.1.1
4 1 =
Yz
ga.
[Ao + [ A i -* A011
=
N = “ [ - A 01 --+ [ i Aill + [ A1 +A 011 % ] , a = [[A [ A d + A o ] . . . [ A d + A g ] . . . ] + [ A d + [ A A o . . . A € .. . ] ] I ( A o . . . A t . . .> 6, 0 < 6 < a. %2,d,* = [[A A o . . .Ag. . .] + A,], ( A o . . .Ag. . .> 6, 0 < 6 < v < 6.
41
FORMAL SYSTEMS FOR a-PROPOSITIONAL LANGUAGES
rules of inference [A0 + A11
Ao, A6
0 <6
f
<
Al.
<6
[ A A o . . . A t . . .],
L,
@a(La)
'@a,
,Z ga(La)
!#a(L')(La)
C
all '@a(Z)
compEete
5.1.2 Distributive Laws.
y y)
on y
yy
[A Eo.
y.
E t . . .]
..
[ A Ed, t
v.
(Eo. . . E c . . .).
6
%J= " A [ v AllYll + I: v C
e<2expv
V
[ A
C
~,~,(,,111. u >,
By 1.2.1,
gY 1,
y)+.
no
yy.
By
E <2
all
y)
gY.
yy
on
y.
y, y
5.1.3 Chang's Distributive Laws.
n,
I: v
[ A
P
V
AE
g Eyy, [YAE]
(dDge): p < y} 6 < y.
[YA~]
y
f
< y, A€,
42
I N F I N I T A R Y PROPOSITIONAL LOGIC
17, a
> yf.
17, y+ = a ,
n,,,
As
y =2 y‘, p < y, v < y ,
by {gt : 6
< y } = y’y’,
dpv
5.1.2 : v
< y,
y’
................................................................, ................................................................. ......
.......
{d,g(p): p < y}
[1A,,]
y’
g&)
= vp
Apg,(,,
< y’.
wa(17,,)(L,)
L,
@(,
p
a 2 y+.
U n,,)(L,) Y
by 5.1.4 Lemma.
23
7 L,
23.
A
=S
Td
d , s*A = 3.5.5. 5.1.5 Theorem.
L,
d
&’
Z
L,
d
La, FA A
@,(Z) (La)
I ~ AA .
43
gy,alI
‘@a,
LlY,
PROOF:
A =SFd by 5.1.4, 230. s s*(Ao) = s*[Ao + All = 1 s*(Ae) = 1 6<6 s*[A A o . . . A t . . .] Ao, . . ., A t , . . . La. A d s*(A) = 1.
s*(Al) = 1, = 1, A, ga(Z)(La).
v a
gY =[
v
[ A dwl1
ClCY
V
n,,. atfiy Bo. vfi < y g(p) = vfl
= 0,
p
s*dpvs = 0.
p
< y)
At, [-At].
5.2 Development of the Formal Systems for a-Propositional Languages A La 1,+, A $1, $2, JV, [2], 149. 5.2.1 Theorem. A La -,, +, FA ‘@a(La). 5.2.2 Theorem. La, Li t-SFA @a(Z)(L:), f X A LL.
FA on 3.5.3
A
L,
1
X,
on X
Li,
LL.
S,XA
< a + 1) X‘ by
(Sf’Cv: Y ?&(Z)(LL).
A X’
N
< cr + 1)
@a(Z)(La),
X‘
L;.
X SfA
44
I N F I N I T A R Y P R O P O S I T I O N A L LOGIC
A
C,
= STA. S:d
wa(Z),
3.5.3,
At
E
V , h(A6) =
LA.
A?
. . ., C,(c), . . . by
~ ~ [ A ... C O
C,(o),
C,
S,X'C,(o),
+
A.
r k AA
La.
E
...I+
SFC,
. . ., S?'C,(n, . . . by
r
5.2.3 Definition. FAA
by
Li
V
h
=S [ d
(g(A5)).As
r, t < 6, 0 < 6 < a,
A. rlkA.
r,
A.
5.2.4 Theorem. (i) ~ F A A o r t d [ A o + A 1 ] fkAA6
PROOF:
r#4
=
r = 4,
rkAA
t < 6, 6
E <6
@,(L')(La), rFdA1.
FI-A[A Ao. . . A t . . .].
0 < 6 < a, FA [A CO.. . . .] + A . COE FA [A CO]-B A
Ca E
FA A ,
r
1,
F #4
CO
k~ [CO+ Ao]
C1
k~
by 9 1 + [Ao + A111
r.
C
Co
+ C' by VZ,
kC+Co,
C' kC+C1 t-d C + A 1 by
a
r#4 C
kd
+A t
r.
Co
+
5 < 6,
all 5 < 6 V FA + A5 5 <6 + k~ C + [A Ao. . . A t . . .I. 5.2.5 Deduction Theorem. !&@)(La), PkdA kdurA. r = 6 < 6} 0 < 6 < a,
kC + Ce
r
C1. C1 %?I. on Cc
a
Cc.
45
.] + A
td[A td
ur A .
:
(D,: < CY + 1 > rtdDv
tdUrA,
u I'.
A
< u by FAD,
D, D,E I',
I'tdD,. FI-dD,.
V2
rkdDI( 5.2.6 Equivalence Theorem.
D,
5.2.4
D, by FtdDP
of d, k [ A D,] +D, by
< Y.
p
A
= D,,, f t dA .
@a(La),
k [ A t)A ] t[Ao t)A11 + [ A 1 t)Ao]. ~ [ [ Ato)A11 A 1-41t)A211 + [Ao t)A21. t-[Ao t)A11 + [ [ ~ A ot]) [ ~ A I ] ] . t-[[Aot)4 1 A [A1 t)Ail] + [ [ A 0 + A11 t)[Ab --+ Ail]. (v) t [ A [A0 t)A b ] . . . [ A t t)A;]. . .] + [[A A o . . . A s . . . ] t)[A A b . . . A ; . . .]I.
< S}. kdUr[A Ah. . A
{As:t
. ;.
d = {[As t)A ; ] : 6 < S}, I' = tdurA; 6 <6 by . .I. FA [ A Ao. . .As. .]+ [A Ah. . . A ;. .] by
.
.
t d [ A A o . . . A s . .] t)
[A Ab . . .A ;. . . I . 5.2.7 Replacement Principle. (C,: Y < a>
< 0,
A
FAA t)EoCb. . .E,CI.
: kd [ A t)A']
.
= EoCo.
..E,
. .E,Cv. . .E,,, FA
C:]
t)
qa(Z)(La).
A = A' on
@a(Z)(La)
3.4.7.
5.2.8 Theorem.
FA
A
La
'$3a(La).
[A A o . . .An] C=
-[A0 + [. . . [A,-1+
[ - I A , ] ] .. .]I. [A Ao. . .A,]
(1, t)
C
@a(La).
46
I N F I N I T A R Y P R O P O S I T I O N A L LOGIC
L, La.
f on
L,
L,
I(@)) = ( p ) , Ao, ..., A n , .. . f"7AoI)
9,
= = [/(A01 --*
/ ( [ A 0+
/([A A o . . .An]) = [l[/(AO)
--*
*
-
.[f(An-l)+
by L,
on /(A)
/(A)
t)
+.
A
1,
La,
/(A).
L,, !#,(La).
A Ff(A) by 5.2.1, t A
FA 5.2.9 Conjunction Theorem.
A , Ao,
. . ., A t , . . .
:
t[AA Z(6,:
p
...A . . . ] c) A
< v>
< E, t-[ A 4 1 4+
kA
A...A...].
< E),
6 = Z<S,:v
v
[ A [ A A,9+cll V C S
{ A t :5
b[
€
8 = 60
€4.
< S } = {&(€) : 5 < q,
t[ A 4 1 t)[ A (iv)
c) [V
+ 61,
k[ A
&.+J
k[ A
[lA8,+J]
€
dt8i
+
41
[[ A e
+ [[
€d
v
C
t)[
4
[ A
:<do
411 €4.
PROOF: 5.2.8
v
€
5.2.2,
Avcol.
yv =
47
r ={ A t : 5 < v
C, = [ A A,+E]
< E by
E
< E.
IrCv I-F[ A C,].
by
.<8
+[
I-[ A At]
v
eia.
A C,] by V i 8
t < 6, 5 = y,.’ + 8 by 2.1.5, T‘= {C,.: Y < E } , I-T~AE tp [ A At] by
v’
8 < 8,. kC,, + A t .
< E,
2.
5 < S by
€4
I-[ A
GI+[
A A€]. t i 8
v i a
by 5.2.2,
-
k[ A [lAtll €4
5.2.6
v
€
At1
4+
[ A [ A
[4v<eA
[l~y”+Jll.
Eib.
v<e
[ A
€
[~~y.+e1111.
[-d] by A
[ V [ V AY,+J] V<8
€
V
5.3 Completeness of the Basic Formal Systems when a = w1 contradictory choice set
fox-inub
5.3.1 Completeness Lemma. p
.Z
< y, I- V A,, V
$a(.Z)(La).
$a(Z) (La)
all 0 < ,u < y
< y,> < a, y, ,< y
y
k [ ~ [V Aov]] v
4a
INFINITARY PROPOSITIONAL LOGIC
PROOF: A
A A
A y
A.
bound co
< y < a.
of
A ([lAo]Ao>;
[lAo], [A0 + A l l ,
<[A0--* AlIAO>, <[A0+ A l l [ l A l I > , ([l[AO + A111 [1AoIA1>; [ A A o . . . A S . ..I, [A Ao. . .Ac. . .], [ d o ] , . . ., [-&I, ..
.
y,
<[1[A A o . . . A S . ..]]A,> by
A,.
,u, 0
yo = 1
1.1,
y
< ,u < y .
( A p , :v A00
< yp>
= [4].
(Apv: ,u < y , v
< yp).
fi
[+I], 4.1.4.
[-A]
F[V Ape. . . A p y .. .]
pa(La). A
A. 0 < ,u < y ,
p
[+lo], [V A,o. . .A,,-1]
[Ao+A1],
yp 5.2.8,
5.2.2. [A A o . . . A S .. .], A ; = [V A,o.. . A p , . . .] A ; = [V [A A o . . . A S . ..][1A,(o)I.. . [ + l v ( ~ ) l . . .I { A t : E < S} = { A , g ) :5 < S'}, A ; = [V [-[A Ao. . . A S .. .]]A,]. A:; 5.2.9. k[A A o . . . A e . . .] - + A y by V2. !-A; by [ A + A'] t)[V [lA]A']. F [l[V [4]]] )Pa(Z)(La) by
FA.
pml(Lml) co-
49
[37].
[33].
5.3.2 Theorem. @wl(Lml) PROOF : (An, : n < w , v < yn> LwI
0
0
< yn
<w
< n < w.
n
< w,
Ano. . . A n v . . .] Aoo.. . A o v . . .I].
A consistent
6
[A A Ao...At.. [ A A At] t[A + [+it]] I- [A+ [A .. [A A
A
6,
. .] by5.2.4,
Ao. . . A t . . .]I.
Ao,.. .] A
vi A,,] A n + l , o . . .An+l,,.. .I], vn+l
all t,
At]]
Ao. . . A t . . .]]I
t- [A+
Aoo. . .Aov. . .I]. Aoo. . . vo AOvo all i < n [AovoA . . . A;. I-A; t)[ A ; A
< yt
< yn+l
[A;
T = {An,,,:n < w}.
A
5.4 Completeness of the Basic Formal Systems with Chang's Distributive Laws of
will a
> w1. A
5.1.3, [
v
Ir
17,. PROOF:
y'
V
17,
[ A APVll
La
a
@a(L!y)(La).
(A,,:
,u < y', v
< y')
La
50
INFINITARY PROPOSITIONAL LOGIC
A,,
= A,o
p
k [ A A,,,,]
t)[
A A,,]
< y’. y by
< p < y.
y’
p
5.2.9
v
V
Ao,,
y by
< y’, y‘ < v < y .
t[ v [ A A,vll P
IIp
A A,rll-
t)
V
(3) k [ V [ A Awl1 P
A,,, =
5.2.9 v
%(nv)(La).
V
(3) by k [ V [ A Apv]] p
5.4.2 Lemma.
y
(Pa(ny)(Lo).
V
< a, y, < y < y , v < y,) t[ v A A,vl1
p
< y,
(A,,: p
P
V
@a(ny)(La).
y
by 5.4.3 Theorem.
y
(PY+(I7,)(L,+) 5.3.1,
Ly+ 1 V A,,
y’ 0
< yf,
y,
< p < y’
< y‘
,u < y‘, v < y,> < y’,
V
,u
< y’, v < y,) 5.4.2
I-[ V C,],
/A
k [ 1 [ V Ao,]]
V
vy+(ny)(Ly+). <[-&,]:
(A,,:
C, = [ A [TALIPI].
P
V
I-[lC,]
0
< p < y’.
5.4.1,
51
BASIC FORMAL SYSTEMS WITH ORDINARY DISTRIBUTIVE
I-[
A
[lC,]] by
l
5.2.9 b[
*
l
[&I1
+
"v
P
-+COI.
I-CO.
( l ) , (2),
k [ l [ v AOVII. V
5.4.4 Theorem. ?&(U
a
{nY: y < .))(La)
an
PROOF:
If
wp,(L,)
on Ly, y
> o,
L,
< a, 5.4.3. 5.4.4
5.4.3
a
on [l].
on
[37] 6.4.
5.5
The Basic Formal Systems with Ordinary Distributive Laws y
a
a
2
By
y)+.
w3a(17,,xpy)(L,). DY !&(By)(La).
(gc: 5 < 2
y)
L,
5 <2
y,
yy
5.1.2.
(1) t[
v
A e
P
9Jy
N
v
k[l[
Arge(/iJ1
A
C qZ e x p y p < y
I-[ V [ A @
Wa(La).
[-4p,(r)lll -P [4 A
"
[lA,Vlll.
P
V,(9Jy)(L,)
VCY
[ - n A ] by A .
on (1)
52
I N F I N I T A R Y PROPOSITIONAL LOGIC
5.5.1 Lemma.
a 2
a
y)+,
17,
@a(gy)(La).
= { f ~E :< Y O
yi;‘ =
y,
yo, y l
yl},
k[ A [ v
v
+[
,
V
5.5.2 Theorem.
a
Siyoexpyl
[ A Au,c(,,ll PCYO
@a(gy)(La). @a({9y:
y
< a))(La) 5.4.4
5.5.1.
5.5.3 Remark. yn, n
y
@,+([
gY 2
< w,
y
< a.
yn
A gYn])(Ly+) n
6.5. a =w
Q
= w1
by do
WDLI((gy: < a})
!#(2expy)+(9Y)
7.1 y
[42].
5.5.4 Theorem. La
y
a
2
y)+, y
V a ( 9 y )(La)
PROOF:
P
= { # A : il < u}
by
P s’(#A)
s(#l) = 1,
= fiA
s’(#A)
<
u y. = [A s ‘ ( # ~ ) . .s‘(#A). = [+A] s(#n) = 0.
.
C,
t
+A ]
A.
. .],
53
BASIC FORMAL SYSTEMS WITH ORDINARY DISTRIBUTIVE LAWS
d
=
{ A: A
t [ C , +A ]
A k[C,
[-A]
s
-+ [ - A ] ]
A
'@&(La)}.
A A
t [ C , -+ A ]
A
ItrA. BF
s
'@&(La).
{st:
r trA.
E <2
all
T},S2 = (s:
s
=
(1) tC, -+ A
C
E
tC8 + [4].
T
[4,]'@&(La),
A
I
u
s
E SZ.
[fiA
5.5.1,
9,, (4)
T).
s E S1.
@&(La),
E S2,
(3)
=
u}.
v
€<2 exp a
S2 = #,
by (I), 5.2.7
t[
C,J-+A.
V €<2 exp a
tA
by (4)
S1 = #,
by br[-[
#
c,,].
by (4),
krA [A0 -+ A ]
v
€<2expa
4
SZ # 4,
S1 = {sp(t):E
I&-[
-+
< y'},
S2 =
54
by (4)
5.2.9, I-F[
V CBp(J.
krA.
E
5.5.5 Theorem. If a
L, $,({gY: y < a})
a
L,.
PROOF:
L,
y
$&2,,)(L,)
CHAPTER 6
R E P R E S E N T A T I O N THEORY FOR BOOLEAN ALGEBRAS
11
a
a-
by p a7
by p
6.1 Boolean Algebraic Equations Corresponding to a-Propositional Schemes --f
by u + b = i~ v b.
7
+, A , 7
by ,l
1,
+, A.
'23
+
v>
s*T
'$3,
'23
4.3
'23, .d
=
p
A
s
'$3
Td Td = 1
'23.
'$3
'23.
56
REPRESENTATION T H E O R Y FOR BOOLEAN ALGEBRAS
‘$3, 3
8
7
8
gY La
y)+,
(7,
8”
d “d La
a2
8, 5.1.4. 6.1.1 Theorem.
Z
La
7
8 Td
=
d E Z.
1
8
s
b E B, s*A 2 b. PROOF: s*(A) 2 b
8,
La by
@a(Z)(La)
krA
{s*C: C E
r}
bound
2‘3 by
r
A.
6.2 Formally Consistent Sets of Formulas formally consistent A
va(Z)(La)
tr[p
k[p A [+]] sistent.
r
r
A
trA.
[+]I,
p
@a(Z)(La),
no incon-
t r [ p A [+]I,
r‘
b[pA
[+]I.
@a(Z)(La)
I’ A
‘$3a(Z)(La)
{A} A Z be @a(Z)(La)
r
r
k[lA].
r
r { A c :5
Ik [A Ao.. . A t . . .] + [ p
A [+]], @a(Z)(La) wa(Z)(La)
[p A [lp]].
1“
< S}, 0 < 6 <
k [A A o . . . A t . . .] +
57
5.2,
5, 6.2.1
@,(Z)
La
(La),
r a
r
A
I-rA
E
r',
r u r' A,A
@a(Z) (La)
A
r
@a(Z)(La) @a(Z)(La)
r
<6
r
r
I' u { A )
Y
F
r
ru
ru
T u {[V A o . . . A t . . .]}
FU{[V A o . . . A t . . .I, A,}
6.3 Algebras of Equivalence Classes of Formulas Modulo a Formally Consistent Set
r
qa(Z)(La).
5.2.6,
A = A'
I-rA t)A'
on
r
IA[r =
{ A ' : A = A'}. 7
on
B ( l a ( L ' ) ( L a ); r)
: =
IAolr + = I[Ao + A (IAol.. . [ A { l . ..>= l[A A o . . . A t . . .]IT.
by A
v
= I[Ao A = I[Ao v
58
REPRESENTATION THEORY FOR BOOLEAN ALGEBRAS
on 'B(@a(Z)(La); r)=
IAIr
I[$
A
< IA'Ir
I-rA + A',
I[$
[+]lr,
v [+]lr.
< lAIlr Y < 6, t-r[C + A t ] t-r[C + [ A A o . . . A t . . .I], l[A A o . . . A t . . . ] I F I[V A o . . . A t . . . ] I T . @a(C)(L,) t-rA I'lkA t)A'.
I[AAo.. . A t . . .]lr
E <6
IAolr,
. . ., IA&, . . .
A'
t)
B(@a(Z)(La);
'%(La;I')
I')
4.2.
4.2.1
6.3.1 Theorem.
@a(Z)(La)
I'
B(Wa(Z)(La); I')
7 = { A : It-rA),
; F)
'%(@a(Z)(La)
7 'B(Wa(C)(La)) = S(@a(Z)(La); 4)
Td
7 E C.
=1
7
23 La
'23.
