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38
Groundwork for weak analysis
 the Journal of Symbolic Logic
, 2002
"... Abstract. This paper develops the very basic notions of analysis in a weak secondorder theory of arithmetic BTFA whose provably total functions are the polynomial time computable functions. We formalize within BTFA the real number system and the notion of a continuous real function of a real variabl ..."
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Abstract. This paper develops the very basic notions of analysis in a weak secondorder theory of arithmetic BTFA whose provably total functions are the polynomial time computable functions. We formalize within BTFA the real number system and the notion of a continuous real function of a real variable. The theory BTFA is able to prove the intermediate value theorem, wherefore it follows that the system of real numbers is a real closed ordered field. In the last section of the paper, we show how to interpret the theory BTFA in Robinson’s theory of arithmetic Q. This fact entails that the elementary theory of the real closed ordered fields is interpretable in Q. §1. Introduction. The formalization of mathematics within secondorder arithmetic has a long and distinguished history. We may say that it goes back to Richard Dedekind, and that it has been pursued by, among others, Hermann Weyl, David Hilbert, Paul Bernays, Harvey Friedman, and Stephen Simpson and his students (we may also mention the insights of Georg Kreisel, Solomon Feferman,
Effective uniform bounds from proofs in abstract functional analysis
 CIE 2005 NEW COMPUTATIONAL PARADIGMS: CHANGING CONCEPTIONS OF WHAT IS COMPUTABLE
, 2005
"... ..."
"Clarifying the Nature of the Infinite": the development of metamathematics and proof theory
, 2001
"... We discuss the development of metamathematics in the Hilbert school, and Hilbert's prooftheoretic program in particular. We place this program in a broader historical and philosophical context, especially with respect to nineteenth century developments in mathematics and logic. Finally, we sho ..."
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We discuss the development of metamathematics in the Hilbert school, and Hilbert's prooftheoretic program in particular. We place this program in a broader historical and philosophical context, especially with respect to nineteenth century developments in mathematics and logic. Finally, we show how these considerations help frame our understanding of metamathematics and proof theory today.
On the Arithmetical Content of Restricted Forms of Comprehension, Choice and General Uniform Boundedness
 PURE AND APPLIED LOGIC
, 1997
"... In this paper the numerical strength of fragments of arithmetical comprehension, choice and general uniform boundedness is studied systematically. These principles are investigated relative to base systems T n in all finite types which are suited to formalize substantial parts of analysis but ..."
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In this paper the numerical strength of fragments of arithmetical comprehension, choice and general uniform boundedness is studied systematically. These principles are investigated relative to base systems T n in all finite types which are suited to formalize substantial parts of analysis but nevertheless have provably recursive function(al)s of low growth. We reduce the use of instances of these principles in T n proofs of a large class of formulas to the use of instances of certain arithmetical principles thereby determining faithfully the arithmetical content of the former. This is achieved using the method of elimination of Skolem functions for monotone formulas which was introduced by the author in a previous paper. As
Saturated models of universal theories
, 2001
"... A notion called Herbrand saturation is shown to provide the modeltheoretic analogue of a prooftheoretic method, Herbrand analysis, yielding uniform modeltheoretic proofs of a number of important conservation theorems. A constructive, algebraic variation of the method is described, providing yet ..."
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A notion called Herbrand saturation is shown to provide the modeltheoretic analogue of a prooftheoretic method, Herbrand analysis, yielding uniform modeltheoretic proofs of a number of important conservation theorems. A constructive, algebraic variation of the method is described, providing yet a third approach, which is finitary but retains the semantic flavor of the modeltheoretic version.
On Tao’s “finitary” infinite pigeonhole principle
 The Journal of Symbolic Logic
, 2010
"... In 2007, Terence Tao wrote on his blog an essay about soft analysis, hard analysis and the finitization of soft analysis statements into hard analysis statements. One of his main examples was a quasifinitization of the infinite pigeonhole principle IPP, arriving at the “finitary ” infinite pigeonho ..."
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In 2007, Terence Tao wrote on his blog an essay about soft analysis, hard analysis and the finitization of soft analysis statements into hard analysis statements. One of his main examples was a quasifinitization of the infinite pigeonhole principle IPP, arriving at the “finitary ” infinite pigeonhole principle FIPP1. That turned out to not be the proper formulation and so we proposed an alternative version FIPP2. Tao himself formulated yet another version FIPP3 in a revised version of his essay. We give a counterexample to FIPP1 and discuss for both of the versions FIPP2 and FIPP3 the faithfulness of their respective finitization of IPP by studying the equivalences IPP ↔ FIPP2 and IPP ↔ FIPP3 in the context of reverse mathematics ([9]). In the process of doing this we also introduce a continuous uniform boundedness principle CUB as a formalization of Tao’s notion of a correspondence principle and study the strength of this principle and various restrictions thereof in terms of reverse mathematics, i.e. in terms of the “big five ” subsystems of second order arithmetic. 1
Synthesis of moduli of uniform continuity by the Monotone Dialectica Interpretation
 in the proofsystem MINLOG. Electronic Notes in Theoretical Computer Science
, 2007
"... We extract on the computer a number of moduli of uniform continuity for the first few elements of a sequence of closed terms t of Gödel’s T of type (N→N)→(N→N). The generic solution may then be quickly inferred by the human. The automated synthesis of such moduli proceeds from a proof of the heredi ..."
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We extract on the computer a number of moduli of uniform continuity for the first few elements of a sequence of closed terms t of Gödel’s T of type (N→N)→(N→N). The generic solution may then be quickly inferred by the human. The automated synthesis of such moduli proceeds from a proof of the hereditarily extensional equality (≈) of t to itself, hence a proof in a weakly extensional variant of BergerBuchholzSchwichtenberg’s system Z of t ≈(N→N)→(N→N) t. We use an implementation on the machine, in Schwichtenberg’s MinLog proofsystem, of a nonliteral adaptation to Natural Deduction of Kohlenbach’s monotone functional interpretation. This new version of the Monotone Dialectica produces terms in NbEnormal form by means of a recurrent partial NbEnormalization. Such partial evaluation is strictly necessary.
Injecting uniformities into Peano arithmetic
, 2008
"... We present a functional interpretation of Peano arithmetic that uses Gödel’s computable functionals and which systematically injects uniformities into the statements of finitetype arithmetic. As a consequence, some uniform boundedness principles (not necessarily settheoretically true) are interpre ..."
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We present a functional interpretation of Peano arithmetic that uses Gödel’s computable functionals and which systematically injects uniformities into the statements of finitetype arithmetic. As a consequence, some uniform boundedness principles (not necessarily settheoretically true) are interpreted while maintaining unmoved the Π0 2sentences of arithmetic. We explain why this interpretation is taylored to yield conservation results.