Chapter 11 Infinitary Predicate Logic

CHAPTER 11

INFINITARY PREDICATE LOGIC

11.1 Description of the Formal Systems for (a,p, 0, n)-Predicate

Languages

The restrictions on a,p, 0 , n are assumed to be in force: a regular infinite, p = 0 or w < p < a, n Q a, o regular infinite and o < p if p singular, o < a if p regular. For an (a,p, 0 , n)-language L, the basic formal system @&g(L) has as axioms all substitutions to L of the axiom schemes of &, 5.1.1, together with all formulas of L of the form 91. [Vv[Ao+ All] + [A0 + [VvA1]],provided that no variable

of Rng(v) is free in Ao.

9 2 . [[VvAo]+ SFF*("'Ao], provided that A0 has no free occurrence of x E Rng(v) bound by FV(f(x)).

The function f in 9 2 may be any function on Rng(v) to terms of L. In addition, w3,p(L) has as axioms all formulas of the following forms where the Te, T i are terms, y is a special two-place operation symbol, rp a (-place or infinitary operation symbol, P a special two-place predicate symbol, Q an q-place or infinitary predicate symbol : 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.. .I]. 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].

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The rules of inference are modus ponens, conjunction, and Generalization: From A0 infer [ V v A o ] .

If 2 is any set of a-propositional schemes, '&@(Z)(L) is that system like WP,b(L) except that instances of 2 in L are added to the set of axioms. 11.1.1 Laws of Independent and Dependent Choices. Let y be an infinite cardinal. Then a formula A of L is an instance of the law of y independent choices if and only if A has form gS7.

.

.

[ A [3voAo].. [ 3 v t A e ] . .]+ t
[3

00.. .v€.

. .[ A f
€
A o . . . A f - ..I],

provided sequences ve have pairwise disjoint ranges and no variable of Rng(v6) is free in A , for v # 5. Similarly, a formula A of L is an instance of the law of y dependent choices if and only if A has form M Z y .

[ A [ZIVOAO]. . .[ V I
V C €

VO.

. . ~ 9...[ 3 v e A t ] ] .. . ] --+ [ 3 V O . . . V € . . .[ A A o . . . A t . . .]I, €
C
provided sequences vc have pairwise disjoint ranges and no variable of Rng(v6) is free in A , for v < 5. Note that L only has instances of %Sy or 9%'GZif?+ y ,< a and y < /?.It must be possible to quantify sequence " of variables. If Z is any set of a-propositional schemes or schemes %S,, or 9%SY with y < CL n p, Va@(2)(L)is that system like WOag(L)except that formulas of L having forms of Z are added to the set of axioms. 11.1.2 Rules of Independent and Dependent Choices. Let y be an infinite cardinal. When /?is small relative to a , the laws W X y and 9%XYare not very useful because of the restriction y < @. They can however be changed in form to rules of inference which will apply to all the languages with y < a. Rule of y Independent Choices: From [ V Ao. . . A t . . .] infer

.

€
[ V [WvoAo]. .[WvcAe].. .] under proviso of %?GZ?,. t
Rule of y Dependent Choices: From [ V A @ .. . A t . . . ] infer

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121

[ V [VvoAo]. . .[ 3 w e [ V v , A ~ ].].]. under proviso of 9 W X y , prob
vided also that A t has fewer than?!, free variables in and that

U {Rng(v,): v

< t}

Rng(w5) 2 F V ( A F )n U (Rng(v,): v < E}. If Z is any set of a-propositional schemes or of schemes WX,,or 9%‘Xy with y < a n B and if SZ is any set of rules of y independent or dependent choices with y < a, then @as(Z; Q)(L) is that system like @a@(L)except that formulas of L having forms of Z are added to the set of axioms and rules of SZ are added to the rules of inference. 11.1.3 Theorem. Let L be an ( a , @, 0 , n)-predicate language. Then instances in L of valid a-propositional schemes and of schemes 9, 89, % ‘ i f y ,9WX,, are all valid. The rules of modus ponens, conjunction, generalization, and of y independent and y dependent choices not only preserve validity, but preserve the property of holding in a given model. PROOF:Suppose formula A is an instance of valid a-propositional schemed. Then if W is the set of all propositional symbols i n d , A has form S y d where f maps W to formulas of L. Let s be any assignment to any model Cm of L. For symbols A t E W let s’(At) = V ( s ,f(A6)).Then s’ is an assignment of W to truth values 0, 1. Therefore s ’ * d = 1. Since V at s is a homomorphism on the algebra of formulas of L and s’* and S y are homomorphisms on the algebra of schemes, it follows that V(s,A ) = s ’ * ( d ) = 1. Hence A is valid. Formulas of forms 91, 9 2 are valid by 10.1.3, 10.1.4. Formulas of forms 9W%‘,, and %?Xy are valid by 10.3.7 and 10.1.3. If Cm is a model and s an assignment making a formula of the form of the conclusion of the rule of y dependent choices false, then formulas [3vt[+le]] all take value 1 for assignments RepbPs, Y = U (Rng(v6): 5 < y}, by 10.1.3. By 10.3.7, [V A o . . . A t . . .] does not hold in Cm. Hence this rule preserves the property of holding in Cm. A similar argument shows that the rules of y independent choices also preserve this property. This is obvious for the other rules. A formal proof in one of the systems ‘@ag(Z; SZ)(L) is a sequence of formulas having length less than a, where every formula is

