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Cycling in proofs and feasibility
 Transactions of the American Mathematical Society
, 1998
"... Abstract. There is a common perception by which small numbers are considered more concrete and large numbers more abstract. A mathematical formalization of this idea was introduced by Parikh (1971) through an inconsistent theory of feasible numbers in which addition and multiplication are as usual b ..."
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Abstract. There is a common perception by which small numbers are considered more concrete and large numbers more abstract. A mathematical formalization of this idea was introduced by Parikh (1971) through an inconsistent theory of feasible numbers in which addition and multiplication are as usual but for which some very large number is defined to be not feasible. Parikh shows that sufficiently short proofs in this theory can only prove true statements of arithmetic. We pursue these topics in light of logical flow graphs of proofs (Buss, 1991) and show that Parikh’s lower bound for concrete consistency reflects the presence of cycles in the logical graphs of short proofs of feasibility of large numbers. We discuss two concrete constructions which show the bound to be optimal and bring out the dynamical aspect of formal proofs. For this paper the concept of feasible numbers has two roles, as an idea with its own life and as a vehicle for exploring general principles on the dynamics and geometry of proofs. Cycles can be seen as a measure of how complicated a proof can be. We prove that short proofs must have cycles. 1.
Forcing in Proof Theory
 BULL SYMB LOGIC
, 2004
"... Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also ..."
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Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbertstyle proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing modeltheoretic arguments.
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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Cited by 7 (5 self)
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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;
An Effective Conservation Result for Nonstandard Arithmetic
, 1999
"... We prove that a nonstandard extension of arithmetic is effectively conservative over Peano arithmetic by using an internal version of a definable ultrapower. By the same method we show that a certain extension of the nonstandard theory with a saturation principle has the same prooftheoretic strengt ..."
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We prove that a nonstandard extension of arithmetic is effectively conservative over Peano arithmetic by using an internal version of a definable ultrapower. By the same method we show that a certain extension of the nonstandard theory with a saturation principle has the same prooftheoretic strength as second order arithmetic, where comprehension is restricted to arithmetical formulas. Mathematics Subject Classification: 03F30, 03H15. Keywords: Nonstandard arithmetic, prooftheoretic strength, bounded ultrapowers. 1 Introduction While it is trivially true that nonstandard methods used inside ZFC will not increase the set of theorems that ZFC proves, the addition of axioms corresponding to transfer and saturation principles to weaker theories may, or may not, increase the strength of the theory. A conservation result for a higher order theory weaker than ZFC was obtained by Kreisel [7]. The prooftheoretic strength of various axioms for nonstandard arithmetic, especially saturation ...
On the Foundations of Nonstandard Mathematics
"... In this paper we survey various settheoretic approaches that have been proposed over the last thirty years as foundational frameworks for the use of nonstandard methods in mathematics. Introduction. Since the early developments of calculus, infinitely small and infinitely large numbers have been th ..."
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In this paper we survey various settheoretic approaches that have been proposed over the last thirty years as foundational frameworks for the use of nonstandard methods in mathematics. Introduction. Since the early developments of calculus, infinitely small and infinitely large numbers have been the object of constant interest and great controversy in the history of mathematics. In fact, while on the one hand fundamental results in the differential and integral calculus were first obtained by reasoning informally