C. Spector. Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles in current intuitionistic mathematics. In F. D. E. Dekker, editor, Recursive Function Theory: Proc. Symposia in Pure Mathematics, volume 5, pages 1--27. American Mathematical Society, Providence, Rhode Island, 1962.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... numbers are given as Cauchy sequences of rationals with fixed rate of convergence in our theories) It is well known that a constructive functional interpretation of the negative translation of AC ar requires so called bar recursion and cannot be caried out e.g. in Godel s term calculus T (see [21] and [15] AC ar is (using classical logic) equivalent to CA ar AC qf, where CA ar : g(x) 0 0 A(x) with A (and AC qf is the restriction of AC ar to quantifier free formulas) and therefore causes an immense rate of growth (when added to e.g. G 2 A ) From the work in ....

Spector, C., Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics. In: Recursive function theory, Proceedings of Symposia in Pure Mathematics, vol. 5 (J.C.E. Dekker (ed.)), AMS, Providence, R.I., pp. 1--27 (1962).

....[12] negative translation and Howard s [8] majorization technique. The rst step of the proof reduces the case with the full axiom of extensionality to a subsystem WEPRA WKL which is based on a weaker quanti er free rule of extensionality only (see below) which was introduced in Spector [17]. From this system, WKL is then eliminated. This elimination actually eliminates WKL via a strong uniform version of WKL, called UWKL below, which states the existence of a functional which selects uniformly in a given in nite binary tree an in nite path from that tree. This yields the following ....

.... primitive recursive functionals in the sense of G odel [6] and coincides with Troelstra s [18] system (E HA ) c The weakly extensional versions WE PRA of these systems result if we replace the extensionality axioms (E) by a quanti er free rule of extensionality (due to Spector [17]) QF ER: A 0 s = t A 0 r[s] r[t] where A 0 is quanti er free, s ; t ; r[x ] are arbitrary terms of the system and ; 2 are arbitrary types. Note that QF ER allows to derive the extensionality axiom for type 0 but already the extensionality axiom for type 1 arguments, ....

Spector, C., Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics. In: Recursive function theory, Proceedings of Symposia in Pure Mathematics, vol. 5 (J.C.E. Dekker (ed.)), AMS, Providence, R.I., pp. 1-27 (1962). 12

....8x A(x; y) 8x (z0 = 0 x 8z 1 A(zz 1 ; z(z 1 ) Proof: By proposition 5.3 and corollary 5. 7 one has qf 8g Pi 8 g Pi ( g) Hence the assumption of the rule to be proved yields Delta AC qf 8g Pi 8u B 0 (u; v; w) From the work of Spector [23] it follows that G n A AC qf 8g Pi 1 CA(g) has (via negative translation) a Godel functional interpretation in G n A (BR 0;1 ) by terms 2 G n R [B 0;1 ] In [2] it is shown that the type structure M of the so called strongly majorizable functionals forms a model of full bar ....

Spector, C., Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics. In: Recursive function theory, Proceedings of Symposia in Pure Mathematics, vol. 5 (J.C.E. Dekker (ed.)), AMS, Providence, R.I., pp. 1--27 (1962).

....x; a) where 8h9xB 0 (h; x; a) is the Herbrand normal form B (a) of B(a) and a are all free variables of B. ar has (via negative translation) a functional interpretation in T by terms 2 F . For ( d ) this is proved in [4] For ( d BR 0;1 ) this follows from [28] using the facts that d has an interpretation in d , that AC ar is derivable in d PA (note that d PA 1 CA and so by iteration using the presence of function parameters in Pi 1 CA also d PA 1 CA and therefore d AC ar ) and that the ....

....(note that d PA 1 CA and so by iteration using the presence of function parameters in Pi 1 CA also d PA 1 CA and therefore d AC ar ) and that the interpretation of Pi uses only B 0;1 and functionals from d . Note that the crucial lemma 1 from [28] (restricted to B 0;1 ) can easily be proved in BR 0;1 . Hence there are functionals Psi 2 F such that ( 8h; aB 0 (h; Psi( Phi A ; Phi (A B) pr ; h; a) a) Thus Psi : h: Psi( Phi A ; Phi (A B) pr ; h; a) satisfies the claim made in the proposition. 2 Remark ....

[Article contains additional citation context not shown here]

Spector, C., Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics. In: Recursive function theory, Proceedings of Symposia in Pure Mathematics, vol. 5 (J.C.E. Dekker (ed.)), AMS, Providence, R.I., pp. 1--27 (1962).

.... numbers are given as Cauchy sequences of rationals with fixed rate of convergence in our theories) It is well known that a constructive functional interpretation of the negative translation of AC ar requires so called bar recursion and cannot be caried out e.g. in Godel s term calculus T (see [23] and [18] AC ar is (using classical logic) equivalent to CA ar AC 0;0 qf, where CA ar : 9g 1 8x 0 Gamma g(x) 0 0 A(x) Delta with A 2 Pi 0 1 ; and therefore causes an immense rate of growth (when added to e.g. G 2 A ) From the work in the context of reverse ....

