for all ; and all formulas A of L(HA ). Classically we have the following well-known situation: (E-)PA + AC = (E-)PA + ACR has the strength of full si

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1 A note on Goodman's theorem Ulrich Kohlenbach Deptment of Mathematics University of Michigan Ann Arbor MI 48109, USA April 1997 Abstract Goodman's theorem states that intuitionistic ithmetic in all nite types plus full choice, HA + AC, is conservative over rst-order intuitionistic ithmetic HA. We show that this result does not extend to vious subsystems of HA, HA with restricted induction. 1 Introduction Let E-HA denote the system of extensional intuitionistic ithmetic and HA its `neutral' viant as dened in [10]). 1 E-PA and PA e the corresponding theories with classical logic. PA (resp. HA) is rst-order Peano ithmetic (resp. its intuitionistic version) with all primitive recursive functions. T denotes the set of nite types. The schema AC of full choice is given by S ; 2T AC;, where AC ; : 8x 9y A(x; y) 9Y () 8x A(x; Y x) and A is an bitry formula of L(HA ). We also consider a restricted (ithmetical) form of AC: AC 0;0 : 8x 0 9y 0 A(x; y) 9f 0(0) 8x 0 A(x; f x); where A contains only quantiers of type 0 (but maybe pameters of bitry type). By the rule of choice we mean ACR : 8x 9y A(x; y) 9Y () 8x A(x; Y x) 1 Note that both theories contain only equality of type 0 as a primitive relation symbol. 1

2 for all ; and all formulas A of L(HA ). Classically we have the following well-known situation: (E-)PA + AC = (E-)PA + ACR has the strength of full simple type theory. 2 Furthermore, even PA + AC 0;0 is not 0 1-conservative over PA : PA + AC 0;0 proves the consistency of Peano ithmetic 3 whereas PA is conservative over PA. These facts e in shp contrast to the intuitionistic case where { in pticul { the following is known: 1) (E-)HA is closed under ACR. 2) (E-)HA + AC is conservative over (E-)HA for all formulas A which e implied (relative to (E-)HA ) by their modied-realizability interpretation. 3) E-HA + AC is conservative over HA. 4 1) and 2) e proved e.g. in [10],[11] using the `modied-realizability-with-truth' resp. the `modiedrealizability' interpretation. 3) for HA instead of E-HA is due to [4],[5] (a dierent proof is given in [7]). The generalization to E-HA was established by Beeson in [1],[2] (using [6] and solving a problem stated by H. Friedman) who refers to 3) for HA as `Goodman's theorem' (see also [9] for an even stronger result). Let us switch now to the subsystems, (E-)PA jn, (E-)HA jn of (E-)PA, (E-)HA where the full schema of induction is replaced by the schema of quantier{free induction QF-IA : A 0 (0) ^ 8x 0 A 0 (x) A 0 (x 0 ) 8x 0 A 0 (x); where A 0 is a quantier-free formula, and instead of the constants of (E-)HA we only have 0 0 and symbols for every primitive recursive function plus their dening equations (but no higher type functional constants). If we add to (E-)PA j n, (E-)HA j n the combinators ; and ;; (from HA ) as well as the predicative Kleene recursors b R (and { in the case of HA jn { also a functional allowing denition by cases which is redundant in E-HA j n ) we obtain the systems, (E-)d PA jn, (E-)d HA jn due to Feferman (see [3]). PAj n, HAj n e the restrictions of PA, HA with quantier-free induction only. Since both the modied-realizability interpretation with and without truth hold for (E-)d HA jn one easily obtains the results 1),2) above also for this restricted system. In this note we observe that 3) fails for HA d jn (and even for HA j n ) and HAj n instead of (E-)HA and HA. 2 Every instance of AC is derivable in PA + ACR and ACR is a derivable rule in (E-)PA + AC. 3 PA + AC 0;0 proves the schema of ithmetical comprehension (with bitry pameters) and therefore is a nite type extension of (a function viable version of) the second-order system ( 0 1-CA) which is known to prove the consistency of PA (see e.g. [3](5.5.2)). 4 HA can be considered as a subsystem of E-HA either by switching to a denitorial extension of E-HA by adding all primitive recursive functions or modulo a suitable bi-unique mapping which translates HA into a subsystem of E-HA, see [10] (1.6.9) for the latter. 2

3 2 Results Proposition 1 Let A 2 L(PA) be such that PA ` A. Then one can construct a sentence ~ A 2 L(PA) such that A $ ~ A holds by classical logic and HA j n + AC 0;0 ` ~ A. Proof: Let A be such that PA ` A. By negative translation (F ; F 0 ) it follows that HA ` A 0. So there exists a nite set G 1 ; : : : ; G k of instances of the schema of induction IA in HA (more precisely G i is the universal closure of the corresponding instance) such that HAj n ` G i A 0 : Let Gi b (x; a) be the induction formula belonging to G i (number) pameters of the induction formula), i.e. (x being the induction viable and a the G i 8a bgi (0; a) ^ 8x bgi (x; a) b Gi (x 0 ; a) 8x b Gi (x; a) : Dene G i : 8x; a9yy = 0 0 $ b Gi (x; a) and consider the formula ~A : G i A 0 : With classical logic (and 0 6= S0) we have ~ A $ A0 $ A. Using AC 0;0 applied to Gi one shows HA j n + AC 0;0 ` Gi 9f8x; a f xa = 0 0 $ Gi b (x; a) : QF-IA applied to A 0 (x) : (f xa = 0 0) now yields G i. Hence HA j n + AC 0;0 ` G i G i and therefore HA jn + AC 0;0 ` G i A 0 : Hence ~ A satises the claim of the proposition. Corolly 2 For every n 2 IN one can construct a sentence A 2 L(HAj n ) (involving only 0; S; +; and logic) such that HA j n + AC 0;0 ` A; but PAj n + 0 n-ia `= A: 3