B(Wa(Z)(La)) 7
@(@&(La))
[33]
I'oC I'1 h(lAI r , )
=
\ AIr,
7
I'1
on
23(Wa(Z)(La);Fo) 'B(Wa(Z)(La); r1)6.3.2 Theorem. I' @a(Z)(La), T, =1 B(@a(Z)(La) ; I') E C. EC
8 = B(@a(Z)(La);I').
s
8.
s*& = 1
59
PROOFS
Z.
lA(r
s
g on s*
A,,
At
V IS,V
A
s(A6) = Ig(A6)
La
7
s * d = ISFd1,
d.
d
E Z,
d
Srd
s*d =
ISrdlr = 1. 6.3.3 Theorem.
T,
CC
r=
b
7
$' 3.
8 by
La {A: A
=
all d
1
E 2.
La $c
cE
s($,) = s*A = l}.
cE
r
8
r).
93
PROOF: b, r s*A
6.1.1,
[ A t)A'] E r, h IAIr s*A r). lAlr = (A'(r by 6.1.1 s*A = s*A'.
'$3,
=
IAIr = IA'Ir.
= s*A'
Fr[A t)A'], h )A)r
A
k
b
1#0
s*A = 1.
FrA,
C
s*
both
7
r)
b(@a(Z)(La); IAIr IAId
b
b(@a(Z)(La)). 6.4 Metamathematical Proofs of Some Boolean Representation Theorems La b('@a(Z)(La);r) s($) = [<$>lr A s*A = IA lr.
r
Td
=
d
1
EZ
r
by 6.3.2.
f by 6.1.1. 6.4.1 Theorem.
La
L'
60
REPRESENTATION THEORY FOR BOOLEAN ALGEBRAS
r r
:
@,(Z)(L,).
Td
=
E Z.
1
23 Td = 1 $'3 {s*A : A E r} 6.4.2 Theorem. Z
all&
E
Z,
bound. y
:
@,(Z)(La) L, y
L,
y %(@a(Z)(La)) y
Td
=
r = 4.
by 6.3.1 by
all&~C,
1
L,
6.3.3.
y
!B())Oa(Z)(La))
by
{/<#)I: # Td
E Z,
=1
by
8())O,(C)(La))
ll-A, by s*A = IAl
s(#) =
y,
IkA
t-A
A 4.3.4. = 1.
FA. !&(Z)
6.4.2 7
Td
=1
5.3,
&EC 5.4
5.5 6.4.3 Loomis Re#resentation Theorem.
PROOF: 5.3.2
6.4.2. 5.4.3
. 6.4.4 Theorem.
y
[ 13
61
:
v
(bfiV:,u < y ,
B
A b p = 1.
V IrCY
V
l7,
'$3
'xf
'$3.
V P
V P
< y}
'$3
V
h A b,,
V
6.4.2.
on 0
A b,,
'$30 ,u < y
V
V
{b,f(,,: p
by 5.4.3 A bBP.# 1,
A bbV.
P i Y
V
f
V 4(b,,) = 1
A
'$30,
V
< y.
-ih(b,fo())= 1 p (b,,: u , < y, v < y )
E y,
5.4.4 6.4.5 Theorem.
a
6.4.6 Theorem.
a
no
5.5.2
7
< a. y
( y , y)-
< a.
6.4.2 6.4.7 Theorem.
y
a
Z :
!JL(.Z)(La)
La
y y pa-
d E Z,
Td = l '$3
Td
y
'$3(&@J(La) ; F ) F 6.3.3. p = { A : IkrA}, I-rA @(@&)(La) ; F ) = '$3(La;F )
=1
'$3
E
by 4.2.1.
F
IkrA
FlkA.
7
r
62
2
'@a(Z)(La)
%('@,(Z)(L,) ; F ) ,
F
@>IT,
by y
Td
=
d
1
E Z.
y
I
y.
4.3.4,
F 5.5.5
6.4.8 Theorem. a
:
'23 '23
(7,
all y
<
6.5 Algebraic Proof of a Completeness Theorem y = U {yn: n
5.5.3.
all n
< w.
?@,+([
A
< w},
2
.9J)
yn
6.4.2,
n
yn-
n
(C,) 6.5.1 Lemma.
y
b#0
(Ci)
V (b,,: v
C
y-
< y> = 1
(b,,: p
p
< y,
< y, v < y )
C u {b} (CJ
(C,).
(b,,: p
< y , v < y>
v < y> = 1 p < y. (Tb,,: v < y> p < y, < y b . . .7b.
y-
b#0
(C,),
. .>
y.
C
ALGEBRAIC PROOF
A COMPLETENESS THEOREM
V
A bLv= 1 =
C
~
b
V
b = 0,
by (Cy).
63
(Ci)
< y, v < y)
(b,,: y
A b@v # 1.
7 b= V P
V
b # 0. y: (1
.
A bc(y y b . . ~ b. ..),
V
p
<
A b#v Ibfio. . .,b,v.
. .)
V
< y.
1.
C' u {b}
C'
may
6.5.2 Theorem.
y = U {yn: n y.
n
PROOF: b # 0 V (b,,: v
< yn+l
yn
vE set for O
n < W. n C
< w,
p
< y.
< w},
yn
<w
(b,,: p
y
=1
On = y n
0C y ,
-
< y.
u{b) U yf, b:,, = V
will
p
y E 0. (1)
choice
C
b' # 0, n b'
< w, A
f on 0, w A (bLf(C,:p E 0,)# 0.
1 = V (b,,: v < y ) = V (bLn: n < w ) b' = b' A A
all p
< y. f
64
b’ # 0, n < w ,m < o b’ 0 # b’ = b‘
A
0 C Om, b’
< A (bLn :p E 0>,
0
C
A (V
(ym, fo on OO
A C # 0.
A
E
On): p
C
E O>.
b A A
o
00 0,
b
@om = { p : p E 00
do
b A A (bLfo(p): p E OO>< A (b;,: p E OOO), 000 b A A
fo(p) = m}.
CO by c k
b A A A
(Pk)
i
( 0 ~+ :j
Ck
i
Ou = b:p E Oi
= k},
/&A) = i}. fn+l, Cn+l (Pn+l). bn = b A A A
on On+l
i
w
bn A A
p E On+l> = bA+l # 0. Cn+l {OU:i i = n 1) = 00, n+l u 01,n U . . . u 02, n+l-i u . . . V On+l,o. bA+l < A
+
+
+
Cn+1 = U {Ci+l: i
j
+
1
Cn+1 C n by
< w}.
C b
A
A A
Cg # 0
n
idn
6.5.3 Theorem.
I,+([ A gYJ)
y =
{yn: n
n <(u
6.5.2
6.4.2.
< w}
2
yn
< o.
< y.
CHAPTER 7
NON-DED UCIBILITY I N INFINITARY PROPOSITIONAL LOGIC
to
7.1 Summary of Results and Open Problems Z, 27
a @a(Z) M
@&‘) La, @a(C)(La)
@a(Z’)
(La)
Lor, !#a(Z’)(La).
@a(Z) (La)
< @a(Z’)
@a(Z)
@a(Z) @a(Z’) @a(Z)
M @a(Z’).
7.1.1 Summary. ( P 4 - J (177:Y y+
@,(lily)
y @a.
2
@&(lily)
< @a(17y+)
y
yn
< @a(lily+)
< a))
< a,
y
y = w ,@ a ( U y ) < o,
yn, n
(178: < y } ) NN @ a ( U y ) .
all
y
M
BI
66 N 0 N-D E D
y , fpa(U (178:/?
< y})
N T A R Y P R O P O S IT1 0 N A L LOG I C
TY IN
< !&(fly)
y
2
y
< a,
2
y
< a,
2
y
< a,
2
y
< a,
rPa(Ps: /? < r})< W P a ( 9 Y ) . y
y
& ( ( 9 s : /? < y } ) Wa(17,)
< &(By)
< &(gY).
y
y
< CL
y
<
$3a(9y) @a(172,,,,). a
>
by
la,
a = y+ yn y.
y
<
2
y,
a
d
a
y
a2
d
y-
pa(&)< wa(ny) <
!&&Z)
!&(fly+).
7.1.2 Algebraic Summary. :
7.1.1
y y y
w
y,, n
n
yn
< y, fly-
y
/? < y
2
<
y,
y
< w,
SUMMARY
RESULTS A N D OPEN PROBLEMS
67
y. y,
y-
4.2.4 do 7.1.2 6
By :
y
B,, all @
B,,
(@,
1< y
all
K.
[ 161.
(@,
all @
y+ = 2
K,
y,
(y+)-
7.2 no by
4.1.2
7 y
10.2. y, y =o
on
68
7.2 Examples of Complete y-Representable, not y+-Representable Boolean Algebras y
y-
By V
A
(1)Y
<
Xv,
V
V (X,,
vfv’cy
P
v
A Xvcp).
P
By X
on y
all
y+
X = {f
:f E X
on {S(g$):i E I }
g
g= y,
y
tl {S(gt): i E
# C$
U
S(g),
:
E
y
: i E I> < y , fl {S(gt): i E I> g = U {gt: i E I } .
U
y+.
y
y
U {gt: i E
gr u gt,
i, i’ E I , :
7.2.1 Lemma.
C
y
no fl C
x -S(d
=
x -fl{wv)}): (P.,v)Eg}= u{X-S({(PI41):
x
-
(p, v)
Ey
x
(IUP)Eg}I
yf,
4))= u {S({(p,v’)}) : v
S({(pU,
# v’
< r+}.
By on
all VE Sc =
4, 7.2.2 Lemma. A {Sf: 5 < 8) By.
Uc Sc, AE Sc =
X.
SCE By,
tl, Se.
n {Sf: t < /?} =
69
COMPLETE y-REPRESENTABLE BOOLEAN ALGEBRAS
4,
< y,
B by
B 7.2.1.
7.2.3 Theorem. $'3, vE
S(g) C S({(p,v)}) v I#
p Ey
-
y. p, g U {(p,v)} a S(g) n S({(p,v)}) # 4. {S({(p,v)}) : p < y}, By. S({(p,v)}) n
V S({(p,v ) } ) = X
A V
S(g)
p = g-l(v).
P
< y+
v # v'
S({(p,v')})
p
< y.
B,
8, (y,
y(y, 2
[40] U {S,,: v
[26].
S,, =
=X
p
23,.
X =
< y+}
h
v)})
< y. y
p
< y , v < y+.
fl {Srch(,) : p
y+,
A {Srch@) :p
4.
(y,
7.2.4 Theorem. B, all j? < y , PROOF : 6.5.1. E B,
(B, (S,,: p
< y. Cu VA
8,.
K.
B, p
< y, v < y> S #4
(Cl) V {S,,: v
< y} =X C
S(g) C
S(g,) :5
S(g) n fl {SBy:v < y ) = X , n fl {S(gs): 5 < p} n S,, # 4.
)3.
p
< y,
Slvr. p.
< p> vB
70
N 0 N-DE D UC I B I L I T Y I N I N F I N I T A R Y P RO P 0 s I T I 0 N A L LOG I C
S(g,). = {Spvu: ,u < y}.
(v,: ,u
C u { S}
By(/?,
(TNV: ,u <
[36]. V {T,,: v
T #4 A C # 4.
< K}
,u < /?,
=
A
C, fl C = 4. U {fl
(8,
C
= fl p<@
- < 8,
U T,, = 4.
Q<X
,u
7.2.2
U (T r
By
U T,,) = T
v
# 4, T E By.
-
n p
T
C
-
v
=4
By,
U T,, (I
T. U TBv v
T
=
T.
CHAPTER 8
S Y S T E M S OF FORMULAS O F I N F I N I T E LENGTH
1,
on
a, 0 ,
on /?, z,
on
on [, 1,
0
< < 0.
0
< q < n. o
<6
irtdividuul symbols.
Expressions
72
FORMULAS
a
u /? u o u n. As
“ [ @ A o . . A t . . .I” ( [ @ ) - ( - < A t : 5 < &)^
0,
of Formulas
+Systems
3. Atomic formulas :
To, T I
[TOPTI], (2’) [QqTo.. .T t . . .I, Qq
(3’) [QTo. . .T E ..
Q
.I,
P
( T t : 6 < 7) 0
< 7 < x, ( T t :5 < 7) A
formulas
(a,/?)-
:
Ao, A1 E A < A t :5
<
!P [A0 !PA11 A .
A @* [ W A o . . . A t . . .] E A . A , 0 < 6 < a,
( A t :6<
(3) @
[ @ A o . .. A ( . . .] E A . (4) A EA, x ZI 0< < ,8, [x z, A ] E A . 8.1.1 Induction Principle for Formulas. A (2), ( 3 ) , (4)
d
(a,
F&$on
algebra of formulas
FaDon,
7aby @
(2),
[ @ A o . .. A t . . .] by
< A t :6 < 6>.
73
(x, v ) , v ,9, A
8.2
[zvA].
Systems for the Interpretation of Formulas
1,
8.2.4, 8.2.5, 8.2.6. 12 8.2.1 Definition. A system for the interpretation (D 0 RB S) D, B
( B 0’)
D R
on on
B on D
fn-
B on
S assignments).
(x, v ) v
,9, B 8.2.2 Recursion Principle for Formulas.
sES
by s* (D 0)
on
B.
(D 0 s
RB
S)
on on
s*((x)) = s(x) V on :
x.
x Faoon
B
74
SYSTEMS O F FORMULAS OF I N F I N I T E LENGTH
q-
T, QQ,
Q,
V(s, = V ( S ,[QQTo.. .Tt.. .I) = . .s*(Tp). . .)) V ( S ,[ Q T o . . .Tp...]) = R(Q)(<s*(To).. .s*(Tt).. .>). h(A) = V(s,A ) (B
SES,
B,
x,
A,
v
V(s,[ x v A ] )= Q(x, v){V(sv,t , A ) : t E DRngcv) sv, t E S } sv,t = t on s V valuation f w c t i o n , B V 8.2.3 Example. LaDon
on
FaDon
+, A,
1,
V.
on Bo O'(A)
7
R Bo 0' Q S ) D , Bo
0, 1, O'(+) lx v y, O'(+ Q ( V ,v )
on a model
D
0,
B:
v
Ap
(1) V ( s , [A0 + A l l ) = 1
V(s,Ao) = 0 V(s, =1 (3) V(s, [A A o . . .At...I) = 1 E < 6. (4) V (s,[VvAo]) = 1 t DRng(").
V ( s ,A l ) = 1. V ( s ,Ao) = 0. V(s, Ap) = 1 Ao) = 1
all
75
8.2.4 Example.
(D 0 C R B
A
0‘
on
S
8.2.3. S), D, B
O’(+), O’(A),
8.2.3 on B
v)
on
Bo.
12.2 8.2.5 Example. 8.2.3, Labon. t) A ,
v,
V,
3. Fabon
FiBon
As
(F,
( F , 0 C R FaSon 0’
c, S = {s}
C(c) = (c) x, R
s(x) = ( x )
H ,
on F, [QwTo.. . T t . . (ToT I ) (Te :E
[QTo.. .T t . . .],
< 17).
O’(l)(
[A
O’(V)(
-
Fabon
V , 3,by
-,,+, A, .
t),A ,
v,
76
SYSTEMS
FORMULAS OF I N F I N I T E L E N G T H
8.2.6 Example.
on on p:
p(A0) = 0 A0 p([Ao!PA1]) = (p(Ao) ~ p ( A 1 ) ) 1,
+
p([@Ao... A t . .
!P
.I) = ( U ~ ( A F+) ) 1
@
E
p ( [ X v A o ] ) = p(Ao)
(F, 0 R a 8.2.5, R
+ 1,
S>,
x S
0,
0,
+ 1, O’(@)( < P O . . . p ~ .. .>) = U pc + 1, €
O’(!P)(<popl>) = (po u p1)
Q(x, v){p}
=p
An
+1
P ! @
x p ( A ) = V(s,A )
on
a.
8.3 Recursion Principle As
on
no
8.3.1 Lemma.
A A: Ao, A1 A A 6, 0
=
=
[ Ao !PA11.
[QdAo. . . A t . . .I.
< 6 < a,
@pa
77
A
@
[ @ A o .. . A t . . .].
E,
x,
0
< E < ,!?,
=
v
A = hvAo].
A0
( a , p)-
8.2. [,
A 8.3.2 Lemma.
no A = Eo^C^E1 Eo = E l = 4
C
A
= C.
PROOF: C A
[TOPTI].
C [Ti, Ti] ([>^To= Eo^([>^Ti,. = T;^(]>^E1 3.3.2
TI^(]> Ti
=
Ti
El = 4. Eo # [, To = Eb^<[)^Tb,
Eo = 4 A = [QTo. . .T t . . .], Q C C = [QTb.. .T ; . . .] Eo = 4. A -(Tl: 6 < q)^<]) = ^ ( T i : 6 < q’>^<]>^E1. To = Ti,.
4,
3.4.3. A
by
Tg = T ; q = q’, El = 4 q < q’ q’ < q 1. A = C. 8.3.3 Definition. A subformula A A proper A. przncifial terms [TOPTI] To, T i , [QTo.. .Tg. . .] To, . . ., Tg, . . . . 8.3.4 Lemma. 8.1 [TOPTI] = [TbP’Ti] To = Tb, P = P’, T1 = T i .
5 < q n q‘.
.
[Q To. . .Tt. ..] = [Q Ti,. . .Ti. .]
78
SYSTEMS
Tt
FORMULAS
= Ti.
A
no
A(1) 8.1 ( l ’ ) , (2’), 3.3.3. 8.3.5 Lemma. by 8.3.6 Lemma. A
< [. . . [ A ,
E
1
k
k < o,
E d
[.
no
A
. .[ A
< K < o.
1
k
A E
<
[TOPTI],
< [. . . [ A k
[Ti,PTi].
E
<[>“Ti, =
<[.. . [ > T o . k+ 1
3.4.3. E
[QTo.. .T t . . .],
A is
< [. . . [ A k
A
E(l) A. Ao, . . ., A t , [ @ A o .. . A t . . .I,
...
A
[. . .[ A
no
k
no
A E A. A0 E A A = [XvAo] A = [A0 !PA1], E [. . . [ A E E A0
<
< [. ..[Ao,
4A.
8.3.8 Lemma.
A by 8.3.2 A
A
E
E
< A,
E
E
< A,
E
8.3.6.
A = {A:A
A
< E’
Y.
no
8.1.1, A
E
E‘
A.
=
@. Ao, A1 E A
k
E
k+ 1
8.3.7 Corollary.
E A.
A
E
79
A by 8.3.2
8.3.7.
A by
8.1
3.3.2.
E
8.3.7
A
0 <E
(4)
A0 ELI, x A = [XvAo].
< p,
no E = [xvEo], Eo
A
x
Eo # 4. 8.3.1, E [ ~ v ’ A b A6 ], ([x) on on v, v’ v = v’. A0 E A. A = [ X V A O ] E’ [Xv’Ab]. v = v’,
<
(a,
v 2.2.19
< Ao, E
=
v‘ E, no
A6 E’
< Eo < Ao,
E‘ Ao, A6 A0 E A ,
A0 = Ab. 8.3.9 Definition.
Ao, A
A = E’
A E A. [A0 !PA11 [ Q A o . . . A € .. .], [XvAo] Ao.
. . ., A t , . . .
8.3.10 Corollary.
3.3.2 ( A t :5
< 6)
< A ; :E
(1) Ao^(u>^A1 = Ab^
v”A0
= v’^Ab
v
= v’
< Sf> v , v’
u, u‘,
A0
= Ah, u = u‘, A1 = A i .
