1-Consistency and faithful interpretations

1 - Consistency and Faithful Interpretations* b y S. F~,F~BMA~, G. KR:EISEL and S. GREY 1)

1. Introduction. l(a) Basic notions: interpretations. This paper is closely related to parts of [3] and [11]. The notation is a combination from those two papers. However, this paper can be read independently b y anyone familiar with the topics covered in Kleene's book [6]. Briefly, the following notions are involved. The theories considered here are formalized within the classical first-order predicate calculus with identity. Each theory is determined b y a set A of senfences, regarded as the extra-logical axioms, and a set K of basic constant (individual, function, and relation) symbols. Pr.4 denotes the set of formulas provable from A; we also write ~--ar if r We put Pr/a[(P,~ ] if the sequence ~ is a proof from A of r Given theories A (with constants K) and B (with constants L) we write B_~A ifL_K a n d P r BC_ pr a . w e w r i t e A = B i f B ~ A a n d A _EB. We write B ~ A if B is relatively interpretable in A. With every particular means of providing a relative interpretation of B into A is associated an interpretation function / which associates with each particu]a formular q) of L a f o r m u l a / r of K in such a way that: (i) [ lareserves =, proloositional connectives, and quantifiers by relativization; (ii) whenever q~ is a sentence and ~---~(I) then ~--.4/~. (Cf.,e.g., [3], w2(d).) The set of all such / is denoted b y {B < A}. ] is called a

/aith/ul (relative) interpretation/unction if (ii)'

/or every sentence r

~---~r i/and only i/~---a/q~).

I f there is such a n / , we write B ~ A ; the set of all such / is denoted by

fB r -% L e t K o be the set of constants O, ', + , . . Let P be the first-order formalization of Peano's arithmetic in K o. This paper is concerned with giving condi*) Eingegangen am 6. 11. 1960. a) The research of the first two authors for this paper was supported by the Office of Ordnance Research, USA. and that of the third author was supported by the Air Force Office of Scientific Research, USA.

1 - Consistency and Faithful Interpretations

53

tions for B r A ; this will be done under the hypothesis that P -~ A. We denote b y Q Robinson's finite set of axioms for a certain subsystem of P. We assume t h a t each syntactical expression is identified with its GSdel-numbet. For the most part we center attention on theories for which A, and hence Pr,t, is r.e. (reeursively enumerable). B y the result of Craig [1], with every such A can be associated p. r. (primitive recursive) A' such that A -- A ' ; there is usually no loss of generality involved in assuming that A is already p.r.. Certain non-r, e. Pr A wiD play an important auxiliary role in our discussion. l(b) Basic notions: k-consistency. GSdel's method for eliminating primitive recursive definitions in favor of arithmetical definitions leads to certain recursive classes of formulas, which we call the PlY- and RE-formulas. A relation is p.r. (r. e.) just in case it can be bi-numerated (numerated) by a PR(RE-) formula in P (equivalently, in both cases, Q). (Cf. [3], w3). Let P U_A. We shall say that a formula q~ of K 0 is in 27~ relative to A if it is equivalent inA to a formula of the form (3Yl)(VY2) 999~ , where ~ is a PR-formula which we precede b y k alternating quantifiers. We say q) is in]/k relative to A if ~ q~ is in Z7k relative to A. When A ~ P, we drop the phrase 'relative to P ' . E v e r y formula in Z'1 is equivalent in P to an RE-formula. For simplicity, we let ~0 ~ - / / o ~ the class of PR-formulas. The following refinement of w-consistency has been introduced in [7]. Let A have constants K 0. A is said to be k-consistent (k _> 1) filer every//2-1 formula qi(x), the condition ~-~(3x)q~(x) implies that for some n, -, ~-A "~ qi(n). More generally, for A with other constants, we can define the k-consistency of A with respect to a n y given interpretation of the set of constants K 0 in A. We shall primarily be concerned with the notion of 1-consistency in this paper. I n this case the formulas r considered above are PR-formulas and, if Q U A and A is consistent, the condition -, ~-A ~ qS(~/) is equivalent to ~--~ q)(~). We use throughout the following the fact t h a t P is 1-consistent. Let the constants of A be K 0. Let A(~~ consist of A together with all sentences (Vx)q)(x) such t h a t q)(x) is a PR-formula and ~--aqS(~) for all n. The following results connecting I- consistency to A (~) will be needed below.

