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Basic Research in Computer Science BRICS RS-99-20 U. Kohlenbach: A Note on Spector’s Quantifier-Free Rule of Extensionality

A Note on Spector’s Quantifier-Free Rule of Extensionality

Ulrich Kohlenbach

BRICS Report Series ISSN 0909-0878

RS-99-20 August 1999

c 1999, Copyright

Ulrich Kohlenbach. BRICS, Department of Computer Science University of Aarhus. All rights reserved. Reproduction of all or part of this work is permitted for educational or research use on condition that this copyright notice is included in any copy.

See back inner page for a list of recent BRICS Report Series publications. Copies may be obtained by contacting: BRICS Department of Computer Science University of Aarhus Ny Munkegade, building 540 DK–8000 Aarhus C Denmark Telephone: +45 8942 3360 Telefax: +45 8942 3255 Internet: [email protected] BRICS publications are in general accessible through the World Wide Web and anonymous FTP through these URLs: http://www.brics.dk ftp://ftp.brics.dk This document in subdirectory RS/99/20/

A note on Spe tor's quanti er-free rule of extensionality Ulri h Kohlenba h

BRICS Department of Computer S ien e University of Aarhus Ny Munkegade DK-8000 Aarhus C Denmark Mar h 1999

Abstra t

In this note we show that the so- alled weakly extensional arithmeti in all nite types, whi h is based on a quanti er-free rule of extensionality due to C. Spe tor and whi h is of signi an e in the

ontext of Godel's fun tional interpretation, does not satisfy the dedu tion theorem for additional axioms. This holds already for 01 axioms. Previously, only the failure of the stronger dedu tion theorem for dedu tions from (possibly open) assumptions (with parameters kept xed) was known.  Basi Resear h in Computer S ien e, Centre of the Danish National Resear h Foundation.

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1 Introdu tion Let E-HA! denote the system of extensional intuitionisti arithmeti in all nite types as de ned in [5℄. Con erning equality, E-HA! only ontains equality =0 between numbers as a primitive predi ate. For  = 0k : : : 1 , x1 = x2 is de ned as 8y11 ; : : : ; ykk (x1 y1 : : : yk =0 x2 y1 : : : yk ). In the ontext of Godel's fun tional (`Diale ti a') interpretation, a variant WE-HA! (weakly extensional intuitionisti arithmeti in all nite types) of E-HA! is of relevan e whi h instead of the extensionality axioms (E ) for all types only has the following quanti er-free rule of extensionality QF-ER:

A0 ! s =  t ; A0 ! r[s℄ = r[t℄

where A0 is quanti er-free, s ; t; r[x℄ are arbitrary terms of the system and ;  2 are arbitrary types. WE-PA! denotes the variant of WE-HA! with

lassi al logi . In ontrast to (E ), Godel's fun tional interpretation trivially satis es QF-ER whi h was introdu ed in [4℄ for that very reason. It has been observed in the literature ([5℄(3.5.15 and 1.6.12), see also [6℄ for orre tions) that WE-HA! doesn't satisfy the dedu tion theorem for `dedu tions from open assumptions' (whose free variables are treated as parameters and hen e are not permitted as proper variables in the quanti er rules as formulated in [5℄).1 The argument pro eeds as follows: onsider f

=1 g `WE-HA! f =1 g;

where f; g are free fun tion variables. QF-ER yields f =1 g `WE-HA! 8z 2 (zf =0 zg ): 1 In

order to avoid this onsequen e, Troelstra uses a weaker form of QF-ER where the premise of the rule is required to be derivable without assumptions. In this paper we deal with Spe tor's original rule and our de nition of WE-HA! thereby diers from Troelstra's de nition in [5℄. The dedu tion theorem for dedu tions from assumption, however, does hold { under an appropriate variable ondition { for the quanti er-free fragment qf-WEHA! of WE-HA! (see [1℄).

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The dedu tion theorem for derivations under assumptions would yield `WE-HA! f =1 g ! 8z 2 (zf =0 zg );

whi h is underivable in WE-HA! as follows from [2℄ and the fa t that WEHA! has a fun tional interpretation in (the weakly extensional version of) Godel's T . This, however, leaves it open whether the dedu tion theorem also fails for assumptions added as axioms, i.e. assumptions whi h impli itly are understood as universally losed. In this note we show that the dedu tion theorem (both for WE-HA! as well as for WE-PA! ) already fails for 01 -axioms.