S = 6‘
At = A ;
A0 = Ab.
8.3.11 Proof of Recarsion Princifile 8.2.2. by
V 3.3.3
8.3.10
80
SYSTEMS
FORMULAS OF I N F I N I T E L E N G T H
on S x Faaon V b E B. ((s, A ) , 4 .
RC
At,
B,
8.2.2 ((s, A ) , b),
A
E
“ ( s , A , b)”
x Fagoz x B acceptable v
Tc, /I,
E
[ T O P T I ]R(P)(<s*(To)s*(Tl)>)) , E R, P. [QqTo.. .T e . . .I, R(QQ)((s*(To). . .s*(Tg). . .))) E R, Qn, (s, [QTo.. .Tg.. R(Q)(<s*(To). . .s*(Tc).. .))) E R, Q, x. (s, Ao, bo) R O’(!P)(
A1, b l ) E R
(s, [Ao!PA1],
5 <6
[@dAo.. . A t . .
@d.
5 <6
(s, Ag, bg) E R
O’(@)((bo... b t . . .))) E R,
0
[ @ A o . . A t . . .],
< 6 < a,
@. (sv, t , Ao, bt) E R
t E DRng@)
[ ~ v A o ]Q(x, , v){bt: t E
x
sv,t
sv,t
E
E
E R,
S x FaDon x B
Ro, Ro
Ro S x Fagon. = {A: A b EB A , b) E Ro}. A bEB [QwTo...Tc.. .] b# RO A , b)).
A
N
A
8.2.2 E
A , b) E Ro. A . .s*(Tc).. .)), 8.3.4 A , b) $ Ro. d
81
by
(a,
A
3.3.4,
[xvAo] A0 E A . bt (bt: t E DRng(v) s w, t E b # Q(x,v ) ( X ) ,
t DRng(v)
s B
=
S, (sW,t, Ao, bt) Ro. X = by (s, A , Q(x,v ) ( X ) )E Ro. Rb = Ro ((s,A , b)). Ri, Rb as s’ E S A & bi,) E Rb all t’ E DRng(“) sit,r Rb C Ro, (s’, h’v’Ab], Q(x’,v ’ ) ( X ‘ ) ) Ro, X‘ = { b i f :t’ DRng(”’) s : . , ~ Rb, (s, A , b). s = s’, x e ‘ , v = zl’, A0 = Ah, Ao~d, X = X‘. b = Q(x,v ) ( X ) , b. Rb. Rb (s, A , b) Ro. A EA. sy,t
N
+
8.4
Replacement of Equivalent Parts
=
coitgruence relation on formulas on
A
/I.
[xvA] = [ ~ V A ’ ]
A‘
x
A0 Ab A4 = A ; all 5
v
A1
<6
=Ai
[Ao!P’A1] = [ A b Y A i ] [ @ A o .. . A c . . .] = [ @ A h . .. A ; . . .] Y, @.
8.4.1 Example.
A = A’
V ( s ,A ) = V(s,A ’ )
8.2.6,
p
A = A’ on 8.4.2 Example.
sES
p(A) = p(A‘)
K
82
8.2.3,
Lafloa
A = A’ all
V(s,A ) = V(s,A ’ )
s
K on 11
A = A’
FA
on
A’
t)
Lagcn.
Cp by A . As 8.4.3 Lemma.
4 < El
1
C,
A,
Cv, 3.4 2.3
C;
E l # 4,
Eo^El
A
A
< A , El
A [TOPTI], E l = <[>^Ei,E i
El by
El
< To.
3.4.3. A [QTo. . .T E .. .]
A.
A o ~ dA , lcd, El
A on
A0
=
[AoYAl], E l = <[>^Ei, E i El
A
(K,
8.4.4 Lemma.
Eo # rp
Eo^E1
4 <. E l
2.3 A = Eo^E1 Eo = $}. A
(C)
C
El
< Ao.
A = { A: A C El Eo = 4.
8.4.3
d
by 3.3.2
4<
8.3.8
A.
8.3.8
83
REPLACEMENT O F EQUIVALENT PARTS
<
Ao E A , A = [XvAo] = Eo"E1, 4 < El C C. [, Eo = 4 Eo = [XvEi, 4 Ei, < Ao. Eo on A = Eo^E1, E l = (A0 - Ei,)^<]> C. on A0 Ei, = 4. A0 < El C, 8.3.8. Eo = 4, A E A. 8.4.5 Replacement Principle. = on A = ^<E,"C,: v < u>^E,,, C, C , = C: v < u, A' = ^<Ev^Cv': v < a>^Ea A E A'. PROOF: u = 0, u#0 Eo = 4, C o = A , u = l , E a = 4 , by 8.3.8. u#O Ea=4, u4 4 A = Ca-l, u = 1, Eo = 4, b y 8.4.4. Eo # 4 E, # 4 A u = 1, Eo = E,, = 4 by 8.3.2 8.3.5. by Ao, A1 A = [Ao!PA1]. A by 3.4.7, 2.3 A = [ @ A o .. .At. . .] A0 A = [XvAo] = ^<E,"C,: v < u>^Eu Eo # 4, E , # 4. CO [, Eo [XvEb, E , EL]. Ei = E, 0 < v < u. A0 = ^<E:"C,: Y < a) "Ei. A0 = Ai, = ^<Ei"CL: v < u)^Ei. = A = [XvAi,]. [XvAb] = ^<E,."C: : v < u)^E,. A' A = A'.
<
<
<
8.4.6 Theorem.
A A.
CHAPTER 9
SUBSTITUTION
(a,j3, o, n)-
8,
8.1. 3,
S$T
<x>,x
3.5, ,
EX
f (x).
by 3.5.2,
bound
9.1 Free and Bound Occurrences of Variables in Formulas L
L
<
x
occurrence
E ( L )= x. position in
< X n
+
L
< bound by X in
I
<
=
A # I$.
x
L
bound by {x}
bound in L
x
free in no E.
Ao, . . ., A t , Y,
9.1.1 Lemma. 0,
. . ., X,
v
F R E E A N D B O U ND OCCURRENCES O F VARIA B L E S I N FORMULAS
85
X
@, : =
1
+
bound by X by X
.
=
bound by X bound by X 2
+
by X
+
+
. .], 2
=
bound by X
L
bound by X 1
L
+1+ 1 < 6,
+
L
bound by X
L
+
bound
L
5 < 1) + L
L bound by X bound by X Xn # 4.
L
L
L
2+ bound
L
bound
by X bound by X . 8. I’ = 2
+
L
by X
.
t < A)
.] =
Xn
+
L
bound =
< L’ <
# 4.
2,
8.4.4
2.3.
= 1,
2.3.3
A’ < 6 (Eo 1 < A’,
5 < A‘))
<2 +
t < 1 + 1) on
+ =2 L’
<
+
L‘.
<2 +
<2 +
:5
+
bound by X
< A’)
<
< A, t < 1‘ + 1) < 2 + 5 < A ) < I’
A’
<
< il)on
< Eo il= A’,
on L’.
:5
- Eon.) L
=
<
=
L’
+
+
- EoA.)
L;
86
4.
Xn A
+
= L
4
A
=
+
<
n X # 4. [,
-
bv
bound by X X n L bound by X
[ pAo] 2
+
+
A
<2 +
=
# 4,
El = 2
+
#
+
- [xv)
1
+
L
A
= [@&.
+
2
A1
x
A.
. . A t . . .],
A.
A x
=
[~vAo],
A A
2
on
+
L
+
x
A0 =
+
L,
L bound by X Ao. 9.1.2 Corollary. (i) A =[AoYA~], x A0 1 L
L
4. 2
L;
- [xv)
[~’v’Ab] A‘
A’
<
A.
x
1
+
x
L
5 < 1) + L
+
+ A1 x
+c x
L
# A,
x
A0
FV(A) A.
4
9.1.1
:
9.1.3 Corollary.
F V ( [ A o Y A 1 ] )= FV(A0) u = u FV(Ac)
(ii) F V ( [ @ A o . . A € .. .])
€
= FV(Ao) ,Flx@on
sentence
no
/?
9.1.4 Corollary.
PROOF:
d = {A:A
no
/?
d by
A
. . A t . . .]
/?
= [@Ao.
/?
. . ., A t , . . .
A€ A
A A1 E
A A}. A . Ao,
A
At, A
A
= [A0
EA.
EA.
A0 E A A0 E
A
87
IN
A =[~vAo]. A FV(A0) C F V ( A ) LJ
/?
Ao, A V(s,A)
A As (a,
EA.
A
La~on
0,
(D 0
p
B
A0
on
R)
A. :
9.1.5 Theorem. of S (*) E
B 0’Q S )
X
p,
ES
on A , V(s,A ) = V(s’,A ) . A = {A: A on A A.
E
E
V(S,
=
A}.
V(s,A ) = V(s’,A ) A
R(P)(<s*(To)s*(Ti)>) = = VS’,
by
3.5.5
A A.
of
Ao E A , A1 E A . on V(s,Ao) = V(s’,Ao) O‘(!P)((V(s,Ao) V ( s ,A I ) > )= V(s’,A ) . [ Q A o . . . A t . . .] @
A, by 9.1.3. V(s,A ) = A EA. A =
on
A =
Ao,
A0 E A ,
A
=
hv A o ] .
s ~ =, ~ V(s,A ) = Q(~,v){V(s,,t,Ao): V(s‘,A)= Q(x,v){V(s;,,,Ao) t E DRng@), t s;,~ on FV(A0) FV(A)u (*), V(s,A ) = V(s’,A ) . A EA. (*) E S,
t
E DRng@),
88
9.2 Substitution for Free Variables X
on X
f
E
S;"E x E X by
f ( x ). 9.2.1 Definition.
SFTE x EX
f(x),
(I,:
v
E by
< a)
X
E E = ^(EV^(E(~,)>: v < a>^Eu
L,,
=
Y
Q a,
= Elc Ev = EIL, - El U
Lg
v
e+
< a>^Eu.
S 7 E = ^(E,^f(E(c)): v 3.5.3
3.5.1
A
9.2.2 Theorem. =
SF[QTo. ..Tt.. .] = [QSTTo...SFTe ...I. SFF f(x)
x
9.2.3 Definition.
EX
( i V :v
< a)
X
E
.
E = ^<Evn(E(~v)): v < a)^Eu.
S F 7 E = ^<E,^fE(s) : v
< a>^E,.
E fiosition for the free szlbstitution of X . X.
decom9.2.1 E=^<E,'^C,: v < all
89
VARIABLES
(x), x x
EX
9.2.4 Lemma.
X
EX
,
no
. v
=
<
C,
+ 1 : 8 < v) +
Y
< a’)
x EX E. = ^(EVn(E(~,)): v < o)^Ea
<
8
X. 2.2.20.
by
+
L, =
< u,
v
< u>^Eo
=
+
.cv)
:8
v
by
v
=
E,
=
<
v < a, u = u’. 5 < 6) 5 < 6) = 5 < 6).
9.2.5 Lemma.
(6,:
x
EX
v
=
< a) 5 < 6).
=
X A A=
+ 1) v< + 1)
g(A) = g(A
-
=
T2 = { ( x ) : x E X } :
2.3.3
AA
= h<Eg(~)+i+v^
<
+ 1).
+ 1))
-
2.3.4,
X g(A)
<
by
5 < A> =
(1)
5<
<
v
<
+ 1:
=
=
=u
=
Y
<
+ < 6,
90
(3) L,(A)
=
to
=
<
(L,: v
9.2.4, by
+ 1 : v < g(A)> +
I
< 6.
A < 6, 5 < A> < ~ ~ ( 1 ) . v < g(A), tI = + 1 :8 < v) + + 1 : 8 < g(A)), 9.2.4. v < g(A), I , < 5 < A> by
5 < A),
A
xE X L,(A).
<
g(A) = g(A + 1).
2.3.3
An
no
An x EX.
xEX tp
<
An, 9.1.2 A
x EX
1,.
5 < A + 1).
v
:5
<
by I
g(A), [,(A) < t,. (5), g(A) # 5 < A + 1) < i g ( n ) < ctt, g ( I ) < g(A + 1).
+ 1) +1+
g(A
-
(g(A)
xEX
+ 1)))
ti,
+
v
< g ( I + 1)
An. An
-
(g(A)
+
+
. . .,
:5
An An -
(1;
:v
<1+ = v
<
ii
A 1) by
An.
L,(A)
<
An
&,(A)
A
XEX
<
An by
+ 1 # 6.
by 9.1.2, xEX An
5 < A) XEX :5
=
+ 1 : 0 < v> +
+ 1) - (g(A) + 1).
<
An X
(g(A
< A)
(4),
i,(~+l)
x EX A v < g(A 1) - (g(A) g ( ~ by ) ( 3 ) ,9.2.3
+
. . .)
< I ) + ii) = L
xEX
5
+
ti
An.
+ 1).
91
9.2.6 Theorem. At, Y,
SFTA = STA.
A
all
@,
x,
SFT[AoYA1] = [SFfAoYSF,XA1] S q Y [ @ A o . . . A a . . .] = [@SF:&.. . S F f A 6 . . .] S F f [ x v A o ] = [xvSF?-Rng(v)A 01.
SF7E
STE
on
by
Ao, A l .
X ([), ( Y ) , (]>,
X
-<
( L ~ :Y
X A0 A0 = ^(E,^(Ao(t,)):
x EX
v
< u)~E,,.
[XvAo]
+
9.1.2,
[A0 Y A l ] . a)
+
Ao,
X 1,:
v
N
< a).
u # 0. by [xvAo] = ^ ( E ~ ^ ( A O ( L ~v )< ) : a>^Ei. = ([x>"v^Eo, EL = E,,^(]), Ei = 0 < v < u. v < a, 8 < v) =2 1 : 8 < Y) =2 tr by 9.2.4. [xvAo] X [~Ao]. SF,X[XvAo] = ^<EinfAo(&v) :Y < = [ x v S F T - ~ ~ ( "01. 'A 9.2.7 Corollary. A SFTA A = { A :A SFTA X f on X = 0,
+
+
+
+
+
+ +
+
(a,
9.2.6 9.2.8 Corollary.
A p
SFT SFTA = S f A A At. SFT on p(A) = p ( S F 7 A ) . 8.2.6.
92
SUBSTITUTION
d
A
of
X
all
p(A) = p ( S F 7 A ) .
f on X d (a,
SFT A A x EX , A
R)
f(x)
x
f ( x ) bound.
d
A = [Vy[Qyx]
x
[Vy[Qyz]]
d
[Vy[Qyy]] 9.2.9
z.
d
y
Faflon SFFA bound by FV(f(x)) all FV(E) 9.2.9 9.2.9 Lemma.
A
R B 0’Q S>.
no
x
FV(E) E.
x
E.) 9.1.1.
on X A no x bound by FV(f(x)) A x EX. A [AOYAI], At no x bound by FV(f(x)) A6 x X , i = 0, 1. A [ O A o . . . A t . . .] .$ < S, A6 x bound by FV(f(x)) A S x X. A [XvAo], A0 no bound by FV(f(x)) A0 x EX 9.2.10 Theorem.
-
no x
f on x
FOR
93
V ( s , S F T A ) = V(s', A ) on F V ( A ) A no FV(f(x)) x EX}. S F T A = S:A
s'
E
x bound by
A 9.2.2 [AOPA~]
3.5.5 A. A [ @ A o .. . A t . . .] At EA. X,f A no x bound by FV(f(x)) EX, At by 9.2.9. s' E S on F V ( A ) , on FV(A6). 6, V(s,SF;"&) = V(s',A t ) . SFT V s s' on A E A. A = [ ~ Ao] v A0 A . X,f A no x bound by FV(f(x)) x E X,9.1.2 x EX s' E S
-
Ao, FV(f(x))n on F V ( A ) . 9.2.6
x
t E DRng(').
+.
= s , , , ~=
SF;"
V(S,SF,XA) = Q(x, v){V(sv,t,SF;"-Rng(w)A~) s V , ~E S). V(s', A ) = Q(x, v){V(s;,,A o ) :s;,, E S}. on A0
-
no x bound by FV(f( x)) x E X by 9.2.9. sv, t X- Rng(v) s v , t on FV(A0). s& on FV(A0) FV(Ao) C F V ( A )u (*), s V , t S s ; , ~E S . A0
(4) V(s,, t , SFJX-Rng(v)A0 ) = V(s&,Ao)
sV,t
(3), (4), V(s,S F T A ) = V(s',A ) .
A
E
EA.
sE
S 8.2.5 9.2.10. 9.2.11 Theorem.
8.2.6.
S)
S.
94
SUBSTITUTION
Faoon. s E S, X f , f' on X s* 0f s* 0f' on FV(A) n X , V(s,S F 7 A ) = V(s,SF;A) A no x bound by F V ( f ( x ) ) by FV(f'(x)) x E X . PROOF: d A X, 1, f' s* 0f s* 0f' on F V ( A ) n X , V ( s ,S F F A ) = V(s,S F F A ) A no x bound by FV(f(x)) by FV(f'(x)) x E X . A V(s,SFFA) = V(s,S f A ) f'. V(s,S F A ) = V(s,S F A ) by 9.2.2 3.5.5. A [Ao!PA1] [ @ A o .. . A t . . .] A~EA. s, X , f . f' s* 0f s* 0f' on F V ( A ) A no x bound by FV(f(x)) by FV(/'(x)). V(s,S F f A t ) = V(s,SFTAd) all t. SFT SF; V s on A E A. A = [xvAo] A0 E A . s, X , f , f' s* 0f s* 0f' on F V ( A ) n X A no x bound by FV(f(x)) by FV(f'(x)) x EX. sv, t t E DR*(o), (1) V ( S ,S F F A ) = Q(x,~){V(s,,t, SFfX-RnB(')A~) : s V , ~E S } (2) V ( S S , F F A ) = Q(x, v){V(s,,t, SFF-Rnp(v)A~): S V , E ~ S}. A0
on x A s* 0f on X n F V ( A ) . sv, t* 0f' s* 0f' s* 0f on X n F V ( A ) . X nFV(A) = ( X n FV(A0). sv,t* 0 f s,,t* 0f' on FV(A0) n ( X 9.2.8 A0 no x bound by FV(f(x)) by FV(f'(x)) x EX N s,,t*
0f
-
-
V(s,SFFA) = V(s,S F F A )
9.3 Substitution for Bound Variables
Y
g
f ( y ) = (g(y)>
y EY,
on Y 9.2.3
A EA.
95
; =
yEY
L
=
L
y E Y. f g
Y
9.3.1 Definition. =
on Y y E Y.
=
(SF;E)(c) = gE(L) = E(L)
yEY
E
on
bound
Y
9.3.2 Definition.
= gE(L) =
on Y
g
yEY
bound
L
9.1.1.
9.3.3 Theorem.
T
=
T.
A
=
Y, @, =
.
[@
. .] =
..
. on
SF:,
9.3.4 Theorem. =
g on Y hgfOvSF,Yn
g on Y , no yEY
no
SB;
bound
by
PROOF : y [x‘d
bound
L L
yE
L
96
SUBSTITUTION
[x'o"C"], g(y) E
bound. yEY
L
by { g ( y ) }
L
Ao.
g(y)
y
SBFAo. bound by { y }
L
y
A0
Ao.
y
bound SBfAo i bound by {g(y)} C = SF,YnRng(')SBgYAo. [Xg'OvC]. L
+
Ao.
g(y) # y .
g(y)
L
2
bound A0
SBFAo. SB,Y[XvAo] =
i
+ 8, 8 <
=2
+
yEY :
yEY
Case I . 8 g(y)
L
y y
= [xg'OvC](~)
Case 2. 8
Ao.
y
SB,Y[xvAo] 8 y q! y
Ao.