1.1 LEMMA. Let P E A. Then A(~ sistent.

is consistent i / a n d only i / A is 1-con-

1.2 THEOREM. Let A be r.e. and 1-consistent. Then there is a ]ormula ~(x,y) in 111 which hi-numerates the relation Pr/,t(to ) [riO, ~ ] in A(~o). The proof of 1.1 is obvious. A proof of 1. 2 can be found, for example, in [4], 4. 6. H v is a / / 1 sentence which expresses t h a t it is not provable in P, then P u {,~ v} is consistent but not 1-consistent by 1.1. Using 1.2 one can construct a sentence v i n / / , which expresses t h a t it is not provable in p(o~). B y 1.1, is true. From this, it is a direct m a t t e r to deduce the following theorem (which is not, however, essential t~ our work). 4~

54

S.Feferman, G.Kreisel and S.Orey

1.3 THEOREM. We can construct a sentence v such that P u {,-~ v} is 1-consistent but not 2.consistent. 82 generally, by using arithmetical truth definitions for formulas of class Z"k and H~, one can construct extensions of P which are k-consistent but not (k A- 1)-consistent. 1 (c) Principal results. Let P E A, and let A, B be r.e. We shall obtain below the following theorems. Provided A is 1-consistent, (2. 8) i/fl is an RE-/ormula which numerates B in P then B ~ A u {Confl}; (2.9) i] B ~ A then B .,= A. Further, (3.2) i / B is 1-consistent, Q E B, and B ~- A then A is 1-consistent. In 2.8 and 2.9, the faithful interpretation function ] is constructed via a certain formula ~(x) such that (~)

t---All (1~ *-* (~(~),

where r is any sentence of the language of B, A 1 ----A u {Confl} in 2.8 and A 1 : A in 2.9. We show that a can be chosen in ~ar a (tel. to A1). The hypothesis of w-consistency has been used in the proofs of a number of metamathematieal results. (Cf. [6], in the following, for references.) I n m a n y of these cases, inspection of the proof shows that the hypothesis of 1-consistency is sufficient. This is the ease, for example, in GSdel's original proof of the existence of undecidable sentences for P. Closely related is the usual proof that every r.e. set can be numerated in P (by an RE-formula). It later turned out that these results could also be obtained under the hypothesis of simple consistency by a variant of the original construction. Rosser's method gives an undecidable sentence for every consistent r.e. extension A of P. It is shown in [2] that every r. e. set can be numerated (again by an REformula, but not the 'standard' one) in any such extension. In contrast, the result. 3.2 here shows that the hypothesis of 1-consistency is essential to the Theorems 2.8 and 2.9 guaranteeing the existence of faithful interpretations. l(d) Discussion. Relative interpretations have been used extensively for establishing formal independence (for example, the parallel axiom) and consistency (for example, the axiom of choice), and for establishing the recursire undecidability of various theories in standard formalization. The Iatter results are based (in part) on the fact that if B -< A and B is essentially undecidable then A is undecidable. Obviously, if B ~ A and B is (simply) undecidable then so is A. However, in the cases (2.8, 2.9) in which we know the existence of faithful interpretations, we have P _EA, so that our results give no new information. Formal undecidability (i. e., consistency and independence) results for a sentence q} with respect to a theory B are traditionally established by obtaining two relative interpretations ]~{B .~ A1}, ge{B.< As}, where ~---A]q~, ~--A/~gq3 and A1, A2 are known to be consistent; of course, it and g are not faithful in this case. Some formal independence proofs by means of a diagonal argu-