4 Proof: It is well-known (see e.g. [8]) that for every n 2 IN there exists an ithmetical sentence ~ A (involving only 0; S; +; and logic) such that PA ` ~ A (and even PAjn + 0 n+1-ia ` ~ A), but PAjn + 0 n-ia `= ~ A. The corolly now follows from the proposition above. Corolly 3 (E-)HA jn + AC 0;0 fails for HA jn, HAj n and therefore also for d HA j n, HAj n. is not conservative over HAjn. In pticul, Goodman's theorem For d HA jn instead of HA j n and without the restriction to the language f0; S; +; g but with A 2 L(HAj n ), the proof of proposition 1 is even more easy: It is well-known (see [8]) that for every n 2 IN there is an instance ~A : 8x < a9ya(x; y) 9y 0 8x < a9y < y 0 A(x; y) ; where A 2 0 n, of the so-called `collection principle for 0 n-formulas' 0 n-cp (A containing only number pameters) such that PAj n + 0 n-ia `= ~ A: However HA d jn + AC 0;0 ` A: ~ applied to 8x9y(x < a A(x; y)) yields a function f such that 8x < aa(x; f x). Applying AC 0;0 the functional max f x = max(f0; : : : ; f x) (denable in d HA jn ) to f and a 1 yields a y 0 such that 8x < a9y < y 0 A(x; y): Put y 0 := max (f; a 1) + 1. So in pticul for n = 0 we have a universally closed instance ~A : 8a; b 8x < a9ya0 (x; y; a; b) 9y 0 8x < a9y < y 0 A 0 (x; y; a; b) ; of 0 0-CP (where A 0 is quantier-free and a 0 ; b 0 ; x 0 ; y 0 e all the free viables of A 0 (x; y; a; b)) such that PAj n `= ~ A. Using coding of pairs and bounded sech (both available in PAjn ), ~ A can be expressed as a sentence having the form 8a 0 8x 0 9y 0 A 0 (x; y; a) 9y 0 B 0 (y; a) ; where A 0 and B 0 both e quantier-free. Furthermore the instance of AC 0;0 dha j n + AC 0;0 is So we have the following 8a 0 8x 0 9y 0 A 0 (x; y; a) 9f8xA 0 (x; f x; a) : needed to prove ~ A in Proposition 4 One can construct a sentence A : 8a 0 8x 0 9y 0 A 0 (x; y; a) 9y 0 B 0 (y; a) (A 0 ; B 0 being quantifer-free formulas of HAj n ) such that to quantier-free formulas without function pam- where AC 0;0 eters. -qf is the restriction of AC0;0 dha j n + AC 0;0 -qf ` A; but PAjn `= A; 4

5 Remk 5 Note that modied-realizability interpretation yields that d HA jn + AC is conservative over HAj n w.r.t. sentences having e.g. the form A : 8a 0 9x 0 8y 0 A 0 (x; y; a) 9y 0 B 0 (y; a). References [1] Beeson, M., Goodman's theorem and beyond. Pacic J. Math. 84, pp (1979). [2] Beeson, M., Foundations of Constructive Mathematics. Springer Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge, Band 6. Springer Berlin Heidelberg New York Tokyo [3] Feferman, S., Theories of nite type related to mathematical practice. In: Bwise, J. (ed.), Handbook of Mathematical Logic, North{Holland, Amsterdam, pp. 913{972 (1977). [4] Goodman, N., The faithfulness of the interpretation of ithmetic in the theory of constructions. J. Symbolic Logic 38, pp (1973). [5] Goodman, N., The theory of the Godel functionals. J. Symbolic Logic 41, pp (1976). [6] Goodman, N., Relativized realizability in intuitionistic ithmetic of all nite types. J. Symbolic Logic 43, pp (1978). [7] Mints, G.E., Finite investigations of transnite derivations. J. Soviet Math. 10, pp (1978). [8] Psons, C., On a number theoretic choice schema and its relation to induction. In: Intuitionism and proof theory, pp North-Holland, Amsterdam (1970). [9] Rendel de Lavalette, G.R., Extended b induction in applicative theories. Ann. Pure Applied Logic 50, (1990). [10] Troelstra, A.S. (ed.) Metamathematical investigation of intuitionistic ithmetic and analysis. Springer Lecture Notes in Mathematics 344 (1973). [11] Troelstra, A.S., Realizability. ILLC Prepublication Series for Mathematical Logic and Foundations ML{92{09, 60 pp., Amsterdam (1992). 5

Invertible elements in associates and semigroups. 1

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