Formalization Principle

FORMALIZATION PRINCIPLE

G . TAKEUTI* Institute for Advanced Study, Princeton, N.J., USA

A set theory T(not necessarily first order) is said to be “sufficiently strong” if T is true and the theory consisting of all true first order set theoretic sentences can be interpreted in T . Similarly, a set theory T on the class L of all constructible sets is said to be “sufficiently strong” if Tis true on L and the theory consisting of all true first order sentences on L can be interpreted in T, First, let us consider giving principles which generate a sufficiently strong set theory. Now imagine the following situation. A basic theory which we start with and several definable (in set theory) principles by which we get a new stronger true theory from a true theory are given. Then the closure of these principles is easily proved to be not sufficiently strong for the following reason. We can formalize the closure in the set theory and truth theory implies that this closure is not sufficiently strong. Therefore we know that they must be highly undefinable if there exist principles to generate a sufficiently strong set theory. Then the question is this. Is there any meaningful principle among these highly undefinable principles? In order to consider this question, we shall examine the above undefinability proof of our possible principles. In this proof, we assume the following principle. PRINCIPLE: We know practically how to formalize a given well-defined theory. This is rather well-supported heuristically : It might be a most successful and most basic practical principle of modern logic. Let us consider the use of this principle as a device to create new systems. In order to be precise, we now state formal terminology. r$’ is understood to be a Godel number of a formula $. If S is a system, then “ S is formalizable”

*

Work partially supported by National Science Foundation grant GP-6132.

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means that the notion Prov,(r$l) "JI is provable in S" is definable. Let us suppose that a method, say E, to expand a system S to a new system is defined by the term of Prov,. Now start with a basic system So and apply E to all systems which are obtained by E from So. We obtain E(S,), E(E(S,)), ... and S , = u n c , E"(So), where En($,) means a system obtained from So by it times application of E. We can again repeat this method to S, instead of So and so on. Obviously we cannot formalize the system thus obtained just because we have no a priori definition of Prov,, though we are confident that we can find an appropriate definition of Prov, for each definite individual system S. I think that such a principle E is a meaningful but undefinable principle. Now we shall check the notion Prov,. Our experience shows that there are many adequate definitions of Prov,. What is the criterion of Prov,? One necessary condition for Prov, is the following. Condition for Prov,: Prov,(r$T)-Sl- $. Is this criterion enough to characterize Prov,? The answer is yes. However, in order to know that Prov, satisfies this condition, we have to know for what formulas $ Prov,(rJIT) and/or Sk I) is true. In many cases, we cannot decide Prov,(r$l) in S. Therefore the adequacy of a certain definition of Prov, seems to be judged mainly by our mathematical intuition on how S is constructed. The difficulty in considering Prov, is, we do not know anything more than the condition. What we know is that there will be one definition of Prov, which will be chosen by our mathematical intuition. Our formalization principle roughly means the following principle. Formalization principle: If we have created a theory S and (r$TISk $1 is definable in our language, then we can gain a right formalization Prov, for S. The important thing to do is to find a good property of a right formalization. In this paper, we are however interested merely in the sequence of axiom systems S, such that a) S,,, =E(S,) and b) Sa= UB<. S, if a is a limit ordinal. (This kind of sequence was considered in FEFERMAN [I9621 for arithmetic and in SWARD[1967] for first order set theories.) Our So has w-rule and our language is a transfinite type theory. Our E is the following. Reflection principle: Expand S by adding Prov,(r$l) if SI-JI, or -I Prov,(r$l) otherwise for all sentences $. We shall prove the following. 1) For every definable well-ordering 4 of w , there exists a definable sequence of axiom systems S,(B < a o ) satisfying a) and b), where a. is the order type of <.

FORMALIZATION PRINCIPLE

107

2) For every definable sequence of axiom systems S,(p < ao) satisfying a) and b), there exists a provable well-ordering < of w in SB+lsuch that the order type of 4 is 8. (A precise definition of a provable well-ordering in a system S will be given later.) 3) There exists a definable sequence of axiom systems S,(,!3
Our language is similar to the transfinite type theory in TAKEUTI[to appear] but the intended interpretation will be different from the transfinite type theory there. We start with the first order language of set theory which consists of individual constants 0, 1, 2, . .. for all natural numbers, numerical variables il, i 2 ,i,, ... which range over all natural numbers, set variables xl, x2,x3, ..., one predicate constant E, and logical symbols. (As a usually abbreviated notation, we use a,p, y , ... as variables for ordinal numbers.) We define typed variables and extend the notion of formula inductively by introducing the “degrees” of (typed) variables and formulas. We define the degree of a first