1 - Consistency and Faithful Interpretations

55

ment have the following structure. For simplicity, assume P E B, though the set of constants of B m a y be larger than K 0; typically, B is some theory of sets. Then we can obtain an 1 + {B < P u via a formula satisf ng (t) above. B y a diagonal construction we can find a sentence ~ of B such that ~-B ~) {Confl} (/) *-* ~ / ~ . (In [8], p. 274, such a sentence r is obtained b y a formalization of Cantor's diagonal argument.) Then if B is 1-consistent, so that B ~) {Confl} is consistent, neither ~-B ~ nor ~-B ~ r However, it is quite conceivable that ~ or its negation is provable in B u {Confl} or even P u(Confl). 2 . 8 (with (t)) shows t h a t it is possible to choose / so t h a t this is not the case. This can be thought of as providing a 'pure' independence result in t h a t the truth of ~ (on the intended interpretation of B) will be undecided. This clarifies [8], pp. 282-284. The difference between ordinary and faithful interpretations can also be given a simple algebraic formulation in terms of the Lindenbaum algebras of the theories involved and, when the interpretations can be made total (no relativization of quantifiers), also the associated cylindric algebras. Namely, when B ~ A, any interpretation function provides a natural homomorphism of B onto a subalgebra of the algebra o f A ; when B ~ A, this homomorphism is an isomorphism. However, algebraic information concerning the algebraic structure of P and its extensions thus provided by 2. 8 and 2.9 is not new. 2. Sufficient conditions for ~ 2). We suppose throughout this and the next section t h a t A is a theory with constants K 0 (we show how to modify this restriction in w4) and t h a t B is a theory in some set of constants L. The theorems 2. 8 and 2. 9 are proved below by a modification of the arithmetized version of the completeness theorem for first-order logic, according to which, ff P E A and fl(x) is any formula which numerates B in A, we have B

A u {Con }.

(Cf., e.g.,[3], 6.2) This theorem is derived in [3] by formalizing the HenkinHasenjaeger proof. For the purposes below we analyze that proof int~ the following two stages. We begin with the assumption t h a t B is consistent. In the first stage we pass to the set C of formulas consisting of all sentences of B vi and a sequence of formulas (3vin)r -~ Sbv~nnq~n, where (3vt.) Cn is an enumeration of all formulas (~v~) r o / L . The jn are chosen in such a way that C is consistent and for any sentence ~ of L, ~-v~y if and only ff~-B[Y. The next stage consists in enlarging G to a complete and consistent set of formulas S. Let O 0. . . . . On .... be an enumeration of all formulas of L. We define formulas go..... Zn .... and sets Sn as follows: (2. 1)

Sn = U ~) (Xo . . . . . Xn-1}, where

2) The material of this section has been reported, in a weaker form, in the two abstracts [5] and [10].

56

S.Feferman, G. Kreisel and S.Orey

Then we put (2.2) S=

U S,.

In the completeness proof the set S can be used to construct a denumerable model for S; in its formalized version, the formula a(x) which defines S can be used to construct an interpretation / of B into A under the formal hypothesis Confl of the consistency of B. I t is shown in the proof of [3], 6. 2, t h a t (r of w I holds, i.e. for all sentences (/) of L

(2. 3)

~-a u (co.~}/r

~ a(~).

Hence this interpretation will not be faithful if it should turn out t h a t for some sentence r we have ~--A u {Confl} a ( ~ ) although ~ ~-B ~ . This can happen,for example, ff for some n, q~ = On and we can prove in A u {Confl} t h a t --n ~-zn N On. The set S described above is, of course, not in general the only comp]ete and consistent extension of C. Most generally, let X be any set of natural numbers. Then define the sequence gn and Sn as follows. (2. 4)

Sn ~--- C U {Z0 . . . . . gn-i}, where

IOn i/~--snO, or -7 ~-s~O, & --1 ~-a~ "" O , & n e X Y'" = [,-~ O , i / ~ s ,

~ On or --1 ~ - z , On & --1 ~-sn "~ On & n t X .