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either an axiom or follows from formulas earlier in the list by a rule of inference. If A is the last formula of such a sequence, A is provable and we write “I-A”. If A is provable when formulas of A are added to the axioms, we write “I-AA”.A formal system @ao(Z; Q)(L) is comfilete if exactly the valid formulas are provable. A calculus @ao(Z; Q) consists of the basic schemes and rules plus those of Z and Q. It is comfilete if and only if pao(Z;Q)(L) is complete for all (a, j3, 0,n)-languages L. There is no point in discussing strong completeness for these systems, since the situation is the same as for the propositional systems. The argument in the introduction to Chapter 5 shows that no formal system for L can be strongly complete if L has a semantically consistent, non-satisfiable set of formulas. Thus no strongly complete calculus exists for a non-limit or for any of the incompact cardinals of 10.2.3. As a corollary to 1 1.1.3 we have 11.1.4 Corollary. Let Z be any set of valida-propositional schemes or schemes %2, or 9%&,for y < a n j3. Let 9 be any collection of rules of independent or dependent choices with y < a. Then provable formulas of an (a,j3, 0,n)-language L are valid. Moreover, if I-AAthen A holds in all models of A .

11.2 Development of the Formal Systems for (a,/I, o, n)Predicate Languages When we treated the a-propositional formal systems, we proved the completeness theorems syntactically and then deduced the Boolean algebraic representation theorems from them. However, the completeness theorems for predicate logic will be proved algebraically, making full use of the representation theorems for pacomplete Boolean algebras. Such basic theorems as the substitution rule for equality and the rule of change of bound variables will also be proved algebraically, making use of the model-theoretic theorems of Chapter 9. 11.2.1 Theorem. If La is an a-propositional language, L an (a,j3, 0,n)-predicate language, and Z a set of a-propositional schemes, then I-A in @a(Z)(La) implies I - S f A in ‘@ap(Z)(L),where f is any function on the set X of propositional symbols in A to formulas of L. PROOF : An easy induction using substitution properties 3.5.3

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123

shows that if A is any formula of La whose propositional symbols are all in X , and if f is any function on X to formulas of L, then SFA is a formula of L. The theorem follows by the argument of Theorem 5.2.2. 11.2.2 Definition. Let r be a set of formulas of L. Then F t d A iff I-AA or there are formulas Cc E r, E < 6, 0 < 6 < u such that [A Co. . .Ce. . . ] + A . Note the parallel in the definitions of F t A and A t - A . Obviously a formal system for L is complete if and only if the two notions are equivalent for all formulas A , sets r. 11.2.3 Theorem. Let r be a set of formulas of L. Then in any of S)(L) of Sect. 1 1.1, the systems (i) r t d A0 and rl-~ [A0 + All implies Ft-d A1. (ii) r t d A t for all E < 6 implies r t - d [ A Ao. . . A t . . . ] . (iii) F t d A and variables of Rng(v) not free in r implies I'td [ V V A ] . (iv) If the rule of y independent choices is in 9, then under proviso of that rule, I ' t d [ V Ao. . . A € .. .] implies €
rt-d[

.

V [WvoAo].. .[WvbAr]. . ]

CCY

if variables of U {Rng(vE) : E < y } are not free in r. (v) If the rule of y dependent choices is in Sa, then under the proviso of that rule, r t - d [ V Ao. . . A € .. .] implies E
I'td

[ V [VvoAo].. .[ ~ z u ~ [ W V , A . .~] ] ] . €
if variables of U {Rng(vc): 5' < y } are not free in r. PROOF: The proof of (i) and (ii) is like the proof of 5.2.4. In case = 4, all of (i)-(v) are ordinary rules of inference. I n case r # 4, the argument in the proof of Theorem 5.2.4 shows that if T t d A then there is a conjunction C of formulas in such that tdC + A . Therefore, if F t d A and variables of Rng(v) are not free in r, t d C + A for a conjunction C of formulas of r and variables of Rng(v) are not free in C. Hence I-dC 3 [ V v A ] by 91 and modus ponens. Hence (iii). [ V Ao. . . A € .. .], where Suppose V&, is a rule of 9 and

r

attached sequences V O , . . ., ve, . . . of variables satisfy the proviso of the rule and are not free in r. Then t d C + [ V Ao. . . A t . . .] for a ICY

€
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I N F I N I T A R Y PREDICATE LOGIC

conjunction of formulas in I'. Then t~[ V Ao[-,C]. . . A € .. . ] using C
a substitution of an a-propositional formula provable in w a , 1 1.2.1 and modus ponens. To the principal subformulas of the new disjunction attach sequences vo, , v l , . . ., v ~ .. . . of variables where x is a variable not in C or in any of the A t or vt. This sequence again has length y since y was an infinite cardinal, and this sequence again satisfies the proviso of %'A?,,. Hence

t A [ v [VvoAo][Vx[1C]]. . .[Vu&]. C
.. I .

