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Compiling polymorphism using intensional type analysis
 In Symposium on Principles of Programming Languages
, 1995
"... The views and conclusions contained in this document are those of the authors and should not be interpreted as ..."
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Cited by 260 (18 self)
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The views and conclusions contained in this document are those of the authors and should not be interpreted as
Effective moduli from ineffective uniqueness proofs. An unwinding of de La Vallée Poussin's proof for Chebycheff approximation
, 1993
"... We consider uniqueness theorems in classical analysis having the form Vu G(u, v1 ) = 0 = G(u, v2 ) v1 = v2 where U, V are complete separable metric spaces, Vu is compact in V and G : U constructive function. If (+) is proved by arithmetical means from analytical assumptions Yx#z ..."
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Cited by 31 (14 self)
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We consider uniqueness theorems in classical analysis having the form Vu G(u, v1 ) = 0 = G(u, v2 ) v1 = v2 where U, V are complete separable metric spaces, Vu is compact in V and G : U constructive function. If (+) is proved by arithmetical means from analytical assumptions Yx#z only (where X, Y, Z are complete separable metric spaces, Yx Y is compact and F : X IR constructive), then we can extract from the proof of (++) (+) an e#ective modulus of uniqueness, i.e. (+ + +) Vu , k v1 ), v2 ) dV (v1 , v2 ) .
FirstOrder Proof Theory of Arithmetic
, 1998
"... this article will deal only with the intensional approach. The reader who wants to see the numeralwise representability approach can consult Smorynski [1977] and any number of textbooks such as Mendelson [1987]. The intensional approach is due to Feferman [1960]. An e#ective unification of the two a ..."
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Cited by 29 (1 self)
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this article will deal only with the intensional approach. The reader who wants to see the numeralwise representability approach can consult Smorynski [1977] and any number of textbooks such as Mendelson [1987]. The intensional approach is due to Feferman [1960]. An e#ective unification of the two approaches can be given using the fact (independently due to Wilkie and to Nelson [1986]) that I# 0 +# 1 and S 2 are interpretable in Q ; since both I# 0 +# 1 and S 2 admit a relatively straightforward intensional arithmetization of metamathematics (see Wilkie and Paris [1987] and Buss [1986]), this allows strong forms of incompleteness obtained via the intensional approach to apply also to the theory Q ; paragraph 2.1.4 below sketches how the interpretation of S 2 in Q can be used to give an intensional arithmetization in Q . The book of Smullyan [1992] gives a modern, indepth treatment of Godel's incompleteness theorems
Pointwise hereditary majorization and some applications
 Arch. Math. Logic
, 1992
"... A pointwise version of the Howard–Bezem notion of hereditary majorization is introduced which has various advantages, and its relation to the usual notion of majorization is discussed. This pointwise majorization of primitive recursive functionals (in the sense of Gödel’s T as well as Kleene/Feferma ..."
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A pointwise version of the Howard–Bezem notion of hereditary majorization is introduced which has various advantages, and its relation to the usual notion of majorization is discussed. This pointwise majorization of primitive recursive functionals (in the sense of Gödel’s T as well as Kleene/Feferman’s P R) is applied to systems of intuitionistic and classical arithmetic (H and Hc) in all finite types with full induction as well as to the corresponding systems with restricted induction ˆ H  \ and ˆ H  \c 1) H and ˆ H  \ are closed under a generalized fan–rule. For a restricted class of formulae this also holds for H c and ˆ H  \c 2) We give a new and very perspicuous proof that for each Φ 2 ∈ T ( P R) one can construct a functional ˜ Φ 2 ∈ T ( P R) such that ˜ Φα is a modulus of uniform continuity for Φ on {β 1 ∀n(βn ≤ αn)}. Such a modulus can also be obtained by majorizing any modulus of pointwise continuity for Φ. 3) The type structure M of all pointwise majorizable set–theoretical functionals of finite type is used to give a short proof that quantifier–free “choice ” with uniqueness (AC!) 1,0 –qf. is not provable within classical arithmetic in all finite types plus comprehension (given by the schema (C) ρ: ∃y0ρ∀xρ (yx = 0 ↔ A(x)) for arbitrary A), dependent ω–choice and bounded choice. Furthermore M separates several µ–operators. 1
Kreisel's `Unwinding Program
 In Odifreddi [53
, 1996
"... Through his own contributions (individual and collaborative) and his extraordinary personal influence, Georg Kreisel did perhaps more than anyone else to promote the development of proof theory and the metamathematics of constructivity in the last fortyodd years. My purpose here is to give ..."
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Through his own contributions (individual and collaborative) and his extraordinary personal influence, Georg Kreisel did perhaps more than anyone else to promote the development of proof theory and the metamathematics of constructivity in the last fortyodd years. My purpose here is to give
Feasible Computation With Higher Types
, 2002
"... We restrict recursion in nite types so as to characterize the polynomial time computable functions. The restrictions are obtained by enriching the type structure with the formation of types ! and terms x r as well as ( and x r. Here we use two sorts of typed variables: complete ones x ..."
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We restrict recursion in nite types so as to characterize the polynomial time computable functions. The restrictions are obtained by enriching the type structure with the formation of types ! and terms x r as well as ( and x r. Here we use two sorts of typed variables: complete ones x and incomplete ones x . 1.
An Arithmetic for NonSizeIncreasing PolynomialTime Computation
"... An arithmetical system is presented with the property that from every proof a realizing term can be extracted that is definable in a certain affine linear typed variant of Gödel's T and therefore defines a nonsizeincreasing polynomial time computable function. ..."
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An arithmetical system is presented with the property that from every proof a realizing term can be extracted that is definable in a certain affine linear typed variant of Gödel's T and therefore defines a nonsizeincreasing polynomial time computable function.
Remarks On Finitism
 Reflections on the Foundations of Mathematics. Essays in Honor of Solomon Feferman, LNL 15. Association for Symbolic Logic
, 2000
"... representability in intuition. (See [2, p. 40].) But our problem is, of course, not the finiteness of a number, but the infinity of numbers. There is, I think, a di#culty with Bernays' notion of formal object, where this is intended to extend to numbers so large as, not only to be beyond processing ..."
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representability in intuition. (See [2, p. 40].) But our problem is, of course, not the finiteness of a number, but the infinity of numbers. There is, I think, a di#culty with Bernays' notion of formal object, where this is intended to extend to numbers so large as, not only to be beyond processing by the human mind, but possibly to be beyond representablity in the physical world. [2, p. 39]. This di#culty ought to be discussed more adequately then + This paper is based on a talk that I was very pleased to give at the conference Reflections, December 1315, 1998, in honor of Solomon Feferman on his seventieth birthday. The choice of topic is especially appropriate for the conference in view of recent discussions we had had about finitism. I profited from the discussion following my talk and, in particular, from the remarks of Richard Zach. I have since had the advantage of further discussions with Zach and of reading his paper 1998; and I use his scholarshi