S is then defined again as in 2.2. Suppose, in particular, that for some arithmetical formula [g(x), X is the set of n such t h a t ~ ( n ) is true. We can then construct a formula a(x) to define the new set S using this formula ~ . I f we are able to insure that no provable information can be obtained, in A u {Con 8} about the various instances ~(n), then no statement, t h a t On e Sn+l, can be proved in A u {Confl} unless already ~-snOn and hence, eventually, t h a t ~--~On (the same applies %o N On). To do this for all n we need at least t h a t

no conjunction ~(~) (i~ ^ ... ^ ~(~) (in) (i k = 0 or 1, g(0) = g, g(l~ = ~ Z) is disprovable in A u {Confl}. As it turns out, more is needed in order to insure the property of faithfulness, namely that no such conjunction is disprovable in (A u {Confl})(o~). Thus we need a result which will guarantee the existence of such formulas ~ . This will be derived from the following theorem, communicated to us by A. Mostowski but not yet published, a) 2 . 5 THEOREM. Let P E_ T. Suppose there is a/ormula ~(x,y) which bi-numerates the relation P r / f [ ~ ) , ~ ] in T. Then we can construct a ]ormula bY(x) such that/or every i o..... in -~ 0 or 1, T u {~(0) (~~..... ~(n) (in)} is consistent. The idea of Mostowski's proof is to generalize Rosser's construction of an undecidable sentence. A formula ~(x) is constructed such that for each n, bY(n) expresses: for any i o..... in-1 and q such t h a t P r / f [ ~ k
a) Added in proof: i~ostowski's result has since appeared in [12].

~(~) (ik~

57

1 - Consistency and Faithful Interpretations

gt(~), q] there exist j0 ..... jn-1 and q' such that q' _< q and Pr/T [ g-] P(k) ~ik) k
-, ~- ~(~), q']. Then it can be shown by induction on n that gt has the required property. Let us call a formula ~(x) completely independent over T if it satisfies the conclusion of 2. 5. We then obtain the following from Mostowski's proof and 1.2. 2.6 COROLLARY, Suppose P U A , where A is r.e. and 1-consistent. Then we can construct a/ormula gt(x) in H e which is completely independent over A( ~ We now turn to the formalization of the heuristic arguments given at the beginning of Chis seer The hypotheses of the following theorem permit us to carry through in one step what would be the main parts of the proofs of 2. 8 and 2.9, ff these were proved separately. 2.7 THEOREM. Suppose P C_A and that A is r. e. Let/~(x) be an RE-/ormula and fi*(x) a n y / o r m u l a . Let A 1 = A u {Confl*}; Suppose that A I is 1-consistent. Suppose that ~,fl* both numerate B in A 1 and that ~--A Prfl*(x) -> Prfl(x). Then we can find [ ~ {B <=AI} and a(x) such that ~--AJq~ +-~a(-~) /or every sentence q~ o] L. I / f l * is in Z,2 then a is in ~8 (7 [ I s (rel. to A1). Remark. W e do not know if the classification for a can be improved even when t3" is in I/1, 2:1, or Z'0. P r o @ L e t C be the set described at the beginning of this section. We can choose the sequence in, jn, ~)n described in the construction of C to be primitive recursive functions of n. Let

Then for any formula ~b (2)

~---vq) i / a n d only i] there is an n such that e-- B An ~ r

Let ~(x) be a primitive recursive term defining An in arithmetic. Then we can construc~ a formula y(x) o f K 0 such that (3)

~-p Prr(x) ~-* (3u)Prfl*(Z(u) ~ x).

From this, and the formally verifiable properties of An, we easily deduce the following three statements. (4)

~---pfl*(x) --~ )/(x) ;

(5) (6)

~-vFm. (x) ^ V.r(u) -~ (3w)[V.r(w) ^ ~((3.u)x .-2 S.b(w,u,x))]: ~ps.t(x) ~ [err(x) ~ Pr~*(~)];

(7)

i[ fl* e Z2 then the [ormula Prr(x ) is in Z2.