Again using an a-propositional scheme provable in ga, tA[3XC]

+ [ v [WvoAo]. . .[ v v t A € ] .. .]. C
[%C] by ordinary propoSince t [ V x [ l C ] ] [4] by 9 2 , k C sitional calculus. Using a valid scheme in + alone and modus ponens, kAC + [ v [VvoAo].. . [ V v , A s ] .. .]. S
Hence (iv). (v) is similar. 11.2.4 Deduction Theorem. In any of the systems waB(Z;LR)(L) of Sect. 1 1.1, rk,4 A if and only if kdvrA for sets f of sentences of L. PROOF:Immediate from 1 1.2.3 by induction on the length of any given proof of A from assumptions A v r. 11.2.5 Properties of Quantification. In laa(1), (i) k[Wv[Ao --* A111 4 [[VvAo]+ [ V v A ~ l l . (ii) k[Vv[Ao A A l l ] t)[WvAo] A A11 if variables of Rng(v) not free in A1. (iii) t [ W v A ] t)[Vvo[WvlA]]if Rng(v) = Rng(v0) u Rng(v1). (iv) k [ V v A ] t)[Vg 0vSF,R"g(")A]if g one-one to variables not free in A , and A has no free occurrence of v(E) bound by g(v(6)) for 5 < Dom(v). PROOF:These proofs are the same as proofs in ordinary predicate calculus. Perhaps, however, for (iv) we should point out that [[Vg0vSF,R"B(")A]+ A ] is an instance of 9 2 . For this, it suffices 0 and that no free occurrence to show that A = SF~(g0")SFRng(")A of g(v(5)) in S F P @ ' ) Ais bound by v(l) for t ~ D o r n ( v )Let . A' = SF,R"B(")A,A" = SFRng(gow)A'. 8-1 If 1 is an occurrence of a variable y qk Rng(v) in A , then L is still an occurrence of y in A' and since the SF-operator does not affect quantifications, L is free in A if

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125

and only if it is free in A‘. Since L is then not a free occurrence of a variable of Rng(g 0 v ) in A ’ , A ” ( L = ) y = A ( L ) I.f L is a free occurrence of v(E) in A , then L is an occurrence of g ( v ( 6 ) ) in A ’ , free by proviso. Hence A”(&)= A ( L )= ~ ( 5 )Finally, . if L is a bound occurrence of v(5) in A , it is still a bound occurrence of v(E) in A’. Again A“(L)= A ( & = ) ~ ( 5 ) Hence . A = A ” . A free occurrence of g(v(5)) in A’ must have been a free occurrence of v(6) in A . If L had been bound by v(5) in A’, it would also have been bound by v ( 6 ) in A , making the occurrence L bound. Hence proviso on 9 2 is satisfied. Consider the relation

A = A’ iff C-A[At)A’] in

@Pa&’;

Q)(L)

on formulas of an (a,#?. 0, +predicate language L. By the Equivalence Theorem 5.2.6 and 1 1.2.1, = is an equivalence relation which is also a congruence relation with respect to negation, implication, and conjunction. By 11.2.5 (i), it is also a congruence relation with respect to the cylindric operations taking A into [ W v A ] ; that is,

A = A‘ implies [WvA] = [VvA’]. It is therefore a congruence relation in the sense of Sect. 8.4. As a

corollary to 8.4.5, we have 11.2.6 Replacement Principle. If A = EoCo.. .E,C,. . .Ea, where (C,: v < a> is a sequence of formulas of L, and if I-A [C, t)C i ] for all v < a in Q)(L), then FAA t)[EoCi,. . .E,C:. . .En]. Consider also the relation

T = T’ iff C-A[T= T’] in @&;

Q)(L)

on terms of L. It is reflexive by 891, symmetric and transitive by 8 9 4 with P = =. It is a congruence relation on the algebra of terms by 6 9 2 and 6 9 3 . Moreover, 8 9 4 and 8 9 5 tell us that this congruence relation on the algebra of terms and the above congruence relation on formulas are linked as follows:

To = Ti, and T1 = T i implies [ T O P T I= ] [Ti,PTi] Tg = T i for all E < implies [QTo. . .Tg. . .] = [QTi,. . . T i . . .] for all special two-place predicate symbols P , all infinitary or qplace predicate symbols Q. Therefore the following describes a system for the interpretation of formulas of L, a system we will call G(?&@; Q)(L), A ) :

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INFINITARY PREDICATE LOGIC

For terms T , let IT1 { A ‘ :A = A‘}. Let

= {TI: T

= T’}, for formulas A , let ( A (=

= {ITI: T is a term of L}. B = { [ AI : A is a formula of L}. S = {s} where s ( x ) = I(x>l for all variables x . C(c) = I(c>l if c is an individual constant. O(y)() = I[ToyT1]I if y is a special two-place operation symbol.

