J. Avigad and S. Feferman. Godel's functional ("Dialectica") interpretation. In S. R. Buss, editor, Handbook of proof theory, volume 137 of Studies in Logic and the Foundations of Mathematics, pages 337--405. North Holland, Amsterdam, 1998.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Remark 1.3.1 and Theorem 2.4.1, the situation can then be depicted as follows # EET (T IN ) Tot) Ext) PA # System T Remark 1.3. 1 where the second wavy arrow refers to a result by Godel, well known as the Dialectica interpretation, which is described for example by Avigad and Feferman [AF98]. In sum, we may thus conclude that the system # T of predicative polymorphism is of the same proof theoretic strength as PA. It is also well known, that PA and System T are prooftheoretically equivalent, which leads us to the conclusion that # is conservative over System T . Therefore, from a ....

J. Avigad and S. Feferman. Godel's functional ("dialectica") interpretation. In S. R. Buss, editor, Handbook of Proof Theory, chapter V, pages 337--405. Elsevier, 1998.

....on the use of the additional axioms for UWKL only. The conservation results in [10] are much more general than the one we mentioned. This makes the proofs more involved than is needed for the special ( 2 )case relevant here. A corresponding simpli cation of our argument has been worked out in [1]. Brackets whose occurrences are uniquely determined are often omitted, e.g. we write 0(00) instead of 0(0(0) Furthermore we write for short k : 1 instead of ( k ) 1 ) Pure types can be represented by natural numbers: 0(n) n 1. The types 0; 00; 0(00) 0(0(00) are ....

Avigad, J., Feferman, S., Godel's functional (`Dialectica') interpretation. In: [2],

....the previously mentioned canonical embedding) the second order system ( n AC) Proof: Similar to the proof of proposition 2.12. In the notation of [15] n CA) is the system n CA 0 full induction. This follows from [3] together with elimination of extensionality (see also [1]) In the notation of [15] this system is n AC 0 full induction. 3 Generalization of WKL to more complex trees: De nition 3.1 The generalization of WKL to n trees is given by n WKL : 8n 9f 1 18 n nA(f n) 9f 1 18n A(fn) where A(k ) 2 n (with arbitrary further ....

Avigad, J., Feferman, S., Godel's functional (`Dialectica') interpretation. In: [2], pp. 337-405 (1998).

....1 WKL suces to prove the uniform continuity of continuous functions : 0; 1] IR (and more general: for continuous functions In the notation of [34] n CA) is the system n CA 0 full induction. This follows from [5] together with elimination of extensionality (see also [1]) 25 from compact metric spaces into Polish spaces) However, in order to show the existence of a modulus of uniform continuity function we apparently need a slightly stronger form 1 WKL : De nition 5.15 Let A(a ) 2 n (with arbitrary parameters) n WKL : 8h 9f 1 18 n ....

Avigad, J., Feferman, S., Godel's functional (`Dialectica') interpretation. In: [3],

....with extended introductory notes by A.S. Troelstra can be found in [46] 122] chapter 3, section 5; this is a very concise and compact treatment of functional interpretation) 101] covers in detail C. Spector s extension of functional interpretation to analysis by means of bar recursion) [3] (a very readable and comprehensive treatment of the whole subject) A variant of functional interpretation which interprets A # A # A in a simpler way so that the decidability of prime formulas is not needed was developed in [31] 13] contains an interesting discussion about the functional ....

....in the sense of Godel s T (i.e. given by a closed term of WE HA # ) We don t include a proof of this theorem here as it will follow from the stronger result theorem 8.8 which we prove in the next chapter. For a nice presentation of the original proof from [71] with some simplifications see also [3]. Corollary 7.22 ( 71] d E PA # QF AC 1,0 QF AC 0,1 WKL is # 0 2 conservative over PRA. Suggested further reading: The combination of negative translation and functional interpretation has been studied as a single interpretation in [113] An extension of functional ....

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Avigad, J., Feferman, S., Godel's functional (`Dialectica') interpretation. In: [24], pp. 337-405 (1998).

....be considered finitary since PA is not finitary. Bernays even defends that Gentzen T I # 0 is less non finitary than the whole body of intuitionistic methods in number theory ( Kle52] pg. 498) 4 For more information on the history of Gdel s Functional Interpretation see [Fef93] Gd90] and [AF98]. 5 The slogan Proof Mining was recently suggested to Ulrich Kohlenbach by Dana Scott to designate the project which was previously called Unwinding of Proofs by Georg Kreisel described below. 3 to obtain that interpretation, he extended the so called # substitution (developed by Hilbert, ....