....negative translation) 2) 9f8x; z A 0 . However in our applications the monotone functional interpretation of (1) would require non computable functionals (since f is not recursive) and the monotone functional interpretation of (2) can be carried out only using bar recursive functionals (see [23]) In contrast to this the bound only depends on a functional which satisfies the monotone functional interpretation of the negative translation of 8x9y8z A 0 (x; y; z) In our applications in section 5 such a functional can be constructed in d PR except for the existence of the limit ....

Spector, C., Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics. In: Recursive function theory, Proceedings of Symposia in Pure Mathematics, vol. 5 (J.C.E. Dekker (ed.)), AMS, Providence, R.I., pp. 1--27 (1962). 34

....of equation will turn out to correspond to the consistency of the schema of arithmetical comprehension #f#x(f(x) 0 # A(x) where A(x) contains only number quantifiers but maybe function parameters. The solution requires so called bar recursion (of type 0) which was introduced by Spector [118] an which goes beyond Godel s primitive recursive functionals. We will discuss this further in chapters to come (see also [82] for a thorough discussion of the modus ponens problem for the no counterexample interpretation) For the time being we confine ourselves with indicating how the above ....

....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 interpretation to classical analysis was given by C. Spector using so called bar recursive functionals ([118], 57] 101] 2 In [38] Spector s approach is 2 For an interesting alternative to functional interpretation in this context see [8] CHAPTER 7. NEGATIVE AND FUNCTIONAL INTERPRETATION 62 extended to still stronger systems using bar recursive functionals of infinite types. A di#erent extension ....

Spector, C., Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics. In: Recursive function theory, Proceedings of Symposia in Pure Mathematics, vol. 5 (J.C.E. Dekker (ed.)), AMS, Providence, R.I., pp. 1--27 (1962).

....corollary 5. 7 one has GnA AC 1;0 qf 8g Pi 0 1 CA(g) F Gamma 8 g Pi 0 k UB Gamma j n ( g) Hence the assumption of the rule to be proved yields GnA Delta AC qf 8g Pi 0 1 CA(g) F Gamma 8u 1 8v tu9w fl B 0 (u; v; w) From the work of Spector [24] it follows that GnA AC qf 8g Pi 0 1 CA(g) has (via negative translation) a Godel functional interpretation in GnA i (BR 0;1 ) by terms 2 GnR [B 0;1 ] In [2] it is shown that the type structure M of the so called strongly majorizable functionals forms a model of full bar ....

Spector, C., Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics. In: Recursive function theory, Proceedings of Symposia in Pure Mathematics, vol. 5 (J.C.E. Dekker (ed.)), AMS, Providence, R.I., pp. 1--27 (1962).

....of the classical axiom of dependent choice allowing for the extraction of witnesses from proofs of 1 formulas in classical analysis. We also give a bar recursive de nition of the fan functional and 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 ....

....(s ) k) if k jsj then s k else (kjsj) s : overwriting with s, i.e. s ) k) if k jsj then s k else (k) s x : s k:x; i.e. s x) k) if k jsj then s k else x] k : h (0) k 1)i 2 k : k = k: Definition 2.1. Spector s de nition of bar recursion [21] reads in our notation as follows: s) G(s) if Y (s 0 ) N jsj H(s; x: s x) otherwise: 1) In his thesis [13] Kohlenbach introduced the following kind of bar recursion which di ers from Spector s only in the stopping condition: s) G(s) if Y (s 0 ) ....

[Article contains additional citation context not shown here]

C. Spector, Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles in current intuitionistic mathmatics, Recursive function theory: Proc. symposia in pure mathematics (F. D. E. Dekker, editor), vol. 5, American Mathematical Society, Providence, Rhode Island, 1962, pp. 1-27.

....and Howard s [8] majorization technique. The first step of the proof reduces the case with the full axiom of extensionality to a subsystem WEPRA # QF AC 1,0 QF AC 0,1 WKL which is based on a weaker quantifier free rule of extensionality only (see below) which was introduced in Spector [19]. From this system, WKL is then eliminated. This elimination actually eliminates WKL via a strong uniform version of WKL, called UWKL below, which states the existence of a functional which selects uniformly in a given infinite binary tree an infinite path from that tree. This yields the following ....

....functionals in the sense of Godel [6] and coincides with Troelstra s [20] system (E HA # ) c . The weakly extensional 3 versions WE PRA # and WE PA # of these systems result if we replace the extensionality axioms (E) by a quantifier free rule of extensionality (due to Spector [19]) QF ER: A 0 # s = # t A 0 # r[s] # r[t] where A 0 is quantifier free, s # , t # , r[x # ] # are arbitrary terms of the system and #, # # are arbitrary types. Note that QF ER allows one to derive the extensionality axiom for type 0, but already the extensionality axiom for ....

Spector, C., Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics. In: Recursive function theory, Proceedings of Symposia in Pure Mathematics, vol. 5 (J.C.E. Dekker (ed.)), AMS, Providence, R.I., pp. 1--27 (1962).