SB:[XvAo]
y SB,YAo
bound
g(y)
yEY
g(y) SBFAo [xg'OvC1(4 = d Y ) . L
8
g(y).
g(y) I$
=
V(s,A ) = V(s,S B T A )
bound (a,p,
0,
12.
tA
SBFA 9.3.5 Theorem. t)
(***)
R B 0' Q S> Fabon A, E Y
g on Y A , V(s,[ x v A ] )= V(s,[ X ~ ' O ~ S F ~ " ~ ~ ~ ( ~ ) A ] A) no yEYn bound by {g(y)). g on Y , V(s,A ) = V(s,S B r A ) E g on Y A. Y ,g d = { A : V(s,A ) = V(s,SBFA)
97
sE
do A). A =A A [Ao!PA1] [ @ A o . .. A t . . .] At EA, A A SB; s on A = [xv Ao], A0 E A , no A. =[x~'O~SF,Y"~"~'~~S byB 9.3.4 ~YAO] V(s,S B r A ) = V(s,[ x ~ S S B ~ A O ] ) . sB,t = t DRngfv). Ao E A , V(sV,t,Ao) = V(sB,t,SBTAo) sv,t E V(s,k v S B f A 0 1 ) = V(s,A ) A E A. 11.2, (***) (***)
8.2.3, 8.2.4, (D 0
9.3.6 Definition. A
12. S) substitution property
R
FaDon Q(x,v) = Q(x, v ' )
v, v'
j3, (*)
s,
Y
ES
j3, (**) sE
S
E
B
Y E S.
1E
D,
on (*) D
(**).
on
D
on y,
D on
by
<
D
9.3.7 9.3.7 Theorem. s
<,
y
S
no (*)
(**).
(D 0 RB Fa~on X
9.2.10. S)
/?,
98
x bound by FV(f(x))
no
A x
X.
9.3.8 Lemma. Y
(*), (**)
S
9.3.6.
g on Y
S
S. by (*),
(**), E
s'.
9.3.9 Lemma.
(**),
s on Y , tg-1 on S. (*)
t'
=
S
X,
9.3.6.
Y
/3.
=s
E
S
S.
sE
S =
Ys
=
by (*).
(*) =
(D 0 R B Faflon V(s,A ) = V(s,S B T A ) all do
S>
9.3.10 Theorem.
g (***)
sE y
B
E
on Y
g
B,
9.3.5.
no
y bound by {g(y)}
Yn V(s,[XVBI) = V(s,[xg'OvSF,Y nRng(v)BI).
B' = SF: nRw(v)B,sv, t = Sg'0v.t' E DRng(O'Ov).
t E DRng@)
on x
V
(1) V ( S , = Q ( x ) { v ( s v , t , B ) :~ v , t S} (2) V ( S [Xg'OvB']) , = Q(x){V(sg,ov,p, B ' ) :S
n F V ( B ) n Y ,X 1 =
X O= t , t' t(x)
x
Xo.
paired
S}.
g ~ O v E, ~ ~
t'(x) = t(x)
S, s ~
-
n FV(B) Y . Xi,t'g(x) = , ~ ~ , ~ , t , t'
xE
99
V(sv,t,B ) = V ( S , , ~B’). ~,~, 9.3.7 V(S,~,,,,,B’) = V(s’,B) s’ s,,,~,, on Y A s , , ~ ~ , ~ X O , s’(x) = S , . ~ ~ , , ( ~ ( X )= ) t’g(x) = t ( x ) = sv,t(z).
-
s’(x) = S = FV(B)
FV(B)
s‘ 9.1.5.
= sv,t(x).
V(sv,1, B ) by
, ~ ~ ~ , ~= ( tX’ ()x )
= t ( x ) = s,,t(x).
s‘(x) = ~ , . ~ ~ , ~= , ( s(x) x)
on FV(B).
sv,t
V(s’,B ) =
t, t’ on
will t‘
sv,t E S
t’
t, t’ t 0gl on
s , , ~ ~E , ~
7s
s v ,t
t on
t, t‘
-
E
0v )
on 9.3.9
T 1 sE S ,
E S.
E
s , , ~ ~ E, ~ , by
9.3.9
A
9.3.8
g, y.
g-1,
9.4 Function Notation for Substitution will
v
A.
“A(v)”
“ A ( v ( 0 ) .. . v ( E ) . . .)”. all A and A. f A (f 0v ) = A (fv(0). . .fv(E). . .) bound
fv@)
A
by
a([).
fv(5).
A
bound by
Y bound A:Y = do
A. v
< u}.
g(y0)
2
-
Y 2 A
all
Z, :0
Y
< v}.
< u, g
g(yy)
100
SUBSTITUTION
on Y, SF~w(")SBFA.
A. no
SBrA
v(E) bound by FV(fv(E)). A FV(SBFA). 9.4.1 Theorem. A(f 0v )
A(f 0 v ) = any SBFA FV(A) = fv(f)
A
RB
S> (*)
(***)
9.3.5.
V ( s ,A ( f 0 v ) ) = V(s',A @ ) ) on F V ( A ) . A
9.4.2 Theorem. A(f 0v ) V(s,A(f 0 v ) ) = V(s',A ( v ) ) on F V ( A ) . = A(v))
by 9.2.10, 9.3.5, 9.3.10.
9.2.10
all s'
E
s' E
< j?, V(s,A ( f 0 v ) ) re-
CHAPTER 10
INFINITARY PREDICATE LANGUAGES
10.1 The (a,p, o, n)-Predicate Languages a
B
< B < a.
/?
0
o
a /3 (a,/?,o,
/I a. FaPo=
n Labon
a 1,
+,
V.
A,
(a,B, m,
Lab
8.1.1 10.1.1 Induction Princifile for Formulas.
A ELI, [ - A ] E A , Ao, A1 E A , [ A o + A11 E A , <6 < < 6, A EA v p, [Vv A ] E A ,
(4)
A
A o . . . A t . . .]ELI,
A Fagon
R>
D
-
Labon
R
8.2.3,
=. s on
D, s*<x) = s ( x )
D
s*
x,
=
102
I N F I N ITARY PREDICATE LANGUAGES
..
To, . . ., Tt, . : (1) s * [ T o y T ~= ] O(y)((s*Tos*Tl>)
all
y,
s*[yTo.. . T e . . . ]
O(y)((s*To...s*Te.. .>)
=
0
< 5 < 0,
y,
V on S x Faflon
To, . . ., Tq, . . . ES, (3) V(s,[ T O P T I ]= ) R(P)(<s*Tos*Tl>) P, (4) V(s,[QTo.. . T t . . . ] ) = R(Q)(<s*To.. .s*Tq.. .>) Q, Ao, . . . , A t , . . . v
O
8, V ( s ,[ d o ] ) = 1 (6) V(s,[A0 + A l l ) = 1 (7) V ( s ,[ A A o . . . A t . . . I ) = 1 5 < 6,
V(s,Ao) = 0 V(s,Ao) = V(s,A l ) = 1 V(s,A t ) = 1
V(s,[ V v A o ] )= 1 t E DRne@). t),A ,
Ao) = 1
3,
V,
v,
by 8,
V‘
8.2.5, [A0 t)A11
Labon”,
V’([Aot)All) (9) V ( s , [A0 t)All) = 1 (10) V ( s ,[A0 A All) = 1 (1 1) V(s,[A0 v All) = 1
V(s,Ao) = 1 (12) V(s,[V Ao. . . A t . . . I ) = 1 V(s,A t ) = 1 V(s,[ S v A o ] )= 1 Ao) = 1.
V(s,Ao) = V(s,A l ) V(s,Ao) = V(s,A l ) = 1 V(s,A l ) = 1
t <6 t E DRng(”)
(a,fi, o, m)-PREDICATE L A N G U A G E S
10.1.2 Remark. a, @,
103
on 0,
on a
n.
o
a
v
a = U {yr: v
< K,
< K}
K
< a,yv < a
[ A A o . . . A t . . ,] €
.
[ A [ A A o . . . A t . . .]. .[ A A o . . . A t . . .I.. .]. 1<1
€<%
€
bound
a+
n
o
< B t9
o
B
11
on o
n
A ;
by [13].
on [17]
As no
a,
/? < a.
9.1.3
a
B
B
by Labon a. A
R)
0,
‘D =
holds
92.
1
%? model I of F all all A satisfiable
0.
valid
‘D. “Il-A”). A
A F s
104
consequence A
ItA a
A
V(s,A ) = 1 A E r. “II-~A”) A strict semantic consequence of { A t : 5 < S} I t [ A A o . . .Ac. . .] + A .
r
A
r r
“rltA”)
r.
r
ally consistent
semantic A.
r
semantica
a.
ty 9.3.6.
9.1.5, 9.2.10, 9.3.10, 9.4.2,
10.1.3 Theorem. s, s’ A, V(s,A ) = V(s’,A ) . 10.1.4 Theorem. A no FV(f(x)) x V(s,S F Y A ) = 10.1.5 Theorem. V(s,A ) = V(s,S B T A ) g do A. 10.1.6 Theorem. V(s,A(f 0 v)) =
on x bound by
A).
A(v)).
a
v(E)
fv(5)
A
t~
A.
bound
9X
10.1.3,
9X. 10.2 Semantic Consistency, Hanf’s Example (w, 4
10.2.1 Example.
L
y
a. (y+, ,!?,0 , n)-
y+
r = {[ v
-
[xu = Yo1
:
-
. . .[xu = yy]. . .I: p < y+} u
V
{&=xu,] : p # PI, p, p‘
< r’}.
105
SEMANTIC CONSISTENCY, HANF’S EXAMPLE
a
81, accessible
o. A
y
-
E,
y XO,
10.2.2 Example.
. . ., xe,
<2
y.
. . ., E < 81.
A0 1.1.4
1.
=[
A0
0
v xo. . .x t . . .
xy
€
t) [
< 81
-
v
XY+l =
C
Cy y.
: xo,
x2
x2)
xo
(xo)
x1
XO,
xo
C(x0,
xo
on (So. . .S,) A (xo, (SO.. .S,)
. . ., x,)
Cz =
A
A xo
C
106
INFINITARY PREDICATE LANGUAGES
:
n
Bo(xo)
SO
Bl(x0, X I ) So = U S1
Cz
E)
C SO
S
So: xo)
=
x1)
B’(x0, X I , x2,
A
5
=
Z xo]
A
5
xl, x2,
xo) A
A Xg
5
+
5
A
A
275,
Bz(x0, X I ) ( S E)
Bz(x0,
Cz
So Bz(x0,
=
A
x4, x 5 ) + [Bo(xo) v r = {Ao,
Az = Bz(x0, el}.
5 xo + A x5 5 xz
A
C
v
Az} u
Z xo
y
r by Ao.
81
C,.
by
81. 81
81
Az.
ro r To
A
K
81.
81
107
S E M A N T I C C O N S I S T E N C Y , HANF'S E X A M P L E
C , E I'o
<
y
Ti,= {+, K+, f }
K.
K+
< t}
S _C U {TL:v
T'
T,,
<
K}.
Tc,t < K+,
K
on
Ti, 0 < t
I'o
C,
T,,,
f
K+
< K+,
f
t < K+
T;= T' = U {TL:t < K+}.
K.
(T' E). of
T'.
(T'E)
T'.
T' f.
K+
Az,
A1
5 < K+ t
v
v
T,, Az
y
Bo(x0). A Ey
yv E y ,
y
< K+,
y y'.
<
{y,,: {yv:v
T,, y' E y ,
K+
T'
K+
v. K+
y = U {y,,: v
< A> E T,,, < A} 1.
< A}
v
Az
< K+
<
y
<2
y', y
T,,
T'.
y = K+,
Az.
y
81. 10.2.3 Remarks. a
A2
>w
La,
a.
[47] a
on
incompact no strongly incompact
(a,
ct. ct
4.1.2 w
a
>w
108
do
CL
>w
on
As
a
[47]. 10.3 A Criterion for Satisfiability [9].
by
a
10.3.4
3
> o.
a
go
10.3.1 Defisition. j3
y K(Y+,
(1)
B)
y+.
K
Y GK
(2) K 10.3.2
E
=K
E K(Y+,
B). B.
y < K ( Y + , B) < y B d K(y+, B). B K ( Y + , 8) = U {y < K(Y+, B)
U K
K
B,
i
= U {y
< B.
E: E
<
< B}.
< K ( Y + , B). E: E
< B}.
K‘
K
< 6 < 0,
K.
g,
g ( f ) = n((y>nf6: t < 6).
B,
f
g(f) by
g
2.
K
=
B.
y
0
iE
=
(fo.. .fs..
K by
.) E
K’b K ~ ,
A CRITERION
U {Sg: i
< p.
<
1. @
<
E
109
SATISFIABILITY
x
< ~ ( y + @). , U (St: i E <
=
< ~ ( y +0). ,
2.
p)
K(Y+,
(y+,
K
10.3.5,
K.
(yf, ~ ( y + fl), ,
no
j3 = w
10.3.3 Example.
p) = y. 5 < y,
K(Y+,
(yf,0 ) -
ug,
A
E , v < y , 8 # v,
u,]
[-ME=
v
[Wxo[
-
[xo = 2401.
-
. .[xo = 2.461. . .I].
€
up K ( Y + , 0)
A
= y.
= y+
K(Y+,
,9) = 2
=,
y.
(y+,
C,
Ty+ up
1
1.1.5 y
by K(Y+,
w
< @ < y+ ut, 5 < y,
(yf,
-
=, C,
A 5 # Y,
[ - t u ~ zuv]
Ci = Vxoxi[[[ A [ixo = us]] A [ A [ i x i = %III E
=7
Y
< y,
--*
c xo t)x2 5 X i ] --* xo 5 X I ] ] ] .
3 X O . . .xn.. .[ A xn+lCx,] n
n<m
C3 =
6,
t
[Wx2[[x2
C2
yf).
A [WXO[~G XO €CY
C4 = [ A O<E
.
[ W xo. .xg.
. .[3xbWxb+lC']]
C<E
C' = [%?-tiiE xp t)[ v
Cg = Wx,q[[ A 7x0 = ug] + [ V [ 3 XO. . . X E . €
o
t i 6
Cis
-
xg+1 = x
. .Vxs+lC']]].
d.
110
INFINITARY PREDICATE LANGUAGES
. .CF. . .E/S),
CO.
A
up CF
by
S CF
8. 1.1.4,
B).
K(Y+,
A
4.1.4. A
A
A A T A A0 ELI A0 E [-A01 $ F, [Ao+A1] ELI i r n p l i e s [ A o ~ A 1 ] ~ r i f f A o ~ r o r A[ A 1 ~A0 r ,...A t ...I [A Ao. . . A F . ..] E r A t E.'l d y
4.3.4 A
5.3.1. no A;
(y+, /?,0 ,
A T
A
T
: T
A T
T.
d
/3
T
(3) = &+,1). 10.3.4 Lemma. A K(Y+,
T f
Ly+Bon
B)
A
= K(Y+, (c)
no E
<
all
= ~ ( y + B). , K
x EX
(l), (2), (3)
T
(4)
(x>
A
d = (SFfFv(OC: T']: T , T' E T}
{[T
A
X.
X
A.
E TFJ'(Q}u
T
A
B)
To
all A.
A
K
E =K
To
9.1.4
111
A CRITERION FOR SATISFIABILITY
A o
< j3.
To
by
A.
=
q~
A =K
v
=K
> 0, T, [ T O Y T I ] [VTO ...Tt ...I Tt E U 1 < v}. >0 =K 1 < v, v
< o}
o = K.
K
(4)
o
A
j3
<
y,
=K
(1)
(3). (2),
o
v
o
< j3.
As
f E
< j3.
o 2 j3. j3 o < y+. do = {SFTvcc)C:C A f E TA,An 1 < v, T,. U {At:6 < v} A , = (SFfFv(') : C A f E =U v < j3}. C A C j3 j3 f v < j3 fE A = {A,: v < j3} u {[T T']:T , T' E T>. (1) (2), ( 3 ) , (4) by on v < j3. A0 TA AA (2), (3), (4) 1 < v, v < j3, A,. A T, AA j3 all X . C A all SFFv(c)C U {FV(f(y)): y E FV(C)}, X j3 by T, A , (4). =K 1 < v, =K 1 < v. U A A )= K A
y
= K.
(3 ). A
10.3.5 Criterion for Satisfiability. Ly+Bon X K(Y+, j3) A. A, 10.3.4. : A K(Y+, /3). A
I' [T
TI
T E T,
A
all
112
INFINITARY PREDICATE LANGUAGES
[TI
T ~EIr
[T€
T;] E r
[Toy T I ] ,[Tby T i ] T, [To [ [ T O ~ T ~ ]= [ ~ b y ~ i E] r. l [pTo. . .T t . . .]. [q~Ti,. . .Ti. . .]
5
23
E
T,
-
[ [ p , ~ o . . . ~ c= . . [. ~p ~ b . . . ~ ; . . . ~ ~ E r .
(3)
Ti]
[To= Tb], [ T I (4)
[Tc= Ti] E r [ Q T ~. ..T . . .;I r.
[ T O P T I ] ,[TbPTi] A, [ T O P T IE] r [TbPTi] r. [QTo.. .Tc. . .], [QTb.. .Ti. . .] 5 <7 [QTo.. .TE.. .] E r
A,
A0 [+lo] 4 r. [ A 0 + A 1 ] ~ dt,h e n [ A o + A l ] E r i f f A o 4 r o r A 1 E r . [ A A o . . . A € .. .] [A A o . . . A € .. .] E T
A0 E A ,
At
E
r.
[WvAo] d ,
[WvAo] E
SFF*(V)A~E r
f
E
TRng@').
PROOF: 9.R
A
r. ;
[WvoCo(vo)]
K
[Wv€C&)] on
t
< v.
v
< K.
...
v
on 5 # v,
< v,
E < v.
l}
d:
.. .[Wv,C,(v,)].. .
gv
=
no
all
Co(g0 0vo) .. .Cv(gv 0 v v )
E
V(s, will
K
[WvtC&)]
=
9.R
X do
10.3.2
E
X g,
113
A CRITERION FOR SATISFIABILITY
0v,) + [Vv,C,(v,)]].
W , = [C,(g,
t, on tc on
by
D E
< v,
V(s,, W,) = 1. t < v, V(si,[Wv,C,(v,)]) = 1, V(s,, W,) = 1 V(s:, [Vv,C,(v,)]) = 0, t on D C,(v,)) = 0. t, t, 0g, t on V(s,, 0v,)) = C,(v,)) = 0 by 10.1.6 10.1.3 on FV(C,). V(s,, W,) = 1 I'= V(s,, = l}. on FV(W,), V ( s K W,) , =1 v < K. V(s,, [Vv,C,(v,)]) = 0, V(sK,C,(g, 0v,)) = 0. no X bound A, 0 v,) = F
tc
5 < v.
s:
tc on t, on
(8). K(Y+,
[T
A
r
b). TI
A
T E T, [T= T']E r
T = T'
T.
on
< ~ ( y f p). , As xo
(3)
=
0,
X:
a c,
= I
\<xo>l y,
O(Y)(
[ T O W TEI ]T, I<xo>l tp,
.
(<do. . de. . .>) =
Te E dE
I To. . .T E .. .] I
[pl To.
. .Te. . .] E T,
l<xo>[
r q-
114
I N F I N IT A R Y P R E D I C A T E L A N G U A G ES
TOE do, T1 E d l [ T ~ P TE ~A ]n r. R(Q)(<do.. .de.. .>) = 1 Tc E dE [ Q T o . ..Tt.. .] E A n I'. R(P)(<dodl>) =1
R(=)
E
on D. s(x) = I<x)l
EX
.