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order variable a (denoted deg(a)) to be 0; and the degree of a first order formula $ (denoted deg($)) to be 0. For every formula (of first order) A(il, i2) having only numerical variables as free variables, variables of type A are introduced and denoted Xf,X:, X t , .... We define deg(XA) to be 1 and extend the notion of formula by adjoining variables of degree 1 and quantifiers with respect to the typed variables to our starting language. The degree of any formula in this language is defined to be the maximum of the degrees of the variables in it. Assume that we have introduced variables of degree n and formulas of degree n in this way. Then for every formula A(il, i2) of degree n having only numerical variables as free variables, we introduce variables of type A . The degree of a variable of type A where A ( i l , i2) is of degree rz is defined to be n+ 1. The notion of formulas is extended by adjoining variables of degree n + 1 and quantifiers with respect to variables of degree n + 1 to the language. The degree of a formula is defined to be the maximum of the degrees of variables in it. The intended interpretation of XA is this. If A is a well-ordering of o and the order type of A is u, then X A is a variable of type c1 and if A is not a well-ordering of o,then X A is a variable of type 0, i.e., a first order variable. In order to stress that A is a well-ordering of o,we sometimes use X “ in place of XA.According to this interpretation, we define that il and i, in XA are bound. The formula “ A is a well-ordering of o” can be easily expressed by using first order variables and A and is denoted W(A). “ W ( A ) and the order type of A is a’’ is denoted IA1 = u, which is also expressed by using first order variables and A . 2. Provable well-ordering

When we talk about a system S of axioms, we always assume that S is some extension of ZF + V = L in our language. S also contains axioms which assert that V i is equivalent to VXEW. We also assume o-rule. Therefore Vi$(i)is provable in Sif and only if all $(O), $(I), $(2), ... are provable in S. DEFINITION. A binary predicate < in our language is said to be a provable well-ordering of o in S if and only if the following conditions are satisfied. 1) Lst- W(<). 2) If m and n are two numerals, then one and only one of n = m, S I- n 4 m, and Sk m < n holds. DEFINITION. Let

< be a well-ordering of o,and n be a numeral. lnI4 is

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defined to be the order type of n in the well-ordering 4 . a=lnl, can be expressed by using first order variables and 4 .

+

THEOREM. Let S be a true theory and and 4 be two provable wellorderings of w in S and m and n be two numerals. Then one and only one of d, JmJ, < JnJ <, or JnJ = JmJ + is provable is S. In1 < JmJ PROOF:In this section, A(n, m), B(n, m ) and C(n, m) are defined to be In1 < Im("., ImlQ.< In(a and In( = Jm1, respectively. We prove by transfinite induction on (mi* that if A (n,m), then S t A (n, m). As an inductive hypothesis, we assume that this is true for every m' Q m. Since A (n, m),there exists a numeral k such that C(n, k ) A k Q m. By the inductive hypothesis, S t A(k', k) for every numeral k' < n. Since k' < n is decidable in S, S k V k ' ( k ' + n-+A(k', k ) ) by using w-rule. Since Sl- k Q m, S t A ( n , m). In the same way, S t B ( n , m),if B(n, m). Since C(n, m ) is equivalent to Vk < n A ( k , m) A Vk 4 mB(n, k ) and k 4 n and k 4 m are decidable in S, S t C(n, m) if C(n, m). Therefore one of A(n, m), B(n, m), and C(n, m) is provable in S, and so A ( n , m),B(n, m) and C(n, m ) are decidable in S. THEOREM. Under the same hypothesis as the previous theorem, one and only one of 14 J
Clearly, if W( G ) , then

< i z ) v (il + i, A V i W( 4 + 1) and J G + 11=I < I+ 1.

A

i 4 i,

A

i,

i (i 4 iz)).

PROPOSITION. If 4 is a provable well-ordering of w in S, then so is < + 1.

+ < + 3, ... are defined to be ( 4 + 1) + 1, (4+ 2) + 1,...

4 2, respectively.

