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Weak theories of nonstandard arithmetic and analysis
 Reverse Mathematics
, 2001
"... Abstract. A general method of interpreting weak highertype theories of nonstandard arithmetic in their standard counterparts is presented. In particular, this provides natural nonstandard conservative extensions of primitive recursive arithmetic, elementary recursive arithmetic, and polynomialtime ..."
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Abstract. A general method of interpreting weak highertype theories of nonstandard arithmetic in their standard counterparts is presented. In particular, this provides natural nonstandard conservative extensions of primitive recursive arithmetic, elementary recursive arithmetic, and polynomialtime computable arithmetic. A means of formalizing basic real analysis in such theories is sketched. §1. Introduction. Nonstandard analysis, as developed by Abraham Robinson, provides an elegant paradigm for the application of metamathematical ideas in mathematics. The idea is simple: use modeltheoretic methods to build rich extensions of a mathematical structure, like secondorder arithmetic or a universe of sets; reason about what is true in these enriched structures;
"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 show how ..."
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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.
Nelson’s Work on Logic and Foundations and Other Reflections on Foundations of Mathematics
, 2006
"... This paper starts by discussing Nelson’s philosophy of mathematics, which is a blend of mathematical formalism and a radical constructivism. As such, it makes strong assertions about the foundations of mathematic and the reality of mathematical objects. We then offer our own suggestions for the defi ..."
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This paper starts by discussing Nelson’s philosophy of mathematics, which is a blend of mathematical formalism and a radical constructivism. As such, it makes strong assertions about the foundations of mathematic and the reality of mathematical objects. We then offer our own suggestions for the definition of mathematics and the nature of mathematical reality. We suggest a second characterization of mathematical reasoning in terms of common sense reasoning and argue its relevance for mathematics education. Nelson’s philosophy is the foundation of his definition of predicative arithmetic. There are close connections between predicative arithmetic and the common theories of bounded arithmetic. We prove that polynomial space (PSPACE) predicates and exponential time (EXPTIME) predicates are predicative. We discuss Nelson’s formalist philosophies and his unpublished work in automatic theorem checking. This paper was begun with the plan of discussing Nelson’s work in logic and foundations and his philosophy on mathematics. In particular, it is based on our talk at the Nelson meeting in Vancouver in June 2004. The main topics of this talk were Nelson’s predicative arithmetic and his unpublished work on automatic theorem proving. However, it proved impossible to stay within this plan. In writing the paper, we were prompted to think carefully about the nature of mathematics and more fully formulate our own philosophy of mathematics. We present this below, along with some discussion about mathematics education. Much of the paper focuses on Nelson’s philosophy of mathematics, on how his philosophy motivates his development of predicative arithmetic, and on his unpublished work on computer assisted theorem proving. We also discuss the
The Riemann Integral in Weak Systems of Analysis
"... Abstract: Taking as a starting point (a modification of) a weak theory of arithmetic of Jan Johannsen and Chris Pollett (connected with the hierarchy of counting functions), we introduce successively stronger theories of bounded arithmetic in order to set up a system for analysis (TCA 2). The extend ..."
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Abstract: Taking as a starting point (a modification of) a weak theory of arithmetic of Jan Johannsen and Chris Pollett (connected with the hierarchy of counting functions), we introduce successively stronger theories of bounded arithmetic in order to set up a system for analysis (TCA 2). The extended theories preserve the connection with the counting hierarchy in the sense that the algorithms which the systems prove to halt are exactly the ones in the hierarchy. We show that TCA 2 has the exact strength to develop Riemannian integration for functions with a modulus of uniform continuity.
Amending Frege’s Grundgesetze der Arithmetik, Synthese Vol.147
, 2005
"... Abstract. Frege’s Grundgesetze der Arithmetik is formally inconsistent. This system is, except for minor differences, secondorder logic together with an abstraction operator governed by Frege’s Axiom V. A few years ago, Richard Heck showed that the ramified predicative secondorder fragment of the ..."
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Abstract. Frege’s Grundgesetze der Arithmetik is formally inconsistent. This system is, except for minor differences, secondorder logic together with an abstraction operator governed by Frege’s Axiom V. A few years ago, Richard Heck showed that the ramified predicative secondorder fragment of the Grundgesetze is consistent. In this paper, we show that the above fragment augmented with the axiom of reducibility for concepts true of only finitely many individuals is still consistent, and that elementary Peano arithmetic (and more) is interpretable in this extended system.
FRAGMENT OF NONSTANDARD ANALYSIS WITH A FINITARY CONSISTENCY PROOF
"... We introduce a nonstandard arithmetic NQA − based on the theory developed by R. Chuaqui and P. Suppes in [2] (we will denote it by NQA +), with a weakened external open minimization schema. A finitary consistency proof for NQA − formalizable in PRA is presented. We also show interesting facts about ..."
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We introduce a nonstandard arithmetic NQA − based on the theory developed by R. Chuaqui and P. Suppes in [2] (we will denote it by NQA +), with a weakened external open minimization schema. A finitary consistency proof for NQA − formalizable in PRA is presented. We also show interesting facts about the strength of the theories NQA − and NQA +; NQA − is mutually interpretable with I∆0 + EXP, and on the other hand, NQA + interprets the theories IΣ1 and WKL0.
OPEN QUESTIONS IN REVERSE MATHEMATICS
, 2010
"... The objective of this paper is to provide a source of open questions in reverse mathematics and to point to areas where there could be interesting developments. The questions I discuss are mostly known and come from somewhere in the literature. My objective was to compile them in one place and discu ..."
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The objective of this paper is to provide a source of open questions in reverse mathematics and to point to areas where there could be interesting developments. The questions I discuss are mostly known and come from somewhere in the literature. My objective was to compile them in one place and discuss them in the context of related work. The list is definitely not comprehensive, and my
INTERPRETABILITY IN ROBINSON’S Q
"... Abstract. Edward Nelson published in 1986 a book defending an extreme formalist view of mathematics according to which there is an impassable barrier in the totality of exponentiation. On the positive side, Nelson embarks on a program of investigating how much mathematics can be interpreted in Rapha ..."
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Abstract. Edward Nelson published in 1986 a book defending an extreme formalist view of mathematics according to which there is an impassable barrier in the totality of exponentiation. On the positive side, Nelson embarks on a program of investigating how much mathematics can be interpreted in Raphael Robinson’s theory of arithmetic Q. In the shadow of this program, some very nice logical investigations and results were produced by a number of people, not only regarding what can be interpreted in Q but also what cannot be so interpreted. We explain some of these results and rely on them to discuss Nelson’s position. §1. Introduction. Let L be the firstorder language with equality whose nonlogical symbols are the constant 0, the unary function symbol S (for successor) and two binary function symbols + (for addition) and · (for multiplication). The following theory was introduced in [35] (see also the systematic [42]): Definition 1. Raphael Robinson’s theory Q is the theory in the language L