We claim that also (8)

Pry bi-numerates P r o in Al(~~

58

S.Feferman, G. Kreisel and S.Orey

For p* numerates B in A1, hence Prfl* numerates Pr B in A 1 ; then b y (2) and (3), Pr~ numerates Prc in A r Thus if (P e Pr o, certainly ~-AI(r P r T ( ~ ). Now suppose r r Pr c, i.e. b y (2), for all n, ~ ~-B(An -~ r Since fl is an REformula, it is seen t h a t the foImnla (3u)Prfl(~(u) .-2 x) is in ~1. B u t fl numerates B in A 1 and A 1 is 1-consistent, hence (Vu) ~ Prfl(~(u) .~ ~ ) is true and thus ~--A~(~o)(Vu) N Pr/~(2.(u) ? ~ ) . But then by the hypothesis connecting fi and fl*, ~-a#~)(Vu) "~ Prfl*(~(u) ~. ~ ) , i.e. ~-Al(w) ~ Pry(P). Thus (8) is proved. Let ~Y(x) be a formula which is completely independent over AlIm), ~ ( x ) in I/2, b y 2. 6. Consider a primitive recursive enumeration On of all foImulas of L. Let the corresponding primitive reeursive term in arithmetic be vq(x). We then define a formula Z(x,y) such tha~ Z(~,~) is true if and only if r = %n; here gn is defined by the condition 2.4 with respect to the set X of n such t h a t ~(~) is true. This is given as follows. (9)

Let 7~(~,u,z) be Prr([~ (z). w ~ ~). w<2

Then let ~(x,y,z) be S.q(z) ^ L(z) = x' ^ (z). x = y ^ (Vu) {u <_ x -~ [~(0(u),u,z) v ~ ( 0 ( u ) , u , z ) ^ ~ ~ ( . ~ # ( u ) , z ) ^ ~ ( u ) ] ^ (z). u = 0(u) v [~(.~0(u),u,z) v ~(#(u), u, z) ^ ~ ( ~ . 0(u),u,z) ^ ~ ~ ( u ) ] ^ (z). u = . ~ #(u)}.

Finally, let %(x,y) be (3z)0(x,y,z). T h e n we can prove b y induction on x in A i, (10) and (11)

~,(Yx)(BY)(3z)[e(x,Y,Z) ^ (VYl)(Yzi~(~(x,Yl,Zl) -~ Y = Yl ^ z = zi) ] ~--x, ~(x,y,z) n u < x -+ %(u,(z).u).

We t h e n let (12)

a(x) ---- (=lw)x(w,x).

I t is then seen t h a t (13)

~--~ Scma;

Here Scma expresses t h a t S is s~rongly complete, i. e. is complete, consistent, 9 vi and contains Sbvi ~ for some j whenever it contains (~v~)~). I t follows from (4), (9) and (12) t h a t (14)

~--~fl*(x) -~ a(x).

Then just as in the proof of [3], 6. 2, we can construct ] such t h a t (1.5)

] e {B -< Ax},

and

(16)

~-~,l~ *~ a@)

for every formula r of L.

1 - Consistency and Faithful Interpretations

59

Now let ~ be a sentence of L and suppose that ~--~J~ but -7 ~ - ~ . Then, as we have already observed, also -7 F--cr We next define a sequence of formulas g~ and sets S~ as follows: (17)

8~ = ~ u {z~ ..... z~-l},

where

: n i/ ~'-x~On or -7 ~--s:On & -~ ~-'8 "~ 0 ,

I

Then it is seen by induction that for every n, (18)

S~ u { ~ }

is consistent.

Let (19)

S'=

U S~. nero

Then (20)

S' is complete and S' = {g~..... X~.... }.

I t follows from (18)-(20) that for a certain m, (21)

~ ~ = Z~.

Define for every k (22)

i~ = l 0 i/Z~ = O~

1l q z l

,,,0~.

Then we claim that for every n (23)

~-a,(o)~(0) (i~ ^ ... ^ ~(n) (in) -~ Z(0, g~) ^ .-. ^ g(n, ~ ) .