D

O(rp)((ITol...IT€l...)) = I[pTo.. . T c . . .]I if rp is a [-place or infinitary operation symbol, 0 < [ < 0 , and [ is the length of the sequence (To. . .T t . . .). R(P)()= I[ToPT1]1if P is a special two-place predicate symbol. R(Q)(. O’(-d(lAl) = I-AI

O’(+)() = \ [ A A o . . . A t . . . ] I Q W , 4({IAI)) = IPvAll.

An induction on terms shows that the unique homomorphism s* on terms such that s * ( \ ( x ) [ ) = s(x) for variables x is the following:

s*T

=

IT1 for all terms T .

An equally easy induction on formulas shows that the valuation function V is the following:

V ( s ,A ) = IA I for all formulas A . 11.2.7 Substitution Rule for @&;Q)(L) of Sect.11.1, if FA SFF A t)SFF A , where f , f’ that A has no free occurrence

W f’(41.

Equality. In any of the systems t d f ( x ) = f ’ ( x ) for all X E X then map variables X to terms, provided of x E X bound by FV(f(x))or by

PROOF: Form G(&@; fz)(L), A ) and let s be the unique assignment in s. Then s* 0f and s* 0f’ agree on X since the values are If(x)I = If’(x)l. By Theorem 9.2.1 1, V(s,S F T A ) = V ( s ,S F F A ) . This is exactly what we wished to prove.

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11.2.8 Rule for Changeof Bound Variable. I-A t)SBFA invas(L) where g is a one-one function on Y to variables not in A . PROOF: Form G())3u~(L), #). According to Theorem 9.3.5, the result will follow once we know k[WVA]

c) [Wg’

0 vSF;””w(%4]

for all one-one functions g to variables not free in A such that A has no free occurrence of y E Y n Rng(v) bound by g(y), where g’ is g on Y ,identity elsewhere. This follows by 11.2.5. Let vo, 01 be Y. chosen so that Rng(v0) = Y n Rng(v), Rng(v1) = Rng(v) Apply 11.2.5 (iv) for vo, then use (i), (iii) and replacement. 11.2.9 Corollary. t [ W v A ] 4 A(f 0v ) in &p(L).

-

11.3 Completeness of the Basic Formal Systems with Chang’s Distributive Laws and the Rule of Dependent Choices for Certain a, The completeness theorems make use of the criterion for satisfiability, 10.3.5. Since that theorem was formulated only for sentences, we must first verify that our systems are complete if only the valid sentences are provable. In case @ = a there is no problem about this for if A is a valid formula and sequence v contains all variables free in A , then [WVA] is a valid sentence. From t [ W v A ] we conclude FA. In case @ < a it may not be possible to form the closure [ V v A ] , but the following lemma says that the familiar device of replacing the variables by individual constants still applies. 11.3.1 Lemma. Let A ( X O , . . .,xg, . . .) be a formula of an (a,@,0 , ~ ) language L and suppose sequence (xo.. .xg. . .> contains all variables free in A . Let L’ be the language like L except that it has one new constant cg for each xg. Then (i) II-A(x0, . . ., xg, . . .) if and only if II-A(co, . . ., ce, . . .). (ii) For any of the formal systems of Sect. 1 1.1, FA (CO, . . . ,cg, . . .) in D)(L‘) implies I-A(xo, . . ., xg, . . .) in ))3as(Z; Q)(L). PROOF:Since no bound variables are changed in passage from A (xo, . . ., xg, . . .) to A (CO, . . ., cg, . . .), A (CO, . . ., ce, . . .) is the result of changing free occurrences of xe to cg. Therefore in any model 1332’ of L’, V ( s , A(c0, . . ., c ~ ., . .)) = V(s,A ( x 0 , . . ., xg, . . .)) for assignment s ( ~ € )= C(c,), by Theorem 9.2.10. Thus A(c0, . . ., c,, . . .)