2 Results Theorem 2.1

su h that

There exists a 01 -senten e A and a quanti er-free formula B

WE-HA! + A ` B; but WE-PA! `= A ! B: Proof: Let ConPA the standard onsisten y predi ate for Peano arithmeti PA. In WE-HA! , ConPA an be written as A : 8x0 (tPA x =0 0) for a suitable losed term tPA of WE-HA! . WE-HA! + A ` tPA =1 01 ; where 01 := x0 :00: By QF-ER we obtain WE-HA! + A ` x2 (tPA) =0 x(01 ); where x2 is a free variable of type 2. Let's assume now that () WE-HA! ` A ! x2 (tPA) =0 x(01): Then a fortiori WE-HA! ` A ! 8x 2 12 (x(tPA) =0 x(01 )) and hen e WE-PA! ` 8x 2 129y0(tPAy =0 0 ! x(tPA) =0 x(01)); 3

where 12 := x1 :S 0 and x1 2 x2 : 8y1(x1 y 0 x2 y). By orollary 3.4 from [3℄ there exists a losed term s0 of WE-HA! su h that WE-HA! ` 8y 0 s(tPAy =0 0) ! 8x 2 1(x(tPA) =0 x(01)): By the omputability of every xed losed term s in WE-HA! , there exists a number n 2 IN su h that WE-HA! ` s =0 n: Sin e (by 01 - ompleteness of WE-HA! ) WE-HA! ` 8y 0 n(tPA y =0 0); we get and therefore

WE-HA! ` 8x 2 12 (x(tPA) =0 x(01))

WE-HA! ` tPA =1 0; i:e: WE-HA! ` ConPA; whi h ontradi ts Godel's se ond in ompleteness theorem, sin e WE-HA! is

onservative over Heyting arithmeti HA (as follows by formalizing the model HEO of all hereditarily ee tive operations in HA, see [5℄). Hen e () above is false. So the theorem holds with B : (x2 (tPA) =0 x(0)) and A as above. ! ! Corollary 2.2 The dedu tion theorem for both WE-PA and WE-HA fails already for losed 01 -axioms. The argument above an be applied also to stronger systems whi h allow a fun tional interpretation by majorizable fun tionals. Then we have to use a onsisten y predi ate for a suÆ iently strong system. Remark 2.3

The failure of the dedu tion theorem for WE-PA! (already for 01 -axioms) might suggest that a system like Troelstra's [5℄ PA! (=(HA! ) ) whi h is neutral with respe t to extensionality but still only ontains equality for numbers as a primitive predi ate, would be more favorable in the ontext of fun tional interpretation. However, we believe that for appli ations to mathemati s and the extra tion of data from given proofs it is Final Comments:

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desirable to have as mu h extensionality avalaible as possible. If we work in WE-PA! + A and want to shift A to an impli ative premise of the on lusion, then we an do this provided that we restri t + to  where WE-PA!  A means that A must not be used in the proof of the premise of an appli ation of QF-ER. This is a less severe restri tion than to work in PA! + A.

Referen es [1℄ Bezem, M., Equivalen e of bar re ursors in the theory of fun tionals of nite type. Ar h. Math. Logi 27, pp. 149{160 (1988). [2℄ Howard, W.A., Hereditarily majorizable fun tionals of nite type. In: Troelstra (ed.), Metamathemati al investigation of intuitionisti arithmeti and analysis, pp. 454-461. Springer LNM 344 (1973). [3℄ Kohlenba h, U., Pointwise hereditary majorization and some appli ations. Ar h. Math. Logi 31, pp. 227-241 (1992). [4℄ Spe tor, C., Provably re ursive fun tionals of analysis: a onsisten y proof of analysis by an extension of prin iples formulated in urrent intuitionisti mathemati s. In: Re ursive fun tion theory, Pro eedings of Symposia in Pure Mathemati s, vol. 5 (J.C.E. Dekker (ed.)), AMS, Providen e, R.I., pp. 1{27 (1962). [5℄ Troelstra, A.S. (ed.) Metamathemati al investigation of intuitionisti arithmeti and analysis. Springer Le ture Notes in Mathemati s 344 (1973). [6℄ Troelstra, A.S., Metamathemati al investigation of intuitionisti arithmeti and analysis. Corre tions to the rst edition. ILLC Prepubli ation Series X-93-04, Universiteit van Amsterdam (1993).

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