I<%)]
all T. [ToyTl]ET, To TI T, s*To = IT01 s*T1 = ITI[, s*[ToyTil = O(y)(lTol = I[ToyTi]I. TO. . .Tt.. .] E T s*T = IT1 T E T. V(s, = 1 C E I' C EA. A E T, on A 0 [QTo. . .Te. . .] = C E A . T, V ( s , = R(Q)(ITol.. . IT€[.. .). R(Q) (4) I' V(s, = 1 C E I'. C = [TOPTI] (3) x
s*(u> =
by
on
C C C E A , C = [A CO.. .Cc. .]. C' A f [A Cb. . .C;. .] 9.2.6 C=
.
C
= SFFv(c"C' f
C' [ A S F ~ v ( c ' ) C i.) .SFrv(c')C;. . .
. .I.
Ce = SFyv(c.)C;
Ce E A ,
[Wv Cb]
CE
C,
E
V(s,
= 1.
T, C E I' V(s, = 1. C = [WvCo] E A . C = SFFv(c')C' C' A f' T. 9.2.6, C' C = [WvSF~v(c')Cb]. f T, S F ~ w ( ' ) C= ~ SF,H;v:/Jo')Cb E A. C by 9.2.8, V(s,SF?(')Co) =1 SF?(')Co E I' by (8)
r
V(s,S F F ( ' ) C O )= 1
f
E TRng(').
115
A CRITERION FOR SATISFIABILITY
X bound C , V ( s ,C ) = 1 V(s,S F F ( o ) C ~=) 1 by 10.1.4. V ( s ,C ) = 0, t on Rng(v) D CO)= 0. f f(y)E t(y) yE s* 0f = t s*T = IT/ on T. V ( s ,SF?(')Co) = CO) = 0 by 10.1.4. V(s,C ) = 1 V(s,SF~*(')Co) = 1 all f E TRng("). [C9(g90v9)
10.3.6 Lemma.
+ [Vv,C,(v,)]]
g9
6 # v,
< v.
6
[VvgCg(ve)]
s'
U
:v
on
< u}.
r
10.3.7 Lemma.
v
W,
F
. . ., A,(?&).. .v,), . . . no
on do s
t < v.
Ag
< u}.
s . .v,)] all
[3v&(vo.
Ao(vo),
1
Y
W
=
r.
10.3.8 (Skolem-LowenheimTheoremfor Sets of Infinitary Sentences). (y+, 8, o, y+ <
r
r
all E
E =
PROOF:
< 8.
F
A
8) =
8, o,
A
r.
10.3.5. 10.3.9 E
D. tl
by
r
(a,/?, o,
y
u
r
y.
PROOF:
cg,
< y.
F
6 # 6'.
A
10.3.5, A
E =y
A [+€
=
8) = y .
(y+, /?,o, y.
y,
y
err]
116
10.3.5
10.4 Algebras of Equivalence Classes of Formulas Modulo a Semantically Consistent Set (a, p,
r L.
0,
A = A'
rltA
A'
t)
on
A
I'
[WVA]
B(L; I') lAlr = { A ' : A = A'}.
on
B(L;r)
+Ir
lAolr
=
=
-+
A (
.. A t . .
7 Volr
= I[Ao A
A
B(L;
r) r
A
v>.
r
10.4.1 Theorem.
Fm
= I[Ao v
lAolr v
B(L;
7
r
F C
s
=
Fm}.
r
Fm, s
:
r)
B(L; F )
7
{(Fm, h ( A ) = {(Fm, : s K
=
4.2.1. :s
'I u { A } Fm). h ( A ) ,h(L-40 A = h(A0) n W l ) , N A o v h ( [ A A o . . . A . . .I) = fl AS): t < S}.
h ( [ l A ] )= K =4
A
%ll} 4 0 )
-
u 4-41)? n
ALGEBRAS
EQUIVALENCE CLASSES
FORMULAS
F
Fit-A
A'
t)
g([AIr)=
=
on B(L; F )
+
T=
N
n
F
g(lA1,) = [Alp on B(L; T)= B(L; +) B(L; r). B(L; r) IAIr < IA'lr r k [ A --+ A ' ] , -
-
XIIT,
I[Tx = x]Ir, I[x = A {IAclr: 5 < S } = l[A A o . . . A F . .
.]IT
I[V A o . . . A t . . .]IT.
V ([A&:
5 < S}
=
no p
Bo,
(a,
10.4.2
L'
0,
r'a I'
B(L';
by 10.4.1.)
p PROOF:
y =U
~ ( y + u) , = U {y
E: E
<
E: E
r')
< As L' X
L L"
(y+,
A, T, d
L".
A
<[Wv,C,(v,)]: v
of
,'l
all 10.3.4 A, X all T A
L". L' L'.
W, =
< y , {W,: v < y } I' u {W,: v < y}. I" Y
B(L', I")
p
F A.
all
d
C
y
0,
10.3.6.
=
L
no 7
by 10.3.2
=y
0,
L'.
r
L
u,0 ,
F
117
L'
7
0 v,) : v < a) 0 v,) --+ [Vv,C,(v,)]] L'. r' =
h Bo. l} u ( A } . 10.3.5 L" A . fi
=
118
INFINITARY PREDICATE LANGUAGES
[T= r] As
A
$,
< 5.
[Ta = T ; ]
E
5
F
all E
<
< 5,
T E T. look hl[T~ =T;]Ir*
-
-
=1
.
C = [ A [Te = T ; ] ]+ [ P I T O.Te.. . . . ] = [vTTi,. . T i . . . ] ecc
L'.
b(L', F), = 1.
IClrt = 1. h ( [ y T o . . T e . . . ] = [vTTi,.. . T i . . . ] l p A AEF
F.
F.
A
by I[VvAo]Ir#< ISFFnB(w)Aolrf all f E TRng(').
F.
T
hl[VvAo]Ir*= 0. W , = [Ao(g, 0 v ) + [WvAo]] E r'. b(L';r'), hlW,.l = -,hlAo(g, 0v)lre = 1.
As (8), WvAo]E [WvAo] $ F, Aa(g, 0v )
r
Ao(g, 0v ) 4 P.
4.3.6
10.4.3 Theorem. :
(iii)
no
p p p
?@s,(n, a) 6.4.2
?@a(n.,a)
6.4.7
CHAPTER 11
INFINITARY PREDICATE LOGIC
11.1 Description of the Formal Systems for (a,p, 0, n)-Predicate
Languages on a,p, 0 , n
p
p
=0 o
(a,p,
<
w a
a
< p < a, n Q a, o
o
p
L, L of L
0,
all
basic formal system @&g(L) &, 5.1.1,
91. [Vv[Ao+ All] + [A0 + [VvA1]],
no
Ao. 9 2 . [[VvAo]+ SFF*("'Ao], A0 xE bound by FV(f(x)).
f
92
no L.
on
w3,p(L)
Te, T i
y
P :
691. [ T Z T ] .
6 9 2 . [[To Tb] A [TI T i ] ]+ [[ToyT I ] [TbyT i ] ] . 6 9 3 . [ A [To-TTi;]. . .[Tc = T i ] .. . ] + TO.. .Te.. . ] = [rpTb...Ti.. 614.[[TO Tb] A [TI= T i ] ]+ [ [ T O P T I-B] [ T b P T i ] ] . 6 9 5 . [A [To = T b ] . . .[Te = T i ] .. . ] + [[QTo... T e . . . ] + [QTb. . . T i . . .I].
120
INFINITARY
A0
[VvAo].
2
'&@(Z)(L) L
2
WP,b(L)
11.1.1 Laws of Independent and Dependent Choices. A L of y independent choices A gS7.
.
law
.
[ A [3voAo].. [ 3 v t A e ] . .]+ t
[3
00.. .v€.
. .[ A
A o . . . A f - ..I],
f
€
no
ve
choices M Z y .
v # 5.
A, A A
Rng(v6)
L
[ A [ZIVOAO]. . .[ V I
law VO.
V C €
€
no
vc
v
A,
< 5.
%Sy 9%'GZ?+ y ,< a "
L
< /?. Z
%S,,9%SY
y
< CL n p,
Va@(2)(L) Z
L
WOag(L)
11.1.2 Rules of Independent and Dependent Choices.
/?
a,
9%XY y y
.
[ V [WvoAo]. .[WvcAe].. .] t
y
y dependent
. . ~ 9...[ 3 v e A t ] ] .. . ] --+ [ 3 V O . . . V € . . .[ A A o . . . A t . . .]I, C
y
y
< a.
y
WXy y < @.
[ V Ao. . . A t . . .] €
%?GZ?,.
[ V A @ .. . A t . . . ]
121
[ V [VvoAo]. . .[ 3 w e [ V v , A ~ ].].].
9WXy,
b
At
?!,
v
U 2
Z 9%‘Xy
y
< t}
F V ( A F )n U
v
< E}. WX,,
< a nB
SZ
y
y
< a,
@as(Z; Q)(L)
Z
L SZ
@a@(L)
11.1.3 Theorem.
( a , @,
L
0,
9,
L
89, % ‘ i f y ,9WX,, y
y
A W A
Syd
W
f
Cm
L.
L.
V ( s ,f(A6)).
1.
V
0, 1.
L
s’*
on Sy
on
V(s,A ) = s ’ * ( d )
Y
=
U
Cm. A A
s’(At) =
W
s’*d =
Cm
s
At E W
9W%‘,,
91, 9 2 %?Xy
[3vt[+le]] 5 < y}, by 10.1.3.
= 1.
by 10.1.3, 10.1.4. by 10.3.7 10.1.3. y 1
10.3.7, [V A o . . . A t . . .]
Cm. y
‘@ag(Z;
a,
SZ)(L)
A
122
by A
A
“I-A”.
A
A
“I-AA”.A
@ao(Z; Q)(L) comfilete A calculus @ao(Z; Q) Z Q. comfilete
Q)(L)
j3, 0,
L.
5
no
L
no
L
no
a
10.2.3. As 1 1.1.3 11.1.4 Corollary.
valida-propositional schemes
Z
%2, 9%&,
y
< n j3.
9 y
(a,j3, 0,
I-AA
A
<
L
A.
11.2 Development of the Formal Systems for (a,/I, o, n)Predicate Languages
will
pabound
will 11.2.1 Theorem.
9. La
(a,j3, 0,
L
Z
I-A f
@a(Z)(La) on X
I-SfA
‘@ap(Z)(L),
A
L. :
3.5.3
123
DEVELOPMENT O F T H E FORMAL SYSTEMS
A
La
X,
on X
f
SFA
L,
L.
5.2.2. 11.2.2 Definition. I-AA [A . .Ce. . . ] +A .
by
r
Cc E r, E
FtA L
I'td
A,
r
11.2.3 Theorem.
L.
E <6 y
I'td
At-A.
r.
S)(L) 1 1.1, rl-~ [A0 + All
r t d A0 r t d At FtdA [VVA]. If
Ft-d A1.
Ao. . . A t . . . ] .
r t - d [A
r
9,
.
[ V Ao. . A € .. .] €
rt-d[
F t dA
L.
< 6, 0 < 6 < u
.
V [WvoAo].. .[WvbAr]. . ]
CCY
:E
U y rt-d
r.
Sa,
[ V Ao. . . A € .. .] E
I'td
[ V [VvoAo].. .[ ~ z u ~ [ W V , A . .~] ] ] . €
: 5'
U
r.
< y}
5.2.4.
PROOF: = 4,
r
C
FtdA td C
+A
r
C
I-dC 3 [ V v A ] by 91
C.
V&,
[ V Ao. . . A € .. .],
9 VO,
r # 4,
TtdA tdC + A .
5.2.4
ICY
. . ., ve, . . .
r.
tdC
+ [ V Ao. . . A t . . .] €
r,
124
I'.
t~[ V Ao[-,C]. . . A € .. . ] C
w a , 1 1.2.1
vo, <x>, v l ,
. . ., v ~ .. . .
C
x y
At
vt.
y
%'A?,,.
t A [ v [VvoAo][Vx[1C]]. . .[Vu&].
.. I .
C
ga, tA[3XC]
+ [ v [WvoAo]. . .[ v v t A € ] .. .]. C
t [ V x [ l C ] ] [4] by 9 2 , k C
[%C] by
+
.
kAC + [ v [VvoAo].. . [ V v , A s ] . .]. S
11.2.4 Deduction Theorem. waB(Z;LR)(L) 1 1.1, rk,4 A kdvrA f L. 1 1.2.3 by on A A v r. 11.2.5 Properties of Quantification. laa(1), k[Wv[Ao --* A111 4 [[VvAo]+ [ V v A ~ l l . k[Vv[Ao A A l l ] t)[WvAo] A A11 A1.
= u t [ W v A ] t)[Vvo[WvlA]] k [ V v A ] t)[Vg 0vSF,R"g(")A] g A, A no v(E) bound by g(v(6))
5< [[Vg0vSF,R"B(")A]+ A ] A = 0 g(v(5)) S F P @ ' ) A bound by v(l) SF,R"B(")A,A" = SFRng(gow)A'. 8-1 1 y qk A, L
92.
no
A' y
A' L
A
=
125
FORMAL
A‘.
0v )
L
A ’ , A ” ( L= ) y
= A(L).
v(E)
A = A’ on
C-A[At)A’]
L
g(v(6))
A, L by A”(&)= A ( L )= ~ ( 5 ) . v(5) A , bound A“(L)= A ( & = ) ~(5). A = A”. g(v(5)) A’ bound by v(5) A’, v(6) A, L bound.
L
A’, bound v(E)
v(6)
@Pa&’;
A’.
A . If L bound by on 9 2
Q)(L) L.
(a,#?. 0,
1 1.2.1, =
5.2.6
11.2.5
A A = A‘
[WvA] ;
[WvA] = [VvA’]. 8.4.
8.4.5, 11.2.6 Replacement Principle. A = EoCo.. .E,C,. .Ea, (C,: v < L, I-A [C, t)C i ] v < Q)(L), FAA t) . .E,C:. . .En].
.
T = T’
C-A[T= T’]
on L. by 8 9 4 P = =. by 6 9 2 693. on on
To = Ti, Tg = T i
T1 = T i
E<
@&;
Q)(L)
by 891, 894
on 895
[ T O P T I= ] [Ti,PTi] [QTo. . .Tg. . .] = P,
S> G(?&@; Q)(L), A ) :
. . T i . . .] qL,
126
INFINITARY PREDICATE LOGIC
T, { A ‘ :A = A‘}.
IT1
= {TI: T
D
= {ITI: T B = { [ AI : A S =
= T’},
A,
( A (=
L}. s ( x ) = I(x>l
L}. all
C(c) = I(c>l O(y)(
O(rp)((ITol...IT€l...))
=
x.
y
I[pTo.. . T c . . .]I 0 < [ < 0, [
(To. . .T t . . .). R(P)(
=
(To. . .T t . . .>.
I[QTo... T c . . .]I Q 0 < 7 < x, 7
O’(-d(lAl) = I-AI O’(+)(
s* on
on s * ( \ ( x ) [ ) = s(x)
s*T
=
IT1
x
T.
on
V V ( s ,A ) = IA I
A.
11.2.7 Substitution Rule for Equality. @&;Q)(L) tdf(x)=f’(x) all X E X FA SFF A t)SFF A , f , f’ X A no x E X bound by FV(f(x))
W f’(41.
G(&@;
0f If(x)I = If’(x)l.
fz)(L), A )
by
s
0f’ on X 9.2.1 1, V(s,S F T A ) = V ( s ,S F F A ) .
127
11.2.8 Rule for Changeof Bound Variable. I-A t)SBFA g on Y A.
PROOF:
#). k[WVA]
c) [Wg’
9.3.5,
0 vSF;””w(%4]
g
all no g on Y ,
A bound by g(y), by 11.2.5. vo,
yEYn =
Yn
=
11.2.5 vo, 11.2.9 Corollary. t [ W v A ] 4 A(f 0v )
A
01
Y.
11.3 Completeness of the Basic Formal Systems with Chang’s Distributive Laws and the Rule of Dependent Choices for Certain a, 10.3.5. @
no
=a
A
v
[WVA] @
A, FA.
t[WvA]
[VvA],
by
A (XO,
11.3.1 Lemma.
. . .,xg, . . .) (xo.. .xg.
L
A. cg
, ~ ) -
L’
II-A(x0, . . ., xg,
xg.
. . .)
.
II-A(co, . . , ce, . . .). 1 1.1, FA (CO, . . . ,cg, I-A(xo, . . ., xg, . . .) ))3as(Z;
no bound A (xo, . . ., xg, . . .) A (CO, . . ., cg,
. . .)
. . .), A (CO, . . ., ce, . . .) xe
1332’
(a,@, 0
. .>
cg.
L’, V ( s , A(c0, . . ., c ~ ., . .)) = V(s,A ( x 0 , s ( ~ € )= C(c,), by 9.2.10.
. . ., xg, . . .)) A(c0, . . ., c,, . . .)
I N F I N I T A R Y PREDICATE LOGIC
128
A (XO,
. . ., xg, . . .)
L’.
A (XO,
Y
A (co,
92’
. . .,cg, . . .)
Y
A(x0, . . ., xq,
. . ., xq, . . .)
. . .).
on Y C, by
C: y by g(y).
L
C: by
cg
C,‘,
xg
C:
xe
(C,”: Y
C,” no
by xe.
< u + 1)
L, C: = SBTA(x0, . . ., x ~ ., . .).
all &a(Z; Q)(L).
A,
L
xe.
A (co,
ce
L
C:(L)= (SB;A)(L).
L
y g(y) C:. (SBTA)(L). kSB;A(xo, . . ., xe, FA (xo, . ., x t , . .) by @aa(Z; Q)(L)
.
C:.
bound
bound
A,
. . ., ce, . . .) = Ca
xe
L
.
A = A’
y
Ca.
yEY
L
C:(L)= D)(L).
. . .) bound
11.1.
t-A
A‘
t)
on
B((PaB(Z;Q)(L)) on
+I
by = l [ l A I l > IAol
+
=
A (IAol.. . IAel.. .) = l[A Ao. . . A e . . .]I.
on
B(%dZ; Q)(Lf): lAol A \Ail = A lAol V lAil = V V (IAol.. .IAgl.. .>= [[V A o . . . A c . . .]I. 1 = [ [ Av
1.2.1
129
BASIC FORMAL SYSTEMS WITH CHANG'S D I S T R I B U T I V E LAWS
11.3.2 Lemma.
L,
L" B = B(@&'; Q)(L)) k d @&(El), PROOF: W At
E
2. L, O'(1) = 1,Or(+) = +, O r ( A )= A. s*&
=
1
W /(At) L s*
s(AC). on
s*a = ISTaI
L,
W.
@,(Z'), s * d = I S y d I = 1 by 11.2.1.
Id
B (B
A
B.
s
7 d'
v>
EZ
Td
=
d' 1
=
D)(L)) =
B B. 6.4.2
@,(lIpa)
B
Z
7
B(@,&'.; D)(L))
< \All
lAol
k[Ao + A l l .
by I[VvA(v)]l < \ A ( /0 v)I
bound
92
I[VvA(v)]I ] A ( /0 v ) l .
bound
11.3.3 Theorem.
py+#7y;Qy)
y
y
E =
E
D,
< B, y
: (y+, B,
A @y+p(17y; Qy)(L). Dy)(L)) T A 10.3.4. y.