-

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I N F I N I T A R Y PREDICATE LOGIC

is valid if and only if A ( X O , . . ., xg, . . .) holds in all models 92’of L’. But clearly this is the same as saying A ( X O , . . ., xq, . . .) is valid. Hence (if. Suppose that
+

+

A = A’ if and only if t-A

A‘

t)

is a congruence relation on the algebra of formulas. Let B((PaB(Z;Q)(L)) be the set of all these equivalence classes. Operations on these classes induced by the propositional operations are

+I

= l [ l A I l > IAol

+

lAll = I[Ao+A111,

A (IAol.. . IAel.. .) = l[A Ao. . . A e . . .]I.

The defined propositional operations yield additional operations on B(%dZ; Q)(Lf): lAol A \Ail = I[Ao A AiIl, lAol V lAil = I[Ao V A i l \ , V (IAol.. .IAgl.. .>= [[V A o . . . A c . . .]I. Let 1 = [ [ Av [d]]I.

Then the defining equations for Boolean algebras all hold, for Theorem I 1.2.1 implies this and more:

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129

11.3.2 Lemma. Let L, be the a-propositional scheme language and L" the set of a-propositional schemes in 2. Interpret L, in B = B(@&'; Q)(L)) letting O'(1) = 1,Or(+) = +, O r ( A )= A. Then if k d in @&(El),then s*& = 1 for all assignments s to B. PROOF:Let W be the set of all propositional symbols in&. For A t E W let / ( A t ) be a formula of L in the equivalence class s(AC). Then s* = s ( A 0 = I/(As)l. An easy induction on formulas of L, shows that s*a = ISTaI for all schemes with symbols in W . Hence if I d in @,(Z'), s * d = I S y d I = 1 by 11.2.1. It is a consequence of the lemma that B = D)(L)) = ( B A v > is an 7 u-complete Boolean algebra. Moreover if u-propositional scheme d' E Z then d' holds in B and, what is the same thing, the equation Td = 1 holds in B. Therefore it is a consequence of the completeness of @,(lIpa) and 6.4.2 that if the Chang distributive laws (5.1.3) of levels less than a are in Z then B is 7u-representable. We use this fact in the completeness theorems. The order in B(@,&'.; D)(L)) is lAol

< \All

if and only if k[Ao + A l l .

It follows by the rule of change of bound variable and 9 2 that I[VvA(v)]l < \ A ( /0 v)I for all substitutions of Rng(v) to terms. It is not difficult to see that I[VvA(v)]Iis in fact the greatest lower bound of all the substitutions ] A ( /0 v ) l . 11.3.3 Theorem. If y exp E = y for all E < B, the calculus

py+#7y;Qy) where D,

contains only the rule of y dependent

choices, is comp'ete. PROOF : We have seen that provable formulas are valid (1 1.1.4). We have seen also (1 1.3.1) that to show this calculus complete it suffices to show that every valid sentence A of a (y+, B, 0 , n)language L is provable in @y+p(17y; Qy)(L). To do this, form the Dy)(L))and choose a set X of ~ ( y f / , I) =y algebra 93 = variables not in A and form sets T of terms, d of formulas in accordance with 10.3.4. Both have power y. If, contrariwise, not !-A then this algebra has a homomorphism to 230 sending l[+l]l to 1 and preserving any given collection of y meets and joins, each with at most y terms. This follows from the y-representability of 'B and the theorem in the Foreward on Algebra. We intend to choose this collection in such a way that for such a homomorphism h, T = {C:hlCI = 1) satisfies 10.3.5 (1)-(8). Since [ - A ] will be in T

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INFINITARY PREDICATE LOGIC

it will follow that [-A] is satisfiable, contradicting the validity of A . It is necessary to first go through the procedure of 10.3.5 for selecting witnesses. This step can be omitted if A contains no quantifications. Make a list of length y of all quantifications in d : *

-

. . ., Cv(gv 0v,), .

.

[ ~ ~ o ~ o (* ~* o [vv,cY(vY)l, ~ l , a,

Select witnessing formulas Co(g0 0vo),

so that each g, is one-one on Rng(v,) to X and variables of Rng(g,) are not in Rng(g4) for 5' # v and do not appear in [Wv,Cr(vc)] for 5 < v . Let W , = [C,(g, 0 v,) + [Wv.C,(vy)]]. We claim that 1[lA] A [A W O .. .W,. . . ] I # 0 in 8. To see this, suppose that this element were 0. Then ki[[iA]

A

[A W o . . .Wv..

.I].