This can be seen by induction, using (9)-(11), (17), (22) and the fact (8) that Pry bi-numerates Pro in Al(w). As a consequence of (13) and (21) we have (24)

~-a,(w)~y(~)(i,) ^ ... ^ ~ ( ~ ) am)-~ a ( N ~ )

l~ow by hypothesis, ~ - a J ~ , hence ~--~ ~(~) by (16). Thus we must conclude from (24) that Al(eo) u {~(~)(i~ ..... ~(~)~im)} is inconsistent, contrary to the choice of ~ . Hence we must have ~ - - ~ . Thus (15) can be strengthened to (25)

1 ~ {B ~ A~}.

Suppose now that fl*e2:~. B y (7h the formula P r r is in 27~,hence the formula g(t,u,z) of (9) is in X~. B y choice ~(x) is in//~. It is thus seen that ~(x,y,z) is a propositional combination of formulas in 2:~, hence is in 27a (also in Ha). Z and then a are obtained from ~ by prefixing existential quantifiers, so that ~ is in 2:3; but by (13), we have ~-a,a(x) ~-+ -'- a(.~ x), so that ~ is also in H a, relative to A~.

60

S.Feferman, G. Kreisel and S.Orey

Our two main theorems giving sufficient conditions for r are now directly obtained from 2. 7. 2. 8 T H E O R E M . Suppose P E A and that A is r.e. and 1-consistent. Let fl(x) be an R. E. /ormula which numerates B in P. Then B ~ A u {Confl}. Proo]. I f B is not consistent then ~--p ~ Confl, hence A u {Confl} is inconsistent, and it is easily seen t h a t the conclusion holds. I f B is consistent then Confl is a true H 1 sentence, hence A 1 ---- A t3 {Confl) is also 1-consistent. Because of this,/~ also n u m e r a t e s B in A i. B y taking fl* ---- ~ in 2 . 7 we obtain the desired conclusion. 2 . 9 T H E O R E M . Suppose P E_ A and that A is r.e. and 1-consistent. iT/B is r.e. and B ~ A then B <=A. Proof. Since, b y Craig's theorem [1], for every r.e. B we can find primitive recursive B' with B --- B', it suffices to assume t h a t B is primitive recursive here. We use the idea for the proof of the 'arithmetica] compactness t h e o r e m ' ([3], 6.9, or [11], 3. 1), b y constructing a fl* which numerates B in A such t h a t ~--AConfl*. We m u s t take particular note of the form of fl*. Let B ~ n be the set of q) in B, q} < n. Let ~(x) be a P R - f o r m u l a which bi-numeratcs B in P. Let (fl ~ y)(x) be fi(y) ^ y _< x. Then b y [3], 6 . 7 (1)

~--~Con~ t n

for every n. L e t (2)

fl*(x) -~ fl(x) ^ (Vy)(y -< x -~ Confl ~ y).

B y the same arguments as in [3], 5.9, (3)

fl*(x) numerates B in A

and

(4)

~-~Con~*.

Clearly,

(5)

~-~PrB*(x ) -~ I ~ ( x ) .

Thus all the conditions of 2.7 are m e t and B ~ A. (It is seen from (2) t h a t /?*~//1, hence certainly fl* ~ Z~. I t thus also follows t h a t / , ~ satisfying w 1 (c) (r with a e 27~ c~ II3, can be found.) As a particular application of 2.9, consider the case t h a t B is e m p t y . Then there exists a relative interpretation function ] into P such t h a t ~--p]r if and only f i t is logically provable. One does not see how to derive even this special consequence of 2.9 without the use of completely independent formulas. 3. A necessary condition for ~ . I n this section we assume t h a t the constants of B are also K o. We need the following 1emma. 3. 1 LEMMA. Suppose P F" A and that ] E {Q -< A}. Then ]or any PR-/ormula q)(x) with one ]tee variable x, we have