L
93 =
do X d
230
!-A 1
0,
n)-
/I) =y
l[+l]l
y
'B
y
on
h,
T = {C:hlCI
=
1)
10.3.5
[-A]
T
130
INFINITARY PREDICATE
A.
[-A] go
10.3.5 no d:
A y *
-
. . ., Cv(gv 0v,), .
.
* *
Co(g0 0vo), g,
a,
[vv,cY(vY)l,
X
on
5' # v do [Wv,Cr(vc)] 5 < v. W, = 0 v,) + [Wv.C,(vy)]]. 1[lA] A [A W O .. .W,. . . ] I # 0 8. 0.
ki[[iA]
A
[A W o . . .Wv..
.I].
all
1 A I- [V [lWO].. .[ l W , ] . . .I g, 0v,
W,
W,
/?
w,
:E
U
5 < v.
Wa
/?
d
< v}
W..
4 k [V [vgo o vo[lWoll...[ 3 w [ v g Y 0v,[-rWPl11-. .I vV) A [-1Wv,C,(v,)1],
t)
do
v.
I- [vgv 0 Vv[iWv]]t)[[vgv0vrCv(g9 0%)I v < y by 11.2.5 11.2.5
FA. \ [ - , A ]A [A W O .. .W,. h :
80
A
.. ] I
[ivv,cv(Vv)]]
kA. # 0.
/[-A ] A [A W O . .. W,. . .]I
by 1
131
BASIC FORMAL SYSTEMS WITH CHANG'S D I S T R I B U T I V E LAWS
v{-~I[Tc=T;]l:t< ( } v I[qTo...Tc ...I = [CUT T i . . . ] I [ y To. . .Tg. . .], [ y Ti,.. .T ; . . .] T. 1 by 6 9 3 . v \[QTi,.. .TL.. .]I To. . .Tc. . .I, [Q Tb. . .T i . . .]
V { ~ ( [ T cTk]l:5 < q) v (-I[QTo.. .TE...]I) all A.
1 by 695. < 6) = \[A A o . . . A t . .
{IAc(:5 A.
y
y
hlC( = 11, h [-A] E F F 10.3.5 [T =TI E r by 891. A0 = [To = Tb] A1 = [ T I = Ti] r, A2 = [ [ T o ~ J T I ][TbyTi]]E r (A01 A \All IA2/ by 6 9 2 h 10.3.5
F
=
<
(2),
I[VvAo]l<
(4),
all f on
ISFPw(')A01
T by 9 2 .
X
T
A.
[VvAo]E F
h SF?(')Ao
E F.
[VvAo]# F, Y W, = [Ao(gv0 v) + [VvAo]] h\W,l = 1 hl[VvAo]l= 0, hIAo(g, 0 v)I = 0. Ao(g, 0v) # F. Ao(gv 0v) = SFEw(')A0. [-A] A. A
11.3.4 Theorem.
a
all
9%9Ppa) y < a, 9 % X P a
y
L
L.
y (0,
n
< a.
n}
y
{a}.
L'
A
x
-
< a, n' = y+
A bound
(a,a, 0 ,
0'
n
o = a.
<
o y y)+, y+, o', o < a, 0' = y+
A
o
< a.) A
n Qy n'
o = a, L'
n
132
I N F I N I T A R Y P R E D I C A T E LOGIC
L’
L.
?&2expy)+,y+(172expy; Q) Q 2
y
11.3.3 A 2
172expy
2 2
y .5W&,‘,.
y
A 11.3.5 Theorem.
E
< a. < 0:
y
j3
j3)
< a,
&&77a; Q)
(K+,
y (a, t!?,
y
j3, o’,
~ ( y f /I), , 0’ = o o n’ = yf n = a.
a
< a.
by 1 1.3.4.
A {a}.
y
~ ( y f j3) , = U {y E : E < j3) < a E
y K(Y+,
< j3
on a , j3,
j3 < a
E
y
all
j3 j3
@aD(I7,a; Q) PROOF: j3 = a a. j3 a
y
Q
y
~ ( y f j3) ,
a
@ a b ( I l 7 a ; Q)
y
y
bound
o
< a,
0‘
o
= y+
A
< a. 0,
n o = a, L’
<
-
L. y A {o, n} y n < a. L’
A n’ = n
L. K E =K E < j3, A !&+,a(17K; QK)(L‘)by 11.3.3. ( P a b ( n 7 a ; Q) (L). 11.3.6 Theorem. a !#ao(IT7a) y, ~ y + o ( 1 7 ~ ) @,,,lo L’
PROOF: 1 1.3.3
17, by 1 1.2.1.
~,,,lo
K
n
=
< a,
a = wi,
11.4
/? = 0
B=w
133
Completeness of the Basic Formal Systems when a = w1, p=w 11.3.3
a = wl, ,fl = w.
Val,, I7, Vmla(4;Lna)
11.3.3 Q, &,la
on 80
on 1
on
Vy+,JIIy)
y
> w,
y
on on !#Dy+,w(17y) y > w
8(walw(L))
11.4.1 Theorem. $3p,l, L L. ~ ( w lw, ) = w
(01,
w ,of
A I-A.
d
A 10.3.4. 8(Vmlw(L)) B‘ = {ICl: FV(C) C
d ICI E B‘
w. 8’
X, C
E d.
I[WvC]I
8’ 8(?J&ala(L)).
=A
I[WvC]I
[WvC] E A ,
B’,
{IC(f 0 v)I : f E
< IC(f 0v)I
8‘.
all f
E
%3(@alm(L)),
on
IC’I E B‘
IC’I
< IC(f 0v)l
%‘, f
~XRng(v),
f I-C’ +
134
INFINITARY PREDICATE
C ( f 0 v)
t C ‘ + [WfO v C ( f 0 v)]. t C ’ + [WvC]. FA. IAl # 1 on 8’ 230 IlA[
h
11.2.5
1 4 # 0. 1
23’: d.
A {IArl: t
< S} = [ [ A Ao. . . A € .. .]I B’
d.
B. T,
11.3.3.
A, 1
B’.
T=
ICI E B ’
=
r
[-,A] E I‘
l}.
[-A]
10.3.5. A.
A
11.5 The Reduction to the Rule of Independent Choices when
p=o @a@(f17a )
ct
> 01,
by
fl = w.
12
> GI,,
y
Q I , ,= ~ U { Q I , ~ :K
Q D , ~
OD,, y
=
u {DO,
11.5.1 Theorem.
K
< y}.
a
pa@(IIya;
QI,,a)
A Q a m ( L) = p a o ( f l 2 exp
A yf
(y+, a
; Q I ,y ) (L)
2
y
< a.
135
X
A
T
w
X
B 2‘3;
U { X t :i C
B
< n}}.
X O , . . ., B(CZaw(L))
. . .,
y.
BL = {ICI : F V ( C ) C
8’= U {BL:n < w}.
ICI
A %’.
7
F V ( [ A CO... C t . . .I) [ < S}
A
=
A 10.3.4.
=y
K(Y+,
%A, 8’
CU
6< l[A C O . ..Ct...]l
23;.
< n}
(Xi: i ‘23’
B.
BL
= fa-
B’
B.
B.
I [ W v C ( v ) ]I
B‘
I[WvC(v)]I = A {IC(f 0 v ) l : f E XRng(v)}
B‘.
B.
on Qz,,
on
D IIWv,C,(vv)]l, v
U { X i :i
< y,
< n}, ID/# 0,
BL,
y
g, on ID1 A A IC,(gv 0vv) + [wv~cv(v,)lI # 0 V
8’
B.
11.3.3.
A
&,(L).
l [ + t ] ]# 0. on B’
h 1
230
B’:
y
1 = I[A A o . , . A t . . .]I v
A. V {IlAtl:5
< S}
A. T
A
1
11.3.3.
y
T= [-A], 10.3.5
hlCl = l} [T =TI T
T
136
I N F I N I T A R Y P R E D I C A T E LOGIC
[-A]
A
A. by
XO. XO
1 4I
0 # ICbl
ICbl
v
Cb V, ID,I = 1
< ID,I.
[Vv,C,(v,)] g, on C," = [Cb A [ A [C,(g,
XO
X Ou X I . #0
< ID,I
V, A
ID,\= 1
IC,"l.
X Ou X 1 by Xz. by will IC,"l 2 IC;l 2 . . . IC:l ... B' V, ID,I = 1 n, v ICi( < ID,I n g on X 7 IC(g 0 v)I v I[VvC(v)]I. h {T
b),
ICil: n
y-
X Ou X 1
C; v
0 v,) +
V-=Y
IC,"l # 0.
[Vv,C,(v,)]]]].
X1
< w}
C;
IC:l < by
CHAPTER 12
N 0 N-DED U C I B ILI TY I N INFINITARY PREDICATE L O G I C
b, 0 , n
on
< t!l < a , o
n
/I= 0
<
o
o
(a,
up
> w.
(a, b, 0 ,
L Qn,
CH,
=[
A [3xQnx]]
n i w
w
< w,
[ 3 x o . . . x n . . .[ A Q n x n ] ] .
-P
nco
CH, ?$3a,#7pa )(L). Theorem A.
n
R
< y < ,!?,y
L
Qc, E
(a,
< y.
CHy = [ A [ I x Q+ c[x 3 ]X O]. . . x € . ..[ A Q ~ x e ] ] . E
P
CH,
C i Y
&fi(flpa)(L)
&&7F
a
CH,
u S X ' , y ) (L).
Q'
A A. CH,
VZY.
[Qrxe] by [Q'xsyr].
138 @aB(f17a; Q I , ~ ~ )
B > o.
L
(E,
B, 0 ,
Q
DCH,
= [ V x 3 y Q x y ] + [ ~ x o .. .x % . - [ A Q x n x n + ~ ] ] It<,
L
Theorem
B >w
(a,
DCH,
Q. @aB(n,a; QI,7a)(L).
< y < /I,y
[ V xo.. .xv. ..3xeCt]] 3
= [Vxo3xlQxoxl] A [ A
DCH,
w
1<E
*<€
5 2 2, A Q ~ v x e ]+ [ A
Ce = [[ A
o<e<e
o<e<'t
v<e
DCH,
A Qxvxell. v<e
@a,9(f17a U %?2fPy;
*I,ra)(L).
DCH, v b = <x&, A0 = [xo
= X y by xo], A1 = [ Q x o x l ] , A5 = Cc
t[ A A o . . . A t . . .] + [ A O<€
€
A
Y
<5
=
A Qxvxt]
@a&).
v<e
[ A A o . . . A t . . .]
0
t-A + Qxrxc,
50
YO
YO
t-A + [ A O<E<€o
< 50.
v
O<€
t-A + Q x V , x ~ , , A
< t < y,
50 # 0, # 1,
C,,, FA + [ A
A Qxvxt].
5 2 2.
50, YO. [ 151
A Qxyxe]. v
0:
= w1.
12.1 Summary of Results and Open Problems m
B = w.@ ,,, @ao(IT,,a)
<
7, E
> 01
139
BOOLEAN ALGEBRAIC METHODS
o
o < @, <@
y
< @ a p ( n / a u {qJf,}).
@a@(n/a)
@aa(17, a U VJf? Y ) < lpaB(17, a U {qJfy}).
y
@ag(qJfy).
@aa(qJfp y )
o
@aa(U/ a w
y
< @ao(lirya u s J f , * Y u { S J f y } ) . Qz,,a) < Bas(lir,a u {m#a}QI,,a). ;
u -@qJf,y)
< @,?Paa(lT,a; o
@aS(n/a
by “ ~ X ” .
“%X”
u gvJf,y; QZ,,a)
y
< @a@(n, a u
{ m z y } ;
QI,,~).
on
A @= @aa(n, a
y
um
x
y
u vz,
< a.
a)
@aa(I7., a U S J f , a )
a
@,,(17,a
u 9%’&.,a)
14
no
12.2 Boolean Algebraic Methods by
will As
CH,
> w1,
(w,
CH, As
12.2.1 Interpreting
@aa(17fa)(Laa)
8, 0 , +predicate Languages in Comfilete
140
(u,/?,0 ,
Boolean Algebras.
23 =
S)
RB
D, O ’ ( 7 )= l, 23.
on = A, Q(W, v )
&‘A!
v),
B
O’(+) = +, O’(A)
0,
A
23
R
L
D.
all
1
R(=) (do,dl) = 1
-
do = dl, 0
=
8.2.2.
23,
on
D
R
0,
algebraic model
V(s,
=
C E .‘l
1
9.1.5
A
9.2.10,
V ( s ,A ) = V(s’,A ) on F V ( A ) . V ( s ,S F T A ) = A) A x bound by FV(f(x)) x E X . 12.2.2 Theorem. (CL,
8, 0 ,
‘$2 = (D 0
A holds F
no
R 23)
L.
A
Wub
A
‘$2.
‘$2.
2
Td
=1
L o
23
12.2.3 Theorem. < /? < a. Lug
9ua(Lus).
PROOF:
2
all& E 2,
‘$2.
u (a,
Qn, n < o.
CH,
8
(w, 01)-
7.2 will
141
G E N E R A L METHODS
bny,
n
< o, v < ~ 0 1 ,
$ d,
n<w
d
=
V { A b n f ( n ) :f c w";. AS m<
Lap
W
V<WI
( ~ 1 0
R '$3)
CH,,
= bnv.
V(s,[ A [ Z I X Q ~ X ] ] )= A V
Qnx): v
n< w
It< w
V b,,
=A
< w1) = c,
V(s,[ZIxo...x n . . . [ A Q n x n ] ] )=
X={Xn:n<W}
n<w
[ A Q n x n ] ): t E of}= d .
V
n<w
CH,
c -+d = T c v d # 1.
12.2.2,
'@ap(LaB)
CH,
[23].
[29], [31],
'@ap(Lap) fa-
A
(D 0 '$3
R '$3)
f
A ) : t E DRnB(")}
V(s,[ W v A ] ) = A '$3.
[31],
12.3 General Models (a,8, 0 ,
L
(D 0 L
Bo
R Bo
A general model
L
S) 0, 1, Or(,) = 1,Or(+) = +,
142 N O N - D E D U C I B I L I T Y I N I N F I N I T A R Y P R E D I C A T E L O G I C
Q(V, v )
O'(A) = A,
D
on
S :
(*)
Y
s, s' E S
p,
E
(**)
E S.
Y
S tE
E
b,
S s*)=,
E
R(=)
on D,0,
will
R,
YJ2 =
8.2.2.
V ( s ,W v A ] )= 1 t E DRng(a)
A) = 1 E
S.
A
A satisfiable in %I2 V ( s ,A ) = 1. holds in YJ2 V ( s ,A ) = 1 12.3.1 Examfile.
CH, == [ A P x Q ~ + x ][ ]3 X O .
sES
all s E S.
.xn..
n
n
.[ A
Qn~n]]
n<m
all
YJ2 = ( w R S> S
V(s,[ ~ x Q ~ x= ] )1
n
V ( S [, 3 XO.. . x n . . .[ A n
no
CH,
YJ2
R(Qn)(i) = 1 co
on
all
n = i, 0
< w,
Q n ~ n ] ]= )
0
n<m
S
n
all
s.
xn, n
< w. 9.3.6,
12.3.2 Lemma.
s, s'
E
9 on F V ( A ) ,
S
V ( s ,A ) = V(S',A ) .
12.3.3 Lemma. sES A) by FV(f(x)),x E X .
< 8, no
A
X
V(s,S F F A ) = x bound
143
GENERAL
on F V ( A ) ,
s’ E S V(s’,A ) 12.3.4 Lemma. s V(s,A(f 0v ) ) =
V ( s ,S F T A ) = @,
v
A).
v
s‘ E
on F V ( A ) , V(s,A ( / 0 v)) = V(s’,A ) . 12.3.5 Theorem. L p, 0 ,
L 9, 89,
W
d
L, s’(Ac) = V ( s ,/ ( A t ) )
d
Syd’
sE
At d
s ’ * d = V(s,S y d ) .
W.
W
91
W by 12.3.3
Syd’
by 12.3.2, (**) S.
W
89
92 no
W.
A.
12.3.6 Corollary.
La@ Qn,1z < o.
CH,
a ) (La@).
12.3.5, 12.3.1.
12.3.7 Lemma.
W
(a,@,
L (*)y
s E S,
all 5
E
Xe X
= U {Xe: 6
.%?ify
0,
< y, @,
t = U {te: E
< y}.
92.
L
10.3.7.
A = [ A [3voAo].. .[ W V O . . €
*<€
.up..
. .P v r A e ] ] .. .] + [ a v o ...v t ...[ A A0 ...A t . . . ] ] 6
€
no
144 N O N - D E D U C I B I L I T Y I N I N F I N I T A R Y P R E D I C A T E
A,
v
< t.
E
S
by
t g on
D
ES,
if se
by (*),, tv on Y
Lao
CH, PROOF: 5 = v, 0
q a o ( I I p a U{W2,,)(Lao). '332 = ( y R S> R(Qt)(v) = 1 S on y y.
y
12.3.7
(*)K
K
< y.
12.3.7
12.3.5,
%wW/a u { '332.
~
~
CH,
Z
P(Laid Y )
'332.
12.4 Non-deducibilityof the Axiom of Dependent Choices in Systems with Distributive Laws and Rules of Independent Choices
DCH, V R o ( l I 2 a ; GI,,,)
C tl
y < p. a(@)=
<
0.
E' E E. E E &(a) I ( E ) = {El: E' by 9 C &(a) i ( U g 9 6 ) = Ug I ( 9 6 ) .
<.
Lao
(a,
(On: n
<
< E}.
< o}, E'
I ( 9 )=
>o
I(E) { I ( E ) :E
E
and 9).
145
NON-DEDUCIBILITY O F T H E AXIOM OF DEPENDENT CHOICES
As
9Jl = (&(a) < S )
Lao
-<
on
S no
&(a)
C
ES
ES
s’
1.
u
=
U
no do
(*)
C
tE (**)
S.
E
no
9Jt
S
12.4.1 Lemma.
DCH, = [ V X ~ Y Q X -+ Y[]3 X O . .
.
. ~ n . .[
n<w
R .l! PROOF : [Qxy] no
= (&(a) sES
A Qx~x~+I]] n
< S) . ES,
E
= En
E‘
[ A Qxnzn+1]
S
DCH,
n
Il,,
9Jl by 12.3.5 6
9Jt &‘(a)
< a.
6 12 on
&‘(a)
=
all
distance-fireserving
A
E
on &‘(a).
.
. .I,
[ v [VvoAo]. .[VvgAg]. t
n FV(A,) = ES,
0
6 # Y,
on
t6
Ae
S
=4
n
by
0 ti
:E
no [ V A o . . . A ( . . .] Eta
< 6))
ES
0,
v
=
5 < 6>
t‘
=
u { t l : 5 < 6). 1 In my paper [ 151 was only stipulated that Rngfs) contain no infinite chains. This was an error. The proof of Lemma 12.4.6 was incorrect.
146
12.4.2 Lemma.
on h(Elm) =
h
(&'(a)
<)
I(E)
m < I(E)C
= &'(a),
I ( 9 )C
IG
=
h(Ejm)
< h(E)
h(E) m
m h(Elm) =
=
lm.
12.4.3 Lemma. <&(a)
<),
on
h 12 0 E S.
E
12.4.2,
no 12.4.4 Lemma.
9 C &(a)
h
on
<)
(&'(a)
0
a
I ( 9 )C no
a
E
r(E) = I(9)).
E &(a)
n
U
E f on a
0
- r(E))
0
=
hf
on
h f l I ( 9 )=
&'(a)
hlI(9).