3. The system So

Our intended interpretation of X " is a variable of type I < 1. Therefore

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X " corresponds to X A in TAKEUTI [to appear], where A(a, j?) is j?=cr+ 1 6 I. Since So is a subsystem of the system in 6.1 in Chapter I in TAKEUTI [to appear] in this sense, we shall simply present many provable formulas as axioms of So.For simplicity, we shall sometimes omit the universal quantifiers in front of the formulas and also omit the type sign if no confusion is to be feared, e.g. VX" ( X E X )means V X " ( X " EX"). DEFINITION. X A= Y Bis defined to be

So consists of pure logic of type theory (of our language), ZF+ V=L, where the axiom of replacement is generalized by introducing arbitrary type variables of our language, w-rule and the following axiom-schemata. (All numbered formulas in this section are axiom-schemata in So.) 1. XE Y+(F(X)-F( Y)), where F is an arbitrary formula and X and Y should be precisely written by X A and Y Brespectively. We always assume that a set variable is a special case of a typed variable and a numerical variable is a special case of a set variable. vz"( Z € YC-)F(Z)). 2. W ( +)+3 Y 3. W(+), X A € Y " + 1 4 Z 4 ( Z E X ) . In general, A , , ..., A,+B is an abbreviation of Al A A A,+B. +

4. 1W(<)+3x(x=X"). 5 . 3XA(x=XA). 6 . I$'(<), I$'(+)), I < l < \ + l + 3 Y " ( X " = Y B ) . DEFINITION. As an application of 2, we know that there exists a unique

y" + 1 such that

vz"(Z € Y t , z

f

x v

z 3 x ") .

This formula is sometimes abbreviated as Y<+'={x, X"} or Y = {x,X}. {{x}, {x,X } } is abbreviated as (x,X). 3! Y " f 2 ( Y " + Z = ( X")) ~ , is provable in So. (I! is read "there exists a unique".)

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T y ( ~ rX4) , is also written T'(X4) O+(T'(X ") < a++V Y E X * (T'( Y )
We fix a Godel numbering of our language. The Godel number of I) is denoted r$l which is a natural number. Such notions as "i is a Godel number A I),, i.e. of a formula", "i is a Godel number of a formula of the form a formula whose outermost logical symbol is A " , etc. are expressed by first order language. Thereforewe use tlr$lA (r$l), V r I ) , A $ z l A (r$, A $,I), etc. to express " V i ("i is a Godel number of a formula" +A(i))", " V i ("i is a Godel number of a formula of the form $1 A $2" +A(i))", etc. respecA $zl, we use a tively. Since r$,l and r$21are easily expressed by using notation like Vr$, A $27B(r$11,r$21) as a special form of Vr$l A t,b21 A (r$l A $zl). The number of logical symbols in $ and the degree of $ (defined in Section 1) are primitive recursive functions of rI)l and so easily expressed by first order language and we denote them as NZ(r$l) and deg(r$l) respectively. grad(r$l) is defined to be w - deg(r$l)+ NZ(r$l). grad(r$l) is also said to be the grade of $. The Godel numbers of the k-th numerical variable, k-th set variable and k-th variable of type A are denoted as rikl, rxkl and r X t 1 respectively. The k-th numeral is denoted as n(k). DEFINITION. Y A= X * 'xis defined to be (3!Z"((X,Z)EX)

A (X,

Y)EX)

V

(lq!Z"((X,z)EX)

A

Y-0).

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DEFINITION. T, (CL, X " + 3 ) is defined to be the conjunction of the following formulas. 1) v rx; E xi"'v Y (( rx; E xi"',Y) E xo Y' rxp E Y' T ?). 2) V r i $1 ( g r a d ( r i $ l ) Ex)). 4) t l " d x , $ l ( g r a d ( r V x , $ l ) ~ ~ ~ V Y(('Vx,$l, " Y)EX HvxVz, ( Z E s(x,rxkl, y)+(r$l, z )EX)). 5) VrVXf$l (grad(rVXt$l)
3 a 3 X " + 3 ( g r a d ( r $ l ) < ~A T , ( ~ , X ) (r$l, A Y)EX). By transfinite induction on grad(r$l), we have

DEFINITION. Let t+b be a formula and X;,', ..., Xt; be all free variables in $. $((Ye)) is obtained from $ by replacing X f ; ,..., Xih; by Y ' r X t ; l , ..., Y'~X;;T respectively. By transfinite induction on the grade of $, we have THEOREM. Let $ be a formula and

<

. .., <, be all types in $. Then the

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DEFINITION. T , (r$l, 0) is denoted T, (r$l). As a corollary of the previous theorem we have THEOREM. Let $ be a closed formula and Then the following is provable in So.

(w(
..., <,,be all the types in $.