1 - Consistency and Faithful Interpretations

61

~-a(3x)(b(x) -*/(3x)r Proo]. Let ~(x) be a PR-formula. I t is well known that

(1)

~ ( 3 x ) q ~ ( x ) ~ Pr[Q]((3x)q~(x))

(eft, e.g., [3], 5. 5). l e t ~o be the conjunction of the axioms of Q, and let 0 be the empty set. Thus also (2)

~-p(Hx)q)(x) --> Pr[0] (k~o -> (Hx)q)(x))

Let N(x) be the formula of K o giving the domain of the relative interpretation f. Let Z(x) be ](x = ~), Se(x,y) be ](x' = y), Sm(x,y,z) be ](x + y = z), and Pr(x,y,z) be ](x. y = z). Let ~1 be the conjunction of the following sentences: (3x)N(x), (H !x){N(x) ^ Z(x)}, (Vx){N(x) --~ (H !y)[N(y) ^ Sc(x,y)]}, (Vx)(Vy) {l~(x) ^ N(y) --> (3 !z)[N(z) ^ Sm(x,y,z)] ^ (H !z)[N(z) ^ Pr(x,y,z)]}. Then by the logical properties of ] as a relative interpretation it is seen that for any sentence O, (3)

~ p P r [ 0 ] ( ~ ; -+ Pr[0](~l -+ / O).

Hence (4)

~

(3x)q~(x) -+ Pr[0](~l ^ 1~0 -~ ](Hx)r

B y using the Hiiber~ first e-theorem and restricted truth definitions, one can show that for any sentence X of arithmetic (5)

~-pPr[0](Z) --~ g

(cf. [9], p. 106). Hence by (4) and (5),

(6)

~-p(3x)q~(x) ^ ~u1 ^/~u o -+/(3x)q~(x).

Since ] is an interpretation of Q into A, we have (7)

~--a~i ^ ]~0

Thus the desired conclusion follows from (6) and (7). Another proof, without arithmetization, of this same lemma could be obtained as follows. Introduce a term x(x) in A/, (A with minimum symbol) such that ~'-~/x Z(r(O)), F---~/xSc(r(x), ~(x')). Call a formula 0 elementary if it is built up from formulas of the form x = y, x = 0, x' = y, x + y = z, x . y = z by means of the propositional connectives; call O bounded prenex if it is obtained from an elementary formula by prefixing quantifiers (Hx) or (VX)x < w. Then one can show that for any bounded prenex formula O(x 0.... , xn_l) we have ~ - ~ O(x o..... xa_l) -+ (]6))(r(xo) ..... T(x,_l)); in particular (and by eliminating the # symbol), if O is a sentence, ~--~ O --> ]0. However, GSdel's elimination method makes (Hx)(r q~(x) a PR-formula, logically equivalent to such a O (cf.[3], 3.9), which again gives the conclusion. Vve can directly obtain the following theorem from 3. 1.