H
< H'
HnF HE HnF HnF' F = cj H = H'
H , H' H = H', F
H'^F'
# cj H < H' H,
+
F(0) F = I$. H'
+
= -
< F'
=
<
F F , F'
< F'. E
HaF
F E &(@). H^F'
<
<
&'(@)
HnF H < H'
H'. F'(0) H , H' HnF -
hf
€I@),
=
< r(El),
147
AXIOM
<
< hf(E1).
EO4 I @ ) , r(E0) = by hf(E0) = 0 - ~(Eo)) 0 = hf(E0) hf(E1). = hr(E1) f 0 (Eo - ~ ( E o ) ) f0 - ~ ( E I ) ) hr(Eo) < f 0 (Eo - ~ ( E o )=) 4. = - r(Eo) Eo r(Eo) < Eo = r(E0). Eo El. hf 12.4.5 Lemma. h on &(a) V ( s ,A ) = V ( h0 s, A ) E S, A la^. d = { A : V(s,A ) = V ( h0s, A ) ES h}. x y Qxy d. d hf(E0)=
-
-
< <
<
<
<
A
<
<
=
[VvAo], A0 E d. &(a) = 0.
V(sv,t,Ao) h 0s v , t =
sv,t
S
=
V ( h0s v , t , Ao) = 0 by 0s, V ( h0s, [VvAo]) = 0. V ( h0s, [WvAo]) = 0.
t on 0s),
0 ES
&(a)
Ao) = 0.
t on
V(s,A ) = 0
0s)lFV(A)).
9=
12.4.2,I ( 9 ) C
=
=
a,
a
k on &(a) V(k 0
0
no 12.4.4
a
I@). Ao) = 0. on F P ( A 0 ) . 12.3.2,
h-1
S
Ao) = 0 V(s,[VvAo]) = 0. 6
12.4.6 Lemma.
sES
[ V [VvoAo]. . . [Vv,Af]. . .I, €4
s'
ES
6
rJn = <&(a) <
V(s,A ) = 0,
n FV(A,) = 4 5 # v. V(s',[ V Ao. . . A ( . . .I) €
A
n = 0.
< a, =
t <6
4 te
148
on V(S,,,~,, A t ) = 0. a
s , , , ~ ~=
&'(a)
9=
0
a
06,E
0
< 6,
a,
12.4.4
he, 5 I(@€) s' .= U {kt 0te: E < 6).
H E - F ~He , E 6 < 6>
v=
Ft E
t'
s' E
0te) : t < 6) u
CU
no
SE
0te))
s,~,~~ E
to,
. . ., tn,. . .
no
HfOnF,,,
Htn E I(9?),F f nE no
I(9).
F f n= 4.
12.4.4
s'
E
V(ke 0s , ~ , ~A~) ,= V(s,,,,,, A ) = 0 by 12.4.5 s' on F V ( A t ) , V(s',A t ) = 0 < 6.
V(S', [ v Ao. . . A ( . .
E
he 0 Sv,,tc
= 0.
t
12.4.7 Theorem. DCH, p a a ( l 7 , a ; QI,, J ( L a ~ ) , ! I> w 12.4.1, 12.3.5, 12.4.6. y @aB(n,a
DCH,
< 6,
on I ( 9 )
h#(9)
u g%=@,y;
< y < p,
w
DCH,
Q~,~a)(Laa)
= [vxoaxl
A
[ A l<€
[ v xo. .
.XI. . .
+
v<€
[ 3 X O . . .xg.. .[ A f
C6 = [[
A
A QxIx~l +[ A A o < e e v<e
o<e<€ v<e
O
A v<€
=
149
AXIOM
0
by by(@) = U {Ow: 11
a
< y}.
by
<
E &,(a), I ( E ) 9 &(a)
<.
by
y
U9
y,
:
U
&,(a).
E
S, no
y by
9},
Lao s on
<
= <&,(a) &,(a)
y.
y = K
< y.
my.
DCH, 12.3.7 E S, 5, no CU
:
s E S,,
a,
9%'&'K
12.4.8 Lemma.
no [Wxo3xlQxoxl] Xa = { x v : v < .$} Cc, E = U (t(x0):8 < E}. E' =
'illt,.
E < K,
u y
y.
5 < K} u
y.
< t(x0)
1
S,, v
E
<E
< 8,
0
s E S,,
< 8 < 6.
&,(a)
E
[ A
o<ee
%Ity.
[ A
S,,
A Qx,.xe]]. v<e
A Qxlxc]
O
S,
v i e
xt
DCH,
y.
12.3.5
my.
(*)K
my. E
Cc
by
Xc C
%T.,I
DCH,
@aS(Upa
u 9 q X p , ;Qz,7a)(Lao) 6
%R., 12.4.2
12.4.9 Lemma.
2'2 ..,
12.4.3
go 12.4.4 y
y =o U ( I ( E )n I ( 9 ) )
< /?. I(9).
I(9) y
9
&,(a)
y
Un,(9) 9.
I(9).
no by
150 N O N - D E D U C I B I L I T Y I N I N F I N I T A R Y P R E D I C A T E LOGIC
{ E E :E < y'}) = E < {Et: E < y ' } E < Ec E < 7'. I(U n , (9 ) ) U n Y ( I ( 9 ) ) . { E E :E < y ' } Et. 12.4.10 Lemma. h is on &,(a) 9 y 9E 9)= { h ( E ) :E E 9}. = &,(a) 9 Un@) y
5 < y'},
U
PROOF:
Uny(9)
h { h ( E ) :E E 9}
< E
=
&,(a).
E
h(E) 9) 9} =
{h(E): E E 9}, 12.4.11 Lemma. h on <&,(a) 0C a a E E &,(a) f on a
<)
< h(U 9)
E E 9. 9))= h ( E ) : E E 9} =
U n y I ( 9 )C no r,(E) = (I@) n I ( 9 ) ) . 0 h,,f
hY,f(E)= h(r,(E))^f 0 ( E - rY(E)) on h,Jl Un,I(9) =
HnF HnF F F'. PROOF:
<
<
HE H'^F'
H
&,(a)
UnJ(9).
< H'
F E By(@). F =4 H = H'
by 12.4.10. 12.4.4
Eo E Un,I(9) r,(Eo) = r,(El). I(E1) n I ( 9 ) r,(Eo) ry(Eo)< E' E'
< Eo
r,(Eo) = ry(E1).
Eo
< r,(E1) Eo Eo < r,(El).
<
Eo E l Eo r$ U n y I ( 9 )
< ry(E1),
r,(Eo) < r y ( E l ) E' E I ( E l ) n I ( 9 ) . Eo, E' Eo E' Eo Un,I(9) E' r,(Eo).
< <
12.4.4
151
12.4.12 Lemma.
h
on
V(s,A ) = V ( h0 A )
&,(a)
:
all E S, 12.4.11
12.4.5
12.4.4. 12.4.10 (h-1). 12.4.13 Lemma.
A
Un,l(9) C 6
6
my.
=
< a,
12.4.6will
Un,I(9)
no
( H E :t
v < v' < y , Ha = U 9 6
< y>
H t E UnyI(9).
I(9)
y
HE
I(9). I(9)
y
y.
by
t < v'
< E,.
<
9t
y
< 5.
El
p <5 Ht, 5 He-1 < He
91.
E,
< E,,
<
v'
<
< p,
= U 9t
9 6
t-1
H,
< E. E , < Ht-1 p < 61 < He-1 < Hq.
12.4.14 .Theorem /I> o
v
v
t-1
< He-1 p
,u
U
Hv < Hva
y,
< 6.
,9~
DCH,
u
DCH, 12.4.7, 12.3.5, 12.4.8, 12.4.13.
=
H E,.
(a,
o
H
< y < /?, y
13
THE DEFINABILITY O F T H E I N F I N I T A R Y FORMAL S Y S T E M S
13.1 The Metalanguage for (a,p , o, n)-Formal Systems
p
p
p
on a, p, =0
o
0,
n
a
< p < a, o p
Q
o
Laoon (o, (a,,9, o,
MLa
< n < a. by
XO,
--
. . ., xn, ..., n < w.
lb, rb, ii, ,E, cj, Zj,
6 < a. IOP, BPR, IPR,
I V , I C , BOP, - =, E,
OP, PR. W(L, g)
=
(a,p, o,
G6deZ-numbering) on
T,
Q,
L Q
R
0
E
A*)
B g(V) IV IC BOP
L
el
g(=)
IOP BPR IPR -
{
L}
:Q
L}
on T a
E
OP PR
L}
{
L} L}.
154
C(x0,
. . ., xn-l)
L,
V
, s,
?( L,,
A C(x0,
. . ., ~
.
Qn,
. . ., xn-l)
C(x0,
Fm} Ae)
A) =
I),
Q".
A,
Fm.
~ - 1 )
A
[3xoC(xo)]
'2R
=
xo
D (D,0, C u {(Ec)},
-
Fm'
D,
s C(x0)
A. A
C(x0)Il
C(x0) '2R, A ) = Vm(s,A e )
R>
L:,
c)
[VxoVxl[[C(xo)A C ( x l ) ] --+ xo = X I ] ] C(x0). LL, E,
L,
E
by
E
[Vxo[xo
A
. . ., xn-l
xo.
L, R , = { ( d o . . .dn-l> : ( d o . . dn-l> Fm' =
L:,
=
Fm. p
by
[ ~ x o .. .x n - l y l j 5 [ Q . ' ~ x o .-.~ n - 1 ]t)C(XO,.
-
- 3
xn-1, y ) ] ]
[Wxo. . .x n - l p y C ( x o ,
. . ., xn--l, y ) ] ] [vxo. . . x n - i y z [ [ C ( x o , . . ., x n - I , y ) A C ( X O ,. . ., xn-1, 41 + y = z ] ] Fm X O , . . ., xn-l, y C(x0, . . ., xn-l, y ) . L,
L:,
0,
q.~n, e
Fm'
=
( d o . . .dn-le>
(D,0 u {(~nO,>},
R),
L:,
A 9"
A. OC
( d o . . .dn-l> C(x0,
. . ., xn-l, y ) ,
A ) = Vm(s,A,) s D, A, C(x0, . . ., xn-l, y )
Fm.
13.2 The Definability of Fundamental Notions of Set Theory L,
(o,
-
=, E, a
%a =
( T a E>
Ta a,
on T a
D
155
THEORY
on
Ta a = y+,
By $ Ta. --
y ETa
13.1.1
13.2.1 Definition. y = xoxl 13.2.2 Definition. y = [xg x XI] 13.2.3 Definition. y [xoTxl]
y = (xo,XI). y = xo x X I . y XI.
xg
13.2.4 Definition. n XO. . .xn-l] -13.2.5 Definition. y = 13.2.6 Definition. y 13.2.7 Definition. y
y
y = {xo, . . ., xn-l}. y
xo
xo.
-
=0 --
y
=
4. y
=
xo.
13.2.8 Definition. 13.2.9 Definition. 13.2.10 Definition.
xo xo
xo
--
13.2.11 Definition. y = [ x o 2 x 1 ]
xo
x1 E
13.2.12 13.2.13 13.2.14 13.2.15
Definition. Definition. Definition. Defination.
C x1 -
xg = E
13.2.16 Definition. -
- -
y =
xo
xg
y = xo(x1)
4 xo
xg
xo 1
xg
c) xo
0
C XI.
xg
xg
by
-
xg
--
=
ti
0.. .n -
by
t)y
-=
-
--
...[
-
xn-l]
=
156 D E F I N A B I L I T Y O F T H E I N F I N I T A R Y F O R M A L S Y S T E M S
--
-
fi]
--
13.2.17 Definition. y
=
-
xo. . .xn-l] = fi
%a.
y = U xo.
U xo
a
_ - _ 13.2.18 Definition. y = xg u x1 t)y = U ~ O X ~ ] . -13.2.19 Definition. y = ll xo y = tl xo x g # 4, 4 _ -- 13.2.20 Definition. y = xo n x1 t)y = n xOxl]. - -- xo -D [[xo = U xo] v [xo = U xo U
G XO]]]
%a.
13.2.21 Definition. y
-
=F .
xo
t)y
--
-
= xo U
xo.
:
by S+O=S
+ S(&) = s(6 +
6
&)
U{~+E:[<E}
+
y =6 E
E
Z(E)
z
E
= y,z(0) = 6 so) E
E E #0
F
z(s(6)) = s(z(E)) = UE
z(E) = Lmm. y = xo
by
13.2.22 Definition. y 1xo xo,
4
XI
-
y = a]
xo
A
= y A [zof0]
A
T x1 -
T
y = xo
XI]
xo A Al(z) A
x1 t)
+ z A x1 C A~(z)]]] Al(z)
xo A -
z
+
XI
XI]+
A[zoT~~]
A2(z)
(i), L,
by
13.2.23 Definition. y x1
C$
13.2.24 Definition. y a
-
-
=~ 0 . ~ 1
--
= Zxg
4
y y
=~ 0 . ~ 1 = Zxo
xo,
xo
DEFINABILITY
157
THE FORMAL SYSTEMS
2.2.8
- 13.2.25 Definition. y = ~ 0 ~ tx)1[ 4 ( x o , x l ) - + y = 01 A [C(xo,xl)-* ---yA y= xo XI] A [WtA(t, y, XO, XI)]] C(x0, XI) = xo A S T XI] A(t, y , XO, x1) = [t 5 xo + - -[yoft] = [xooft]] A [t Z x1 -+ xo t]] [xlvt]]. xo y = ^xo
+
z
t t
z(s(t)) = z(t)^xo(t). z(t) =
s(t) E
t
tE
=
z(0) = 4
y=
E
L,
13.2.26 Definition. y
--
=n x ~
y
xo is
= n~~
13.3 The Definability of the Formal Systems (a,p,
MLS
0,
g
L
L
A
L.
A
by
ML"
W(L, (x)
13.3.1 Definition.
(c)
x
xo t)[3x1[[lCxl v IVxl] A
--
xo =
13.3.2 Theorem.
%Ill.
L
3, 3.1, XI
L
E E E Rng(x1)
x2 E
xl(x2)
: x&z)
= ([)^~~(x~)~
xl(x2) = ( [ q ~ ) ~ ( ~ x g ) ^ ( ] )
xz. 'p
p
158
0<5
< 0 , x3
5
=
q~
x3
o
by xz) =
5 x2 A x4 Z x~ A BOPxg -
xz) =
x3 A
-x3 C
A
x3 C
x3 A -E5A
E
o = a.
x3 will
o =a
A (XO)
no
=
T,.
a
A
IOPxe --A
A
--
ZZ
o
--
A
- --
x ~= )
0
- -
=
A
1x3
A
x1 A
xo 5 v
5
-
xi +
xz) v
v
13.3.3 Definition.
by -
xo
xo
L. by
13.3.4 Definition. 13.3.5 Definition.
xo
xo
-
xo
13.3.6 Theorem. 3,N , %,
xo
L L
= [[A [Ad --*
a
. [A A o . . . A t . .
.]I,
0<6
< CL.
159
gl
A
x1
+ [>^
A = <[[
xz
x3
x ~ ^ (--+
x1
= <[>^xz-<-+>-
x4
ML" C
-
A
- --
+ xo - ---
A - x ~ , C
01 A
x1 A
Vl(x0) t)
-
x1 --+
x2 A
x3 rbitb"xz^
- - A(x1, x 2 ,
7x41
-
XI A
- --
=
0
9, N ,V2 13.3.7 Theorem.
0
<6
< y < ci 17y
L
[V [A
L
/c
<APv:,u < 6, v
<
b)
0
< 6 < a,
u
Il,, y
17y
< y < a.
0
L
[V [A P-=Y
...A,,. .
...[-[A
= [-[A
V
p
< y , v < y>
A
A
x1
y
xz
y
y
x3 E y
x~ E y
E
x4,
x4 =
<[1 [
=y
=
y,
x2
= <[ -I [
MLS.
by y by y #0
x1
13.3.8 Theorem. [A [ P
yy
= {gc: 5
v
--+
V
<2
y),
[
v t i 2 exp Y
[A
P4Y
2
y
160
:
A
x1
y y
x2
x3
E y,
on y
x z ~ y . y
y
= ([ ' I[
x5 = <[
xg E
<['I[
y
y
y
x4
x7(x8) =
on y
y.
x7
xg
A =
x2 E y.
xg(x2) =
-+ [ 'I[ A>^(^x~)^<]]]>.
([ [
L
13.3.9 Theorem. 895:
. .] +
. .[Tt
[[A [To
[[QTo... T E . ..] + [QTi,.. .TL..
Q
< u. n
=
0
..T i . . .>
< q < n,
of
A x1, x2
+[ [
A = ([ [
+[
x3 x4 E
= ([>^~1(~4)~(~>^~~(x4)~(]>.
n
=u
9,%A?,, 9V.8, bound
13.3.10 Lemma.
n
161
_.-
-
t)
- -
r b " ~ A]
-
x4 A 1 x 4
--
A ~ [ X IA
v
[XZ
lb q = x 4 = ~ 5 ~
0A x4 5 B A 5 x4+ 01 A- x5 A - x3 A S G x s A x4 xz A x2 5 x3 T 2 + x4
-
A,
xo
X
xz
-
<XOX~XZ)
A.
xz E
= ct
x4
ML"
bound by X
5 81
bound
9,
9.1.
13.3.11 Lemma.
5Xz
5 &X o
t)
5
A
A
xo Set1
C(XO,X I ) =
xo
5
A
xz =
xz =
x1
{L: L
xo}
xo
4
x1
13.3.12 Defilzition.
XI
= FV xo
t)[ l
xo + x1 5 U]
A
xo --* Wx2[xz F x1 t)
A
xo
x1 FV(A).
mxo
x1
all 62,
SFT A
= ^<E,^fA(c) : v
< @>^En
< a>
all X
A Ee(,) = All8(,)- (AIb^
162
( E V : v< u
+ 1).
13.3.13 Definition. [C(xo, X I ) +xo
x2
xg
-
x1 t) [ - C ( x o ,
C(x0, X I ) =
xl"x21
x1)
+ x2
x1 A
01 A
x3 A
xo
x1=x31.
13.3.14 Lemma.
xo
x4
x4)
+1
x5
= x o l x 4 ( s ( ~ )) ( . O ~ X ~ ( V ) ^ < ~ O ( X ~ ( V ) ) > ) s(v) E x ~ ( v= ) x o [ x 4 ( ~) xolU Rng(x4l~) v = U v E Dom(x4)
XB(S(V))
x4) =
+ 1,
+ 1.
4.
:
13.3.15 Definition.
-x5 =
~ 0 x 4
x5
x4).
13.3.16 Lemma.
A , x1
xo
on
x2
SFFA,
SF(x0, X I , x 2 ) X = ( x : g(x) E X I ) f = SF(x0, X I , x2 ) +. SF
x2
x1
0g on X .
:
g 0 SFTA = n<x5(V)^x2(x0(x4(V))) : v
< (T>^x~(u)
x4 0
XI),
=
x5 =
13.3.17 Definition.
x4).
x3
Z SFxoxlx2
x3
= SF(x0, X l r x 2 ) .
13.3.18 Theorem. L 9 1, 9 2 , V X , , 9V%,, o < y 9VX.,.
V%,.
9 2 , W%y 9VX,.. A = [[VvAo] + SF,R"p@)A0], Ao xE bound by FVf(x).
92
no
T
< a,
x
E
FV(T)
x
163
D E F I N A B I L I T Y O F T H E FORMAL SYSTEMS
T. FVTxo t)
x1
-
01 A
92
- -
Zb Ib x1 A
-
--
rb A
-
XI +
E x3 A E
xo
- --
92(xo) t)
XS+
E
by
A
- &F
EB A
x3 A
x1
5
A - -
+
/? = u
0A xz A
A
Z
FV(T)
t)
E
1 ~ x 2A
-
-
XZFVT[X~~~[X~~~X~]]X
1E
x
B
9%Sy
A
. .[
=[A C
VO.