Since the underlying propositional logic is complete, substitutions of valid y+-schemes are all provable. Using such a scheme,

1 A I- [V [lWO].. .[ l W , ] . . .I

To each W , attach sequence g, 0v, for an application of the rule of dependent choices, They have pairwise disjoint ranges and variables of Rng(g.) are not free in Wa for 5 < v . Moreover each formula of d has fewer than /? free variables. It is therefore possible to associate to each W , a sequence w, of length less than /? containing all variables of U {Rng(gs): E < v } free in W.. Then

(a) 4 k [V [vgo o vo[lWoll...[ 3 w [ v g Y 0v,[-rWPl11-. .I

Since k[lW,] t)[C,(g,O vV) A [-1Wv,C,(v,)1],andvariablesof Rng(g,) do not appear in v. or C,, I- [vgv 0 Vv[iWv]]t)[[vgv0vrCv(g9 0%)I A [ i v v , c v ( V v ) ] ] for each v < y by 11.2.5 (ii). By 11.2.5 (iv) and ordinary propositional calculus and generalization, it is apparent that the negation of each of the principal subformulas of the disjunction (a) is provable. By the deduction theorem, FA. But we were assuming not kA. Therefore we must have \ [ - , A ]A [A W O .. .W,. . . ] I # 0. We choose the collection of meets and joins to be preserved by a homomorphism h to 80taking /[-A ] A [A W O . .. W,. . .]I to 1 as follows :

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131

The joins v{-~I[Tc=T;]l:t< ( } v I[qTo...Tc ...I = [CUT i,... T i . . . ] I for all terms of form [ y To. . .Tg. . .], [ y Ti,.. .T ; . . .] in T. These joins are 1 by 6 9 3 . The joins V { ~ ( [ T cTk]l:5 < q) v (-I[QTo.. .TE...]I) v \[QTi,...TL.. .]I for all atomic formulas of the form [Q To. . .Tc. . .I, [Q Tb. . .T i . . .] in A . These joins are 1 by 695. Themeetsh {IAc(:5 < 6) = \[A A o . . . A t . . .](forallconjunctions in A . There are at most y such joins and meets each with at most y terms. Let F = {C: hlC( = 11, h being be the preserving homomorphism. Then [-A] E F and F satisfies 10.3.5 (1)-(8). Formulas [T =TI E r by 891. If A0 = [To = Tb] and A1 = [ T I = Ti]are in r, then A2 = [ [ T o ~ J T I ][TbyTi]]E r because (A01 A \All < IA2/ by 6 9 2 and because h preserves finite operations and order. Hence 10.3.5 (1). The verification of the other conditions is similar. The preservation of the special joins and meets is needed for (2), (4), (7). Checking (8) for the quantifiers, note that I[VvAo]l< ISFPw(')A01 for all f on Rng(v) to T by 9 2 . The proviso is satisfied because terms in T have variables in X and such variables are never in quantifier-sequences of formulas of A . Since h preserves order, [VvAo]E F implies that all substitutions SF?(')Ao E F. Conversely, if [VvAo]# F, then since there is Y such that W , = [Ao(gv0 v) + [VvAo]] and since h\W,l = 1 and hl[VvAo]l= 0,it follows that hIAo(g, 0 v)I = 0. Hence Ao(g, 0v) # F. But in this case Ao(gv 0v) = SFEw(')A0. Therefore condition (8) holds. We then conclude [-A] satisfiable, thus contradicting the validity of A . Therefore A must be provable. 11.3.4 Theorem. If a is strongly inaccessible then 9%9Ppa)is complete. is the set of all y-distributive laws for y < a, 9 % X Pthe a set of laws of dependent choices for all y < a.) PROOF: Let L be any (a,a, 0 , n)-language, A any valid fromula of L. Let y be the least cardinal upper bound of the length of A and of ( 0 , n} {a}. Then y < a and o < y if o < a and n Q y if n < a. Let L' be the ((2 exp y)+, y+, o', n')-language with the constants of A where 0' is o if o < a, 0' = y+ if o = a, and n' is n if x < a, n' = y+ if n = a. Then A is a formula of L' and every

-

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formula of L’ is a formula of L. By 11.3.3 A is provable in

?&2expy)+,y+(172expy; Q) where Q consists of the rule of 2 exp y dependent choices. But 2 exp y


and instances of schemes of

172expy are provable from the distributive law of level 2 exp y and

the rule of 2 exp y dependent choices is provable from .5W&,‘,. It follows that A is provable in the given system. 11.3.5 Theorem. If a is inaccessible and ~ ( y f j3) , < a for all y < a then @ a b ( I l 7 a ; Q) is complete, where Q consists of all rules of y dependent choices for y < a. Moreover if j3 is regular the assumption y exp E < a for all y < 0: and E < j3 suffices. If a strongly inaccessible @aD(I7,a; Q) is complete for all j3 < a. PROOF: If j3 = a the hypothesis on a , j3, reduces to the strong inaccessibility of a. In this case the completeness follows by 1 1.3.4. For j3 less than a and regular, ~ ( y f j3) , = U { y exp E : E < j3) < a for all y < a if and only if y exp E < a for all y < a, E < j3. Hence the reduction in this case. Moreover, if a strongly inaccessible and j3 < a then K ( Y + , j3) < y exp j3 < a for all y < a by 10.3.2. Therefore it remains only to show that &&77a; Q) is complete for j3 < a, provided that ~ ( y f j3) , < a for all y < a. Suppose A a valid formula of an (a, t!?, 0,n)-language L. Let y be the least cardinal upper bound of the length of A and of {o, n} {a}. Then y < a and o < y if o < a and n < y if n < a. Let L’ be the ( K + , j3, o’, n’)-language with the constants of A where K = ~ ( y f /I), , 0’ = o if o < a , 0‘ = y+ if o = a, and n’ = n if n < a, n’ = yf if n = a. Then A is a formula of L’ and every formula of L’ is a formula of L. Since K exp E = K for all E < j3, A is provable in !&+,a(17K; QK)(L‘)by 11.3.3. This proof will also be a proof in