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3.2 THEOREM. Suppose P E_ A, Q E_ B and that B is 1.consistent. Then B .*= A implies that A is 1-consistent. Proof. Suppose ] e {B ~ A}. Suppose q~(x) is a PR-formula and that ~--a (3x)q)(x). Then ~-a](3x)0b(x) by 3. 1. Hence ~-~(3x)(/)(x) by the faithfulness of 1. Thus it cannot be that ~--Q ~-- (/)(n) for all n, so ~--oO(~) for some n. But then also ~--aq~(~). A is consistent since B is consistent and B ~ A. Thus there is an n such that not ~-~ ~-- q~(~). This shows that A is 1-consistent. 3.3 COROLLARY. Suppose that P U A, Q F" B, A, B are both r.e., B is 1consistent and B -< A. Then B <=A i] and only i] A is 1-consistent. This is immediate from 2.9 and 3.2. Thus, under the hypotheses of 3.3 we have obtained a necessary and sufficient condition for ~ . Remarks on 3. 1 - 3 . 3 . (1) Suppose P E A 1 E_A 2 and that A 1 is r.e. Then P ~ A 2 implies P ~ A 1. For, by 3.2, A 2 must be 1-consistent, hence so also is A1; but P -< A 1 by the identity interpretation, so 3.3 shows that P ~ A r (2) Let ~(x) be a PR-formula which bi-numerates P . P -< P u { ~ Conn} by the identity interpretation. But, since P u {--~Con~) is not 1-consistent, 3.2 shows that -~ (P ~ P u {N Con~}). (3) The lemma 3.1 applies to 271 formulas; it cannot be improved, at least in the 27k, H~ classification. Take z(x) as in (2). I t has been shown in [3], 6. 6, that there is an / in {P u {--- Conz} -< P}, hence ] e {V -- {--~Corm} < P t3 {Con~}}. Let A = 39 u {Conn). A is consistent; thus ~--aConn but ~-~/Con~. Thus Con~ is a / / 1 formula O such that -7 ~-~O --> fq). (4) The Theorem 3.2 does not hold if we replace 'l-consistent' throughout by '2-consistent'. For consider the formula ~ (~- N ~) constructed in 1.3 such that P u ~[P} is 1-eonsisimnt but not 2-consistent. Let .4 ~ P u (Con~,kP}. Then also .4 is 1-consistent but not 2-consistent. However, by 2. 8, the 2consistent theory P has P ~ A. 4. F i n a l r e m a r k s . The extension of our results to arbitrary `4, with set of constants not necessarily K0, can be found as follows. Consider a fixed g {P ~ ,4}. We call `4 I-consistent (with respect to g) if there is no PR-fo~mula (P(x) such that ~--~g(~x)q)(x) and ~--AN gO(n) for all n. Call A reflexive (with respect to g) if ~--~gCon[F] for every finite F C `4. Replace ' P ~ `4' by ' P ~ `4' throughout the statements of our theorems. B y suitably rephrasing 2.7 it is seen that 2. 8 holds if we put '`4 u {g Con~}' for '.4 u {Con~}'. 2. 9 holds under the additional hypothesis that A is reflexive. 3. 1 holds if we write ~-~g(~x)O(x) --> ~g(~x)O(x). 3.2 holds when we replace 'Q _~ B' by 'Q -< B'. With this change, also 3.3 holds under the hypothesis that A is reflexive. Stanford University The University, Reading University of ~ n n e s o t a

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BIBLIOGRAPHY [1] Craig, W. On axiomatizability within a system. Journ. Symbolic Logic, v. 18 (1953), pp. 30-32. [2] :Ehrenfeucht, A. and Feferman, S. Representability of recursively enumerable sets in formal theories. Arch. fiir Math. Logik u. Grundlagenforschung, v. 5 (1959), pp. 38-41. [3] Feferman, S. Arithmetization of metamathematics in a general setting. Fund. Math. V. 49 (1960), pp. 35-92. [4] Feferman, S. Transfinite recursive progressions of axiomatic theories. To appear in Journ. Symbolic Logic. [5] Feferman, S. and Kreisel, G. Faithful interpretations (Abstract), Notices A.M.S., v. 6 (1959), p. 516. [6] Kleene, S.C. Introduction to metamathematies. Amsterdam 1952, x + 550 pp. [7] Kreisel, G. A refinement of w-consistency, (Abstract), Yourn. Symbolic Logic, v. 22 (1957), pp. 108-109. [8] Kreisel, G. Note on arithmetic models for consistent formulae of the predicate calculus. Fund. Math. v. 37 (1950), pp. 265-285. [9] Kreisel, G. and Wang, H. Some applications of formalized consistency proofs. Fund. Math., v. 42 (1955), pp. 101-110. [10] Orey, S. Faithful relative interpretations. (Abstract), Journ. Symbolic Logic v. 24 (1959), p. 282. [11] Orey, S. Relative interpretations. Zeitschr. f. math. Logik u. Grundlagen d. Mathematik, Bd. 7 (1961), pp. 146-153. [12] Mostowski, A. A generalization of the incompleteness theorem. F u n d . Math. v. 49 (1961), pp. 205-232.