<

+3~l
+

(T<(r$3-

+3
$1.

5. Transfinite applications of reflection principle In this section, the system of axioms S is always an extension of So and “a formula is provable in S” means that it is provable from S by using logical inferences and w-rule. Let S(r$l) express that $ is an axiom of S. Then Prov,(r$l), which means “$ is provable in S”, can be expressed by using first order language and S(il). We fix one such uniform way to define Prov,(i,) from S(il). Now we shall consider So. So(i,)can be easily expressed by the first order language. Therefore there exists a first order formula To such that

-

so t To ( r s o ( i I)’, r$3 so (r$l). Moreover the following is provable for every

< in So.

I < I > 0 (T<(rso(r$l)’) * so T $ l) ) . +

Now consider an arbitrary system S. Let s” be the system E ( S ) , where E is the reflection principle in the introduction. Then is expressed by

s(i,)

S(i,) v 3r$l[‘‘r$lis closed”

A

v

((Provs(‘$l) A i, = rProvs(r$l)l) (-I Provs(r$l) A i , = r i Prov,(r$l)l))).

Therefore, there exists a primitive recursive function f such that r s ( i I ) T =f(rs(il)’). We fix one of such functions f. Using a theorem in the previous section, we have

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THEOREM. Let G l y...) <,, be all the types in S(i,). Then the following is provable in So. (W(<,)-+l
+

be two well-orderings of w and the order type of any Now let < and type in 4 be less than I 4 I and I < I and 1 Q I be limit ordinals. We can easily construct a new well-ordering 4 . of w such that 4 . is expressed by using first order language, 4 , and Q and 1 Q .I = I Q 1 I< I. We fix a method to construct such a well-ordering 4 . and denote it Q 4 . In the same way, for every j , we fix a method to define j , and a well-ordering 4 . such that j , and Q. are expressed by using first order language, 6 , and Q and ljol< is zero or a limit ordinal and I j l - 1 j,l is a natural number and I Q .I = I 41 + 1 j,l< + 6 . )j l - I j,\ *. In this section we denote this well-ordering Q . by
+

+

DEFINITION. R" (h, i,, i3) is defined to be the conjunction of the following formulas. 1) V j l ( V j 2 ( j l <- j,)-+h'j, =rSo(Q1), wherej,< j , is an abbreviation of j , 4 j , v j , =j,. In this section, we assume that j , isthe first element in 4 . So we use h'j, ='So (it)' instead of 1). 2) Vj,< i2Vj2 ('7, is the successor of j , in <''+h'jz=f(h'j1)}. In this section we denote "the successor of j , in <" ( j , + I)<. So we use V j < i 2 ( h ' ( j + 1)" = f ( h ' j ) ) instead of 2). 3) V<' i2(Jj I 4 EK,,+h'j=Sub;?) Subn(i')i3) 13 where KII is the class of all limit ordinal numbers. S" (i,, i,, i3) is defined to be 3 h ( R " ( h , i,, i,, i3) A (li,/
Let m be rS" (il, i,, i3)l and S" ( i l , i2) be S" (il, i,, m). Then 3) becomes the following in S" (il, i,). i.e.

Vj

< i,(ljl

E K , -+~ h'j

=

Sub::"

Sub%m)rS< (i,, i,, i 3 ) l )

V j e i2(Jjl"~KII+h= j ' r S 4 ( i l ,n(j))').

Therefore we have the following.

FORMALIZATION PRINCIPLE

115

PROPOSITION. Under the assumption in this section, the following is provable in So.

s" ( i l , i 2 ) + + 3 h ( h j ' , = rso(il)l A vj-+ i z ( h ' ( j + 1)" A

A A

=j(hj')) V j q i z ( I j l c e K I I + h ' j = ' S " ( i , , n(j))l) (li21< e K I 1+ 3 j < i2T
DEFINITION. R" (A, i) is defined to be h'jo = rSo(il)lA V j -+ i ( h ' ( j + I)< =f(h'j)) A V j < i(lj1,

E K ~ hj' ~ = - ~' S 4

(il, n(j))l).