-

( P a b ( n 7 a ; Q) (L).

11.3.6 Theorem. If a is inaccessible, !#ao(IT7a) is complete. For any infinite cardinal y , ~ y + o ( 1is7 complete. ~) Also @,,,lo is complete. PROOF:Quantificational schemes were not needed to carry out the steps of the proof of 1 1.3.3 for quantifier-free formulas. Schemes of 17,are known to be provable in basic propositional calculus Therefore instances of such schemes are provable in basic predicate calculus by 1 1.2.1. Hence ~ , , , l ois complete.

B A S I C F O R M A L SYSTEMS W H E N a = wi, B = w

11.4

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Completeness of the Basic Formal Systems when a = w1, p=w

For a = w l , the completeness theorem 11.3.3 applies only for /? = 0 and ,fl = w. We have seen that the completeness of Val,, is a corollary. Moreover since instances of schemes of I7, are provable in basic calculus, 11.3.3 tells us that Vmla(4; Lna) is complete, where Q, is the rule of w dependent choices. But we can say more; &,la itself is complete. We will give a proof that depends only on the existence of a homomorphism on a given Boolean algebra to 80carrying a given non-zero element to 1 and preserving a given denumerable collection of joins and meets. See the Foreward on Algebra for a proof of this theorem. The collection of joins and meets can be chosen so that each has denumerably many terms as well. This suggests that Vy+,JIIy) may also be complete for y > w , for the algebras of equivalence classes have the analogous property for collections of y joins and meets. However the argument we now give does not generalize, for the homomorphisms are taken not on the algebras of equivalence classes 8(walw(L)) but on subalgebras of them. The question of the completeness of calculi !#Dy+,w(17y) for y > w remains open. 11.4.1 Theorem. $3p,l, is complete. PROOF: Let L be any (01, w ,of n)-predicate language, A any valid sentence of L. We must show I-A. Choose a set X of ~ ( w lw, ) = w variables not in A and form sets T of terms, d of formulas, in accordance with 10.3.4. Both have power w. Form the algebra 8(Vmlw(L))and let 8 ’ be the subalgebra with underlying set B‘ = {ICl: FV(C) C a finite subset of X}. Since formulas of d have only finitely many free variables, and all are in X , ICI E B‘ if C E d . The algebra 8 ’ is clearly not w-complete, nor is it an w-regular subalgebra of 8(?J&ala(L)). For in B’, if [WvC] E A , (a) I[WvC]I

=A

{IC(f 0 v)I : f E XRng(v)}, with respect to 8‘.

We have I[WvC]I < IC(f 0 v)I for all f E XRng(v) in %3(@alm(L)), but in that algebra, the meet on the right side of (a) is a conjunction, not a quantification. To see that (a) holds in %‘, note that if IC’I E B‘ and IC’I < IC(f 0v)l for all f ~ X R n g ( v )then , such an f can be chosen one-one to variables not in C‘. Then from I-C’ +

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C ( f 0 v) it follows t C ‘ + [WfO v C ( f 0 v)]. By 11.2.5 (iv) and replacement, t C ’ + [WvC].Hence (a). Suppose, contrariwise, not FA. Then IAl # 1 and 1 4 # 0. Let h be a homomorphism on 8’ to 230 taking I l A [ to 1 and preserving the following collection of meets and joins in 23’: The meets of form (a) for quantifications in d. The meets A {IArl: t < S} = [ [ A Ao. . . A € .. .]I for conjunctions in d. These meets are the same is B’ as in B. The joins arising from infinitary terms in T, infinitary atomic formulas in A , as in the proof of Theorem 11.3.3. These joins are still 1 in B’. Let T = {C: ICI E B ’ andhlCl = l}. Then [-,A] E I‘ and r has properties (1)-(8) of 10.3.5. Hence [-A] is satisfiable, contradicting the validity of A . Therefore A must be provable. 11.5 The Reduction to the Rule of Independent Choices when

p=o

The question of the completeness of @a@(f17a ) is open for ct > 01, but we can show that there are cases where the rules of dependent choices in the completeness theorems can be replaced by rules of independent choices when fl = w. It is a theorem of Chapter 12 that such reductions are not possible when > w. To simplify notation, let

G I , , contain only the rule of y independent choices Q I , ,= ~ U { Q I , ~ :K Q D , ~contain

OD,, y

=


only the rule of y dependent choices

u {DO,

K

< y}.