.... z (i.e. z is the Gdel number of that computation) The predicate T (x, y, z) is decidable (even primitive recursive) and the predicate T # (x) #yT (x, x, y) is undecidable (but recursively enumerable) The material of the following section on Proof Mining is substantially based on [AF98], BS95] BSBar] Bus95] Koh98a] Koh93a] Tro73] and [Tv88] 4 2 Proof Mining The general purpose of Proof Mining is to extract from a given proof of a formula A in a system A some constructive content. By constructive content we normally mean a realizing term for the existential ....

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J. Avigad and S. Feferman. Gdel's functional ("Dialectica") interpretation. In S.R. Buss, editor, Handbook of proof theory, volume 137, pages 337--405. Elsevier, North-Holland, Amsterdam, 1998.

....study the relationship of our variant of bar recursion with others. x1. Introduction. In [21] Spector extended G odel s Dialectica Interpretation of Peano Arithmetic [9] to classical analysis using bar recursion in nite types. Although considered questionable from an intuitionistic point of view ([1], 6.6) there has been considerable interest in bar recursion , and several variants of this de nition scheme and their interrelations have been studied by, e.g. Schwichtenberg [18] Bezem [7] and Kohlenbach [13] In this paper we add another variant of bar recursion and use it to give a ....

J. Avigad and S. Feferman, Godel's functional ("Dialectica") interpretation, Handbook of proof theory (S.R. Buss, editor), vol. 137, Elsevier, North-Holland, Amsterdam, 1998, pp. 337-405.

.... can be built for a variety of logics including proper fragments of Int, classical logic ( 19] 81] 82] Abstract computational and functional semantics for Int which did not address the issue of the original BHK semantics for Int were also studied in [64] 87] and many other papers (cf. [17], 20] 96] Kuznetsov Muravitsky Goldblatt semantics for Int is based on a nonconstructive notion classically true and formally provable incompatible with the BHK semantics. In particular, it does not contain any BHK constructions or proofs whatsoever. As far as S4 is concerned the ....

J. Avigad and S. Feferman, Godel`s functional ("Dialectica") interpretation, Handbook of proof theory (S. Buss, editor), Elsevier, 1998, pp. 337--406.

....there is in fact an interesting kind of reverse mathematics for such principles which naturally takes place over a conservative finite type extension of RCA 0 as base system. 1 As a natural candidate we propose the system RCA # 0 : E PRA # QF AC 1,0 , where E PRA # is Feferman s ( 4] [1]) restriction of E PA # with quantifier free induction and predicative primitive recursion only. 2 We will show that RCA # 0 is conservative over RCA 0 so that for principles which can be formalized already in RCA 0 nothing is lost by using RCA # 0 as the base system. In this paper we show ....

....0 are equivalent to (# 2 ) # ## 2 #f 1 # #f = 0 0 # #x 0 (fx = 0 0) # form a rich and very robust class. We conjecture that one get s further interesting and robust classes by considering other functional existence principles than (# 2 ) like the existence of the Suslin operator ([1], 4] Suslin) #S 2 #f 1 # S(f) 0 0 # #g#x # f(gx) 0 0 ## 1 Here (and also two sentences below) we again identify the o#cial formulation of RCA 0 (from [16] with its (inessential) variant with function variables instead of set variabales. As soon as we have defined that variant ....

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Avigad, J., Feferman, S., Godel's functional (`Dialectica') interpretation. In: [3], pp. 337-405 (1998).

....Symbolic Logic 1 2 ULRICH KOHLENBACH system which can be (e#ectively) reduced to PRA allows one to extract at least a primitive recursive algorithm. In the other direction, e.g. our analysis of proofs in approximation theory (which used the principle of the attainment of the maximum of f # C[0, 1], see [20] led us to an elimination procedure of weak Konig s lemma WKL over a variety of subsystems of arithmetic in all finite types thereby contributing to 1) above (see [19] Likewise our treatment of e.g. the Bolzano Weierstra principle in [26] via an elimination technique of Skolem ....

....to f yields ## # 1 1#x 0 (#(#x) 0) # #x## # 1 1(#(#x) 0) i.e. ## = #(# min n[#(#n) 0] 1 is uniformly continuous on # : # # 1 1 . This argument can be adopted to real functions encoded as in reverse mathematics to show in that context that e.g. every continuous function f : [0, 1] # IR is uniformly continuous. Together with QF AC 0,0 one even gets a modulus of uniform continuity (see proposition 4.10) In our direct type 2 treatment of continuous functions # : IN IN # IN as functionals # 2 satisfying #f 1 #n 0 #g 1 (fn = gn # #f = #g) the binary tree to ....

[Article contains additional citation context not shown here]

Avigad, J., Feferman, S., Godel's functional (`Dialectica') interpretation. In: [3], pp. 337-405 (1998).