PROPOSITION. Under the assumption in this section, the following are provable in So. 1) 3hR" (h, i). 2) A " ( h , , il), ~ " ( h , i, 2 ) , j ~ . i l , j ~ i z ~ h ; j = h ; j . 3) R 4 ( h , i ) , j < C j l 2.L I j l I ~ ~ K I I ~ I j l 1 4 = 0 5 QZ (l.1 4 j z 5 j - 1I h l 4 EKII)-, "h'j is a Godel number of a formula $ and every typed variable in $ is in S" (il, i2), i.e. in 4 and T, j , " . PROOF:1) is proved by transfinite induction on i. 2) is proved by transfinite induction on j . The first part of 3) is proved by transfinite induction on j . The second part of 3) follows from the definition of R" (h, i),fand S" (il, iz). PROPOSITION. Under the assumption in this section, the following is provable in So. R" (hl, i ) , j 4 i+(S" (il,j)++T
DEFINITION. Sp (il) is defined to be Sq(il,i). The system of axioms consisting of closed formulas $ satisfying S: (r+l) is denoted as Sfl . THEOREM. Under the assumption in this section, the following are provable in So. 1) s,m-so(il).

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2) j , = ( j + 1)4-*(SJ:o(il)~E(Sj4) (il)), where E ( S ) (il) is S(i,)v Y $ l ( “ r $ l is closed” A

((Prov,(r$l)

A

i, = rProvs(r$l)l) v

( Provs(r$l) i A

i,

= r i Provs(r$l)l))).

3) lil, EK,,+(S,? (i,)-3j < isJ:(i,)). PROOF:Proof is by transfinite induction on j , and i. 1) is obvious. Proof of 2): By the previous propositions, we may assume R“ ( h , j l ) and so Sif (il)++

-

T,,, (Sub~~”)h’j,)++ T,,, (Sub;!“’f(hj’)) E (T,J1 (Sub;,“” h’j))(il)

+-+ E

( S ; ) (i 1).

Proof of 3): By the previous propositions, we may assume R“ (h, i ) and so Sl< ( i , ) - 3 j

< iT
Let < be the usual ordering of w . Let i be the integer satisfying lil + < = I < 1 + 1. S , is defined to be Sl? <. The theorem means that if < be a wellordering of o,then there exists a sequence Q of systems S, such that Q starts with So and the successor of S in Q is E ( S ) and the a-th member of Q is Us<, S, if a is a limit ordinal and the length of Q is 1 < 1. Now, we shall prove an additional property of S“ . =

+

PROPOSITION. There exists a provable well-ordering $ of w in E ( S ) , whose order type is greater than the order type of any provable well-ordering of w in S. PROOF: We shall prove this by enumerating all provable well-orderings of w in S . Let j be Godel’s pairing function of w . ( j is a 1-1 map from w x u onto w.) Then define < as follows. 1. If el < e , , thenj(e,, n) < j ( e 2 , rn) for any n and m . 2. If e is not a GiSdel number of a provable well-ordering in S, then j ( e , n) < j ( e, m ) if and only if n < m. 3. If $ is a provable well-ordering of w is S, then j(r$’, n) j(r$l, m) if and only if n t m . Such an ordering < is easily expressed by using Prov, and shown to be a provable well-ordering of w in E ( S ) . It is obvious that < satisfies the condition in the proposition.

+

THEOREM. In SG+ 1 ) 4 , there exists a provable well-ordering of w whose order type is greater than I iI 4 .

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FORMALIZATION PRINCIPLE

PROOF:This is proved by transfinite induction on li‘ proposition.

and the previous

6. The theory on L and remarks

Since the truth definition of the first order sentences can be expressed by using second order language and the second order language is a part of our language, we can define the following notion P in our system. P(n) is defined to be “n is a Godel number of a first order formula $(xl) in which x1 is only free variable and 3!ct+(a)”. Now 4 is defined as follows. 1) If i P(n) and i P(m), then n < m is defined to be n
4) ‘+l(Xl)T

< r+Z(xd1 if ~(r+l(xl)l) Amfh(xl)l)

< is easily expressed in

A

~

1W ~+ )l ~(a) A $2 (a)).

3a3B(u < P A

$1

(a).

$2

(PI).

our language and is a well-ordering of w. It is known that F”l
-+

.

~

r

*

l

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References 1. FEFERMAN, S., Transfinite recursive progressions of axiomatic theories, J. Symb. Logic 27 (1962) 259-316. 2. GODEL,K., Remarks before the Princeton Bicentennial Conf. on Problems in mathematics, in: The undecidable, ed. M. Davis (Raven Press, New York, 1965) pp. 84-88. 3. SWARD, G., Transfinite sequences of axiom systems for set theory, Thesis, University of Illionois, 1967. 4. TAKEUTI, G . ,The universe of set theory, to appear.