Then we can prove 11.5.1 Theorem. If a is strongly inaccessible, then p a @ ( I I y a ; Q I , , a ) is complete. The proof combines techniques of the completeness theorems already given and needs only to be outlined at this point. Consider a valid sentence A in a formal system Q a m ( L) = p a o ( f l 2 exp

; Q I ,y ) (L)

A itself being a ( y + , w)-sentence, where 2 exp y < a. Passage from yf to a needs to be made so that the y-distributive law will be

R E D U C T I O N T O T H E R U L E O F I N D E P E N D E N T C H O I C E S 135

available. Choose a set X of K ( Y + , w) = y variables not in A and form sets T of terms, A of formulas in accordance with 10.3.4. Split X into w pairwise disjoint sets X O , . . ., Xn, . . ., each of power y . Form the algebra B = B(CZaw(L)) of equivalence classes and let 2‘3; be the the subalgebra with underlying set BL = {ICI : F V ( C ) C U { X t :i < n}}. Let 8’= U {BL:n < w}. Remember that formulas C in A have finitely many free variables, so such elements ICI are in %’. B is 7a-complete and y-distributive as are the 23;. If 6 < a and F V ( [ A CO... C t . . .I) C U ( X i : i < n} then l[A C O . ..Ct...]l = A {ICsl: [ < S} in %A, in ‘23’ and in B. The BL are therefore f a subalgebras of 8’and of B. However B’ is not even o-complete and not even an w-regular subalgebra of B. It is easy to see that for I [ W v C ( v ) ]I E B‘ (a) I[WvC(v)]I = A {IC(f 0 v ) l : f E XRng(v)} in B‘. The meet on the right is a conjunction in B. The rule Qz,, imposes additional structure on the meets (a). Given any formula D with free variables in U { X i :i < n}, ID/# 0, and collection IIWv,C,(vv)]l, v < y, of elements of BL, the rule of y independent choices tells us that for one-one functions g, on Rng(v,) to X,+1 with pairwise disjoint ranges we have ID1 A A IC,(gv 0vv) + [wv~cv(v,)lI # 0 V
in BL+1, hence also in 8’and in B. The argument is similar to the one given in the proof of 11.3.3. Suppose, contrariwise, the formula A not provable in &,(L). Then l [ + t ] ]# 0. We wish to show that there is a homomorphism h on B’ to the two-element Boolean algebra 230 sending l[+l]I to 1 and preserving the following collection of y joins and meets in B’: a) The meets (a) arising from quantifiers in A . b) The joins 1 = I[A A o . , . A t . . .]I v V { I l A t l : 5 < S } for conjunctions in A . c) The joins arising from infinitary terms and atomic formulas of T and A as in the proof of 11.3.3. They are all 1 in 8‘. Note that each one of these joins and meets has at most y terms. With such a homomorphism we can let T = {C: hlCl = l} as before, and Twill be seen to contain [ - A ] , formulas [T =TI for T E T and to satisfy conditions 10.3.5 (1)-(8). We can then conclude that

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[-A] is satisfiable, contradicting the validity of A . Therefore A must have been provable. To see that there is such a homomorphism, begin by listing all joins of type b) and c) arising from formulas with free variables in X O . By the y-distributive law there is a formula Cb with free variables in X Osuch that 0 # ICbl 1 4I and for each join V, ID,I = 1 in the collection there is v such that ICbl < ID,I. Then list meets of type a) arising from quantifier-formulas [Vv,C,(v,)] with free variables in X Oand attach to them one-one functions g, on Rng(v,) to X 1 with pairwise-disjoint ranges. Let C," = [Cb A [ A [C,(g, 0 v,) + V-=Y

[Vv,C,(v,)]]]]. Then IC,"l # 0. Then list all joins of type b) and c) arising from formulas with free variables in X Ou X I . Use the ydistributive law to find a formula C; with free variables in X Ou X 1 such that ICil # 0 and for each join V, ID,\= 1 in the collection there is v such that lCil < ID,I A IC,"l. Then form C; for quantifications with free variables in X Ou X 1 choosing witnessing formulas by substitutions to X z . Continue by ordinary induction. We will have a sequence IC,"l 2 IC;l 2 . . . IC:l of non-zero elements of B' such that for each join V, ID,I = 1 of type b) or c) there are n, v such that ICi( < ID,I and for each meet of type a) there is n and there is g on Rng(v) to X such that IC:l < 7 IC(g 0 v)I v I[VvC(v)]I. The homomorphism h determined by a maximal ideal containing {T ICil: n < w } preserves all joins and meets a), b), c).

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