....the use of the additional axioms for UWKL only. The conservation results in [10] are much more general than the one we mentioned. This makes the proofs more involved than is needed for the special (# 0 2 )case relevant here. A corresponding simplification of our argument has been worked out in [1]. 3 Every #( #) recursive function is provably recursive in E PRA # i UWKL M. Acknowledgement: This paper was prompted by discussions the author has had with Gerhard Jager and Thomas Strahm who asked him about the status of the uniform weak Konig s lemma in a fully extensional (classical) ....

Avigad, J., Feferman, S., Godel's functional (`Dialectica') interpretation. In: [2], pp. 337-405 (1998).

....Research in Computer Science, Centre of the Danish National Research Foundation. 1 PRA allows one to extract at least a primitive recursive algorithm. In the other direction, e.g. our analysis of proofs in approximation theory (which used the principle of the attainment of the maximum of f # C[0, 1], see [20] led us to an elimination procedure of weak Konig s lemma WKL over a variety of subsystems of arithmetic in all finite types thereby contributing to 1) above (see [19] Likewise our treatment of e.g. the Bolzano Weierstra principle in [26] via an elimination technique of Skolem ....

....to f yields ## # 1 1#x 0 (#(#x) 0) # #x## # 1 1(#(#x) 0) i.e. ## = #(# min n[#(#n) 0] 1 is uniformly continuous on # : # # 1 1 . This argument can be adopted to real functions encoded as in reverse mathematics to show in that context that e.g. every continuous function f : [0, 1] # IR is uniformly continuous. Together with QF AC 0,0 one even gets a modulus of uniform continuity (see proposition 4.10) In our direct type 2 treatment of continuous functions # : IN IN # IN as functionals # 2 satisfying #f 1 #n 0 #g 1 (fn = gn # #f = #g) the binary tree to ....

[Article contains additional citation context not shown here]

Avigad, J., Feferman, S., Godel's functional (`Dialectica') interpretation. In: [3], pp. 337-405 (1998).

....numbers and A is any formula 1 , are studied. In [5] Feferman noticed that the proof theoretic strength of such systems can be determined by functional interpretation based using his non constructive operator and his classical results on the strength of systems based on this operator (see [1] for a survey of those results) In this note we show that a similar use of functional interpretation combined with the majorization arguments which we developed in [8] can be used to determine the strength of systems which instead of NOS are based on the weaker schema of lesser numerical ....

....system based on LNOS and the full axiom schema of choice AC which allows to prove the version of Konig s lemma studied in [6] and is # 0 2 conservative over PRA. In the following HA # and d HA # are the systems of arithmetic in all finite types denoted by WE HA # and WE d HA # in [1], where, however, the quantifier free rule of extensionality is defined as # A 0 # s = # t # A 0 # r[s] # r[t] where A 0 is quantifier free. 2 d HA # contains only recursion on type 0 and induction restricted to # 0 1 formulas. d HA # is the still weaker system with ....

Avigad, J., Feferman, S., Godel's functional (`Dialectica') interpretation. In: [3], pp. 337-405 (1998).

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J. Avigad and S. Feferman. Godel's functional ("Dialectica") interpretation. In S. R. Buss, editor, Handbook of proof theory, volume 137 of Studies in Logic and the Foundations of Mathematics, pages 337--405. North Holland, Amsterdam, 1998.

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J. Avigad and S. Feferman, Godel's functional ("Dialectica") interpretation, Handbook of proof theory (S. Buss, editor), Elsevier, 1998, pp. 337--406.

No context found.

J. Avigad and S. Feferman. Godel's functional ("Dialectica") interpretation. In S. R. Buss, editor, Handbook of proof theory, volume 137 of Studies in Logic and the Foundations of Mathematics, pages 337--405. Elsevier, North-Holland, Amsterdam, 1998.

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J. Avigad and S. Feferman, Godel's functional ("Dialectica") interpretation, Handbook of proof theory (S. R. Buss, editor), Studies in Logic and the Foundations of Mathematics, vol. 137, North Holland, Amsterdam, 1998, pp. 337--405.

No context found.

J. Avigad and S. Feferman. Godel's functional ("Dialectica") interpretation. In S.R. Buss, editor, Handbook of proof theory, volume 137, pages 337--405. Elsevier, NorthHolland, Amsterdam, 1998.

No context found.

J. Avigad and S. Feferman. Godel's functional ("Dialectica") interpretation. In S.R. Buss, editor, Handbook of proof theory, volume 137, pages 337--405. Elsevier, NorthHolland, Amsterdam, 1998.

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J. Avigad and S. Feferman, Godel's functional ("Dialectica") interpretation, Handbook of Proof Theory, S. R. Buss ed, North-Holland Publishing, (1998) 337-405.

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