. .v*. . .
. +
r
[
~
V
... O ~5 ...[ A
€
=
+
< 6.
v
n
= (b
-
A = [[A
A0 ...
€
t#v
- .vv. . .
n FV(A,)
. . v c . . .[-[A
-
+
Ao. . . A t . .
HS,
x1
y
x~
x&)
y,
/?,
n
=
+
t # v,
= -I
[
x3
y
5 > 0,
= <[1[ x3(5) =
<[Wn(^xzlt)^<~1 [ MLa.
/? = u
on
xz(5) 9%'Xfa
x1 13.3.19 Theorem. QD,
o
< y < u,
y by
G
,
GI,.,
,
GD,.,
cr)
< y < u.
GI,,,
164 D E F I N A B I L I T Y OF T H E I N F I N I T A R Y FORMAL SYSTEMS (~0x1)
Q(x0,
Q(x0, X I )
xo, x1
C1
CO,
9. 9 = no,,. CO [ V A @ . .Ac..
C O by
PROOF: COby
9
..
[V ecv
= q4
.
t#
Q
n
n FV(A,) =
Y
< 6,
2 FV(At)n U .<€
..
[-[A
. .
CO=
.
Co by 9
=
x~
y y,
x g , x4
/?,
< t,
= q4
= t# 2 F V ( x z ( t ) )n U
n
Co = <[ [
n
x5
y
= <[
C1 =
xs(O) = <[1 [ W)^
y
x6
<[7 [
> 0, x6@) = <[1 [ 1[ /? = OL
on
13.3.20 Theorem.
!&&:
9)(L)
L.
11.1
PRV(x0)
PRV(x0)
xo
PROOF:
[ V>-
' I
ML". xz(t),
%Q(L,g )
MLa
xo
13.3.6, 13.3.7, 13.3.8, 13.3.9, 13.3.18, AX(x0) xo xo
13.3.19
AX(x0)
S(x0,XI)
S(xo,
XO,
C O , C1
-
v Az(x1,
A
x1 -P
v
G
v
9.
CO by
PRV(xo) = 3 x l [ C q x1 A V x ~ [ x Gz A
(~0x1)
x1
v
G x3 A
xo G
-
x1
CHAPTER 14
INCOMPLETENESS I N INFINITARY P R E D I C A T E LOGIC
1960, (y+,
on LO Z, La
a (a,
8,
(a, a, a,
6 xc, 5
< a,
s.
p
< a.
g
LO
La
-
-
I
1, I, 'I,+, A , v, I I l l I I I
0 3
6 9 12 15 18 21 24
=>
I
Z, sq,
.............. s...
0,1,2,
I
I l l
1 4 7 xo,
6.3
X I , x2,
I
l
l
2
5
8
+1
.............. X f ... I 5.3
+2 LO
%a = a,
La
< T a E) '22a =
167
0,
. . ., 6, . . ., E>
<s(xo) . . .s(x.$). . .>.
. S t . . .>) = <So.. .SF.. .>, X O . . .xe. .
-
[%a
s(Xd)
=
La LO
La.
MLa,
13,
La by
< o, -
6 < a, e(Zb) = 4, e(rb) = 12, e ( E ) = 51, e ( i ) = 81, e(E) = 21, e(4) = 24, e(G) = 0. y MLa s Ta, %R(LO, "t(La, %Ra. A MLa E o ( A ) , Ea(A), Sm(L0, %R(La, g) : e(xn) = X n
e
rt
=6
= - - -
., +,
13.2.
V(s,A )
Ta, A MLa, V(s,A ) %R(LO,g) V ( s , Eo(A)) %RJn, "t(La, V(s,E a ( A ) ) m a . Eo, Em M La by : s
Et([lAI) = Ea([Ao + &([A Ao. =
Eo, Em.
=
[Ez(Ao)+ Ea(A0). * i = 0,
= [A
A
168
I N C O M P L E T E N E S S I N I N F I N I T A R Y P R E D I C A T E LOGIC
Eo(A)
Ea(A).
on
V(s,Eo(A))
%V(La,g)
V(s,A )
%Va
V(s,A ) %V(lO, g) V(s,E a ( A ) )
%a.
LO
PRV(x0) E
Ta,
g)
PRV(x0)
PRV(xo), %Va by
S
Eo(PRV(x0))
'9Xa
by
go on a.
14.1 The Basic Undelinability Argument
%Va.
d
ma.
by
A
[45]
m
on Ta --
m(S) S
.
S(0). . . S T ) . . ]
4 1. . . (S(5)
6 (S(0) 13.2
1.
.
M(xo, -- xz) 2
+
[x2 of
= [-C(xo)
--
+- -
[xo O f x 3 1 . 3 W x 3 [ ~C 3
m
=, 5
102, by A
+ xz
01
A
[C(xo) +
[ x_z q-o ] 5 9 A [ ~ ~
xo]] = 12 A
+ 111
Wx3[x35
C(x0)
-
-
xz
-
x2
A
o f6A - xo + [ x of~ 2
+
xo A 1 x 0
x3]
0A
xo -B O r d [ x o O f x 3 ] ] .
SF
13.3.16. g) by
E, x g , X I , x2, x3
(SOS~SZS~)
S3
%Va by = SF(S0, S1, SZ). Call
169
SO
x1, x ~x3). ,
on
Sz
S1
La, S3 = g 0SFT A f = g-10 SZ 0g on X .
S1 xz, x 3 ) X = { x : g(x) E XI,
14.1.1 Theorem.
A = g-10 SO, La
by
A
%Ra.
N(xo, xi)
xz)
=
21,
A
La.
C
(SoS1)
La,
S1 = g 0 SFPIC,
9' 2,. SFrlA SFP'A
B
f(x0) =
--
2 X2]],
Xi)]].
SO
--N(x0, XI)-YXa gC(0). . . .].
D(x1) La A = [Wxl[N(xo, XI) + -,D(xl)]]. -f(x0) = gA(0).. .g A ( t ) . . .]. B g 0A A. B ma A. B %Ita
A B= 'illla
ma.
no
D(x1). 14.2 The Incompletenessof Definable Formal Systems when a Nonlimit, /?= a
r
IJn,
La
d
5,.
LO
r
by on
A
La
YXa
6 by
8
< LY LO.
La
EQo(x0) = [Wxl[lxlZx~]] 0 <6 < EQd(xd) = 3 x 0 - - - X V . [[wXd+l[lXd+lc XO]] A [ A [Vxd+l[xd+l XV t) l
S
EQd(xd)
5,
S = 6.
170 I N C O M P L E T E N E S S I N I N F I N I T A R Y P R E D I C A T E LOGIC
6 < a, 6 # 0, SEQd(xo, . . ., x6, . . ., xd) = vxd+l[xd+lzxd tJ [ v [vx~2xd+3[[vxd+4[xd+4 cxd+2t)EQ&d+4)]] A P C6
[Vxd+4[xd+4gxd+3 t)EQ&d+4) v xd+4 xt]]] -b Vxd+4[xd+4zxd+lt) (so. . .St. . .SO) sEQd(x0, xd+4 xd+2 V Xd+4 xd+3]]]] . . ., x t , . . ., Xd) Sa Sd {{E}, {t,St}} < S, only Sd = .Sc.. .>. r' all [vxd[xd ZE B t)EQd(xd)]] [VXo. . .X t . . .Xd[Xd = xo. . . X € . , . t)SEQd(X0, . . ., x t , . . .,xd)]] 6 < a. 14.2.1 Lemma. A brvr, A $aa(La)PROOF: L"
I"
1332&.
A
1332,
La
<xc)
G.
7
F on LO
La
=
E< F(<xt>)(xo) = [xo F(<&)(xo) = EQd(xo), 6 < a F ( [ q To. . .Tc.. .])(xo) = [ V xp0.. . x P t -. .[[ A F(T€)(x,,)]4 €ca
all
SEQd(xpo... x p t . . .xo)]], ( x p e :6 < Tt T = {T: T
6 4
xo.
Vxo[F(T)(xo)t)xo
La
all
$aa(La)}.
La
[q
(1) I-rnxo
T
SEQd(T0, . . ., Tt,
t)
I",
. . ., X O ) .
r]
T
T.
T F(T)(xo) = [xo r ] . T = <6>,[F(T)(xo)t)xo TI r'. T by 11.2.9 T= To. . .Tc. ..] Tb E T all E
T.
< 6)
< S.
171
INCOMPLETENESS O F D E F I N A B L E FORMAL SYSTEMS
< 6.
all 5'
Ta
(3) kp*F(Tf)(Ta) c) T f
kp*[ A F(Ta)(Tf)] by (3) M a
(4) kpF(T)(xo)+ xo
T
by
(1)
(2).
1 1.2.9, and
-
(7)kr.Xo
Xpo.
.
.XPe.
..
r' SEQd(Xw, . . .,XPe, . . ., X o ) .
t)
on xp0, . . .,
(6),
T + F(T)(xo).
(8) kpxo
T E T.
(4)
EoCo. . .EvCv.. .E,,
A
r'
ma,A'
A
< a>
Cv [To Z T1]. C: = [3xoxl[F(To)(xo)A ~ x o x ~ [ F ( T o ) ( xA o ) l-p Vxo[F(Ts)(xo)t) l-rrCv c) C i all v < u. kr#A t)A' A' = A' LO no
6 92,.
LO,
S,. l-r,pA. 14.2.2 Lemma.
r' 'Baby
PROOF: by
all
q.
6, 6 < a,
[To TI] F(Tl)(xl)A xo E X I ] F(Tl)(xl) A xo Z X I ] xo E Tg],i = 0, 1, EoCi,. . .E,C:. . .E,,.
'B,.
La
E
r. 6,
La.
r'
no
LO.
A'
bound
EQ&)
sq
172
< 6) U
{x": v
d
by X,
K
26
K
#6
+ 1.
bound = =
. .[-[A
.Xv. .<8
6 # 0, K 2 6,
K
#6
CV=
-
-cv-
+ 1,
XV] --*
cv =
# 1.
K
c:][c:*
+
ci =
A
.
0 < < 6, v = 6,
--*
0
0
< V < 6.
tlCV
#(S)
4
p(S) = ( Y - 3
q'(S0,
0E
Q'(xo, X I ,
Xz
x2)
13.2. = S2
S2 # So Q(X~,
=
A
q(S0,
+ 1, 4
S
6=
by
XQ A
T 21 12
A
+
1212111,
C,,
S2)
0E q
01 A
X3
E
SO
+ 1, SO
6=
E S2
+ 1,
by wX4te'(X0,
sg[[XOT - -
--
[[X1cXO*XQ=[Seq12924[[xot
--
<S) 4,
O r d q A O C X A~ X ~ Z X O
S2
x 2 , x3) =
<S)
-
01 A [D+
- xz
=
v
P(xo,X I ) of Cg, 1, 4
+[XjOf 9 i39 " X O T 1 1 3 F 21 0 CSeq4 9 Is9
D
K
+
E
( x V :v
+
3 T 21 12 12 ---3 [ X I : ~ T 21 12 12 12 1211 A
189 [[xo
X0
--+ X 3
+
I]TQ T 21
173
T
x1 by x 2 ,
D
=
-xo A
x1 A
x2
A
Xz
Xo
r(S0, SZ) SOE S 2
+1
1A
A
T
+ 1, 4
-
A 1 x 2
Xo
+
EQso(xS,) SO,SZ
# SO
S 2
Y
[D+[[Xo 0 A V [TKO 0A A [ x s f 01 O A
--- 01 A
R ( x o , x 2 , X 3 ) = [7D+X3
-
15 9 5 3 [ X 2 : 3 T 21 12 12 -
[xoT
A
-+ Q(xo,
-Z [KO
9
9
3 2 12 1s
D
~ 2 [x5 ,
A A
07 + T 21 4 i39 [[xo7
ZI“ x o t 1212 13
xz A xo E x 2
xo A
=
SEQd(x0,
T
A
92456
XQ
- -
9 15 9
~ 4 -
T 21
-,xz 5 xo +
. . ., xc, . . ., Xd)
6
Y
no
EQ((%d+4)
SEQd(xo, . . ., xt, . . ., x d ) ,
EQd(xd)
r‘.
r
14.2.3 Lemma.
6 no
LO La.
%Ra
S, by 14.2.2
r u r‘
F u r’
La
paa(La) MLa %R(La,g)
fl
r‘. PRV(x0) B(xo)
V
%R(La,
&,(La)
PRV(x0)
by
13.3.20 by
174
INCOMPLETENESS I N I N F I N I T A R Y P R E D I C A T E LOGIC
FU
r'
E,
E a ( [ B y ] )= G
C(e(y)),C(x0) Ea(PRV(x0)) 14.2.1 La
1332, by
r u r'
!)3aa(La).
A,
1132,,
14.1.1
r
C(x0) no 14.2.4 Lemma. S, up
PROOF:
LO
no
B
PRV(xo)
%a
MLa
%(LO,
LO
Val(x0)= Eo(PRV(xo))
LO
A
%a.
LO
It [B -P A ] , 'I
%a
1132,
---
.
-
C(XO) = W X ~ [ X=~[[SqggB(O). , .gB(t).
--
. -
--13.2.25,
r. I'
no
B
no
PRV(x0).
S, up
a
1.1.5, LO
no
14.2.5 Theorem. no
a,
Lo
no
1132, by PROOF:
La. 1.1.5.
14.2.4,
no 1132a)
LO
t-pA
LO.
(a,
A
Sa.
a
r'
LO
all
A
LO,
REFERENCES
C., O n the Representation of a-complete Boolean Algebras, 85 pp.
A., Introduction to Mathematical Logic, 1956.
E.,Infinite Formulas in Complete Theories, 23, p. 361. K.,0ber Formal Unentscheidbare Satze der Principia Matheund matica und Verwandter Systeme I , 38 pp. GODEL, K.,The Consistency of the Continuum Hypothesis, 1940. Models of Languages with Infinitely Long Expressions, (
p. 24. G.,Eine Bemerkung zu Henkin's Beweis fur dze Vollstandigkeit des Pradikaten-kalkiils der Ersten Stufe, of
[ 101 [111 [ 121
18 pp. 4248. The Completeness of the First-order Functional Calculus. 14 pp. L.,A n Algebraic Characterization of Quantifiers, 37 pp. L.,Boolean Representation through Preositional Calculus, 41 pp. L.,The Representation Theorem for Cylindrical Algebras, Mathematical Interpretation of Formal Systems, in 1955.
176
[ 141
L.,Some Remarks on Infinitely Long Formulas, Infinitistic Methods, on pp. Zur Axiomatik der Verknupfungsbereiche, 16 pp.
C.,Independence Proofs in Predicate Logic with Infinitely Long 27 pp. Expressions, C.,A Note on the Representation of m-complete Boolean Algebras, of 14 pp. The Completeness of a Certain Class of First-order Logics with Infinitary Relations, 7 p. 363. C.,Introduction to Metamathematics, 1952. Une Gdn&alasation de la Notion de Corps, 9, 17 pp. The Theory of Transfinite Recursion, 67 pp. S., A Formal System of First-order Predicate Calculus with Infinitely Long Expressions, 13 pp. C. A. Some Theorems about the Sentential Calculus of Lewis and Heyting, 13 pp. A., Proofs of Non-deducibility in Intuitionistic Functional Calculus, 13 pp. A,, A n Undecidable Arithmetical Statement, 36 pp. S., Distributivity in Boolean Algebras, 7 pp. S., Distributivity and the Normal Completion of Boolean Algebras, 8 pp. S.,A Generalization of Atomic Boolean Algebras, 9 pp. S.,Representation Theorems for Certain Boolean Algebras, 10 pp. Algebraic Treatment of the Functional Calculi of Heyting and Lewis, 38 pp. A Proof of the Com+leteness Theorem of Godel, 37 pp. Algebraic Treatment of the Notion of Satisfiability, 40 pp. On the Lattice Theory of Brouwerian Propositional Logic,
177
REFERENCES
189
L.,On Free Xc-complete Boolean Algebras, 38
pp. On the Metamathematics of Algebra,
1951. A New Characterization of a-Representable Boolean Algebras, 61 pp.
of
D.,The Independence of Certain Distributive Laws Algebras, pp.
ifi
Boolean 84
The Sentential Calculus with Infinitely Long 6 pp. Boolean Algebras, und 1960. E. C., A Distributivity Condztion for Boolean Algebras, 64 pp. E.C., Higher Degrees of Distributivity and Completeness in Boolean Algebras, 84 pp. Axiomatic Set Theory, 1960. Sur les Classes Closes par rapport d Certaines Opt%ationes <?mentaires, 16 pp. Grundziige des Systemenkalkiils, 25 pp. Der Wahrheitsbegriff in den Formalisierten Sprachen, vol. 1 pp. Undecidable Theories, 1953. Remarks on Predicate Logic wath Infinitely Long Expressions, 6 pp. Some Problems and Results Relevant to the Foundations of of Set Theory, pp. Expressions,
INDEX O F SYMBOLS
SET-THEORETIC
@* Y) n V N
ns f-l
flS
fog
S x T ST
XI XVI XVI XVI XI XI XI XI XI XI XI XI
9s
D( ...> Lim
<...> 0 0
u+
exp Ta
z<...> ^< ...>
<,<
XI XI XI xi1 xi1 x11 x11 x111 9 10 10 12
ALGEBRAIC 7
A V
A
XVI XVI XVI XVI
v +.
0 1
XVI
55 XVI XVI
FORMAL
1 A V
A
V
30 31 31 30 31
--+ t)
Q 3
30 31 101 102
180
I N D E X OF SYMBOLS
31, 31, 39, 39,
103 104 122 122 65 65 108 41 76 134 134 134 134
XVI 33, 116 57 128 14 40 120 41 131 120 131 119 18
72 86 40 30 101 101 167 167 153 153 40 40 41 41 41 119 120 122 122 119 29 18 26 88,95 95 74
SUBJECT I N D E X
105
-
-
41 122 10
18
72 140
-
81 41
72
-
-
23
-
17
-
bound
--- on
17
-
31, 104
- 31. 104
- 56
40
31, 104
119 1 19
- by X
47 -
84
32, 84
-
7
111
40, 41 122
88 41
47
Wt
-
-
155 158 7, 169
-
-
---
7
-
7, 39, 122
120 120
182
SUBJECT I N D E X
--
41 Chang's -
18
--
41
30
--
17, 72
72, 101 12
3
40 XVI XVII XVII 39, 121 39, 119 56
a-
2 a-
-
-
18
-
30
8
35
-
73 101 in b 1 4 0 73
-
-
30 72, 101
XV 84
-
XVI XI11
142 120
B
a -
XVII 7, 153 7, 154
XI XI11
XVI
6, 153
a
XI11
?.tf 'iUl
74 103
36 103, 142
-
142 41
a--
fa-
XVII XVII
---
--
x
XI11 XI11 107 107 56 120 120 17, 7 1
hh-
-
26, 84
101 71 XV 25, 77 79
SUBJECT INDEX
-
183
25
142 77
31, 104 3 I04
30 30 30
-
86 XI1
XI11 71
-
39, 122
A
-
71 71
39. 122
XVII 76
31, 104
---
18 73
-
7, 39, 122
-
XI11 XVI XVII XVII XVII
-
26
a-f a--
-
IV XVII 77
26, 27
83
-
-
15
88 bound 94
sys97
XVIII XVIII
af a-
17 3
r
-
31, 103
31
F 103
74 3
b
36