....; where A 0 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 classical logic. In contrast to (E) G odel s functional interpretation trivially satis es QF ER which was introduced 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 corrections) that WE HA doesn t satisfy the deduction theorem for deductions from open assumptions (whose free variables are treated as parameters and hence are not permitted as proper ....

Spector, C., Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics. In: Recursive function theory, Proceedings of Symposia in Pure Mathematics, vol. 5 (J.C.E. Dekker (ed.)), AMS, Providence, R.I., pp. 1-27 (1962).

....System T, in connection with what came to be known as the Dialectica Interpretation [ Godel, 1958; Godel, 1990 ] Godel presented his results as a contribution to a liberalized version of Hilbert s programme. 6 Godel s work was later extended to the bar recursive functionals by Spector [ Spector, 1962 ] who used them to give a constructive consistency proof for classical analysis. However, the first full blown generalization of ordinary recursion theory to higher types was made by Kleene in the late 1950 s (see [ Kleene, 1959; Kleene, 1963 ] for a formulation in terms of computation schemes) ....

C. Spector. Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics. In Recursive Function Theory, Proc. Symposia in Pure Mathematics V, pages 1--27. American Mathematical Society, Providence, RI, 1962.

....type, which we now call Godel s System T, in connection with what came to be known as the Dialectica Interpretation [7, 8] Godel presented his results as a contribution to a liberalised version of Hilbert s programme. 1 Godel s work was later extended to the Bar Recursive Functionals by Spector [27] who used them to give a constructive consistency proof for classical analysis. However the first full blown generalization of ordinary recursion theory to higher types was made by Kleene in the late 1950 s (see [11, 12] for a formulation in terms of computation schemes) In Kleene s theory, a ....

C. Spector. Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics. In Recursive Function Theory, Proc. Symposia in Pure Mathematics V, pages 1--27. American Mathematical Society, Providence, RI, 1962.

....recursive encodings of tree ordinals Marc Bezem Utrecht University Wilfried Buchholz Ludwig Maximilians Universitat y 1. Introduction We ask the attention for a definition schema from higher order subrecursion theory called bar recursion. Bar recursion originates with Spector [12], where bar recursion of all finite types is shown to characterize the class of provably total recursive functions of analysis. This class has also been characterized by Girard [3] as those functions which are definable in the second order typed lambda calculus (2, or the polymorphic lambda ....

C. Spector. Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics. In J.C.E. Dekker, editor, Recursive function theory, Proc. Symp. in pure mathematics V, pages 1--27. AMS, Providence, 1962.

....y Thierry Coquand Chalmers University of Gothenburg z Abstract We present a possible computational content of the negative translation of classical analysis with the Axiom of Choice. Our interpretation seems computationally more direct than the one based on Godel s Dialectica interpretation [10, 18]. Interestingly, this interpretation uses a refinement of the realizibility semantics of the absurdity proposition, which is not interpreted as the empty type here. We also show how to compute witnesses from proofs in classical analysis, and how to interpret the axiom of Dependent Choice and ....

....can be generalized to the case of the axiom of Dependent Choice, and in this case, reduce intuitionistically its correctness to a principle of bar induction. We also give an interpretation of the Double Negation Shift and try a comparison with Spector s bar recursive interpretation of DNS [18, 10, 11], which suggests a computational content of the negative interpretation of Axiom of Choice based on Godel s Dialectica translation. We end by an heuristic explanation of our realizability interpretation, based on a game theoretical analysis of proofs. Dip. Informatica, C.so Svizzera 185, 10149 ....

[Article contains additional citation context not shown here]

C. Spector. Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics. Recursive Function Theory, J.C.E. Dekker ed., Proceedings of Symposia in Pure Mathematics V, AMS, p. 1 - 27, 1961.

....r[t] where A 0 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 classical logic. In contrast to (E) G odel s functional interpretation trivially satis es QF ER which was introduced 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 corrections) that WE HA doesn t satisfy the deduction theorem for deductions from open assumptions (whose free variables are treated as parameters and hence are not permitted as proper ....

Spector, C., Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics. In: Recursive function theory, Proceedings of Symposia in Pure Mathematics, vol. 5 (J.C.E. Dekker (ed.)), AMS, Providence, R.I., pp. 1-27 (1962).

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C. Spector. Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles in current intuitionistic mathematics. In F. D. E. Dekker, editor, Recursive Function Theory: Proc. Symposia in Pure Mathematics, volume 5, pages 1--27. American Mathematical Society, Providence, Rhode Island, 1962.

No context found.

C. Spector, Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles in current intuitionistic mathematics, Recursive function theory: Proc. symposia in pure mathematics (F. D. E. Dekker, editor), vol. 5, American Mathematical Society, Providence, Rhode Island, 1962, pp. 1--27.

No context found.

C. Spector, Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics, Recursive Function Theory: Proc. Symposia in Pure Mathematics, J. C. E. Dekker ed, Vol 5, American Mathematical Society, Providence, Rhode Island, 1-27.

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