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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.
On Feasible Numbers
 Logic and Computational Complexity, LNCS Vol. 960
, 1995
"... . A formal approach to feasible numbers, as well as to middle and small numbers, is introduced, based on ideas of Parikh (1971) and improving his formalization. The "vague" set F of feasible numbers intuitively satisfies the axioms 0 2 F , F + 1 ` F and 2 1000 62 F , where the latter is ..."
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. A formal approach to feasible numbers, as well as to middle and small numbers, is introduced, based on ideas of Parikh (1971) and improving his formalization. The "vague" set F of feasible numbers intuitively satisfies the axioms 0 2 F , F + 1 ` F and 2 1000 62 F , where the latter is stronger than a condition considered by Parikh, and seems to be treated rigorously here for the first time. Our technical considerations, though quite simple, have some unusual consequences. A discussion of methodological questions and of relevance to the foundations of mathematics and of computer science is an essential part of the paper. 1 Introduction How to formalize the intuitive notion of feasible numbers? To see what feasible numbers are, let us start by counting: 0,1,2,3, and so on. At this point, A.S. YeseninVolpin (in his "Analysis of potential feasibility", 1959) asks: "What does this `and so on' mean?" "Up to what extent `and so on'?" And he answers: "Up to exhaustion!" Note that by cos...
A Theory of Hyperfinite Sets
, 2005
"... We develop an axiomatic set theory — the Theory of Hyperfinite Sets THS, which is based on the idea of existence of proper subclasses of big finite sets. We demonstrate how theorems of classical continuous mathematics can be transfered to THS, prove consistency of THS and present some applications. ..."
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We develop an axiomatic set theory — the Theory of Hyperfinite Sets THS, which is based on the idea of existence of proper subclasses of big finite sets. We demonstrate how theorems of classical continuous mathematics can be transfered to THS, prove consistency of THS and present some applications.
INTRODUCTION TO THE COMBINATORICS AND COMPLEXITY OF CUT ELIMINATION
"... Abstract. Modus Ponens says that if you know A and you know that A implies B, then you know B. This is a basic rule that we take for granted ..."
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Abstract. Modus Ponens says that if you know A and you know that A implies B, then you know B. This is a basic rule that we take for granted
On the Lengths of Proofs of Consistency  a Survey of Results
"... This article is essentially a part of my thesis for the degree DrSc (Doctor of Sciences). Therefore it mainly surveys my articles [41, 42, 43, 28, 29, 44, 22], and it is structured according to the requirements for such theses. I made only minor changes in the original text and added a few further r ..."
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This article is essentially a part of my thesis for the degree DrSc (Doctor of Sciences). Therefore it mainly surveys my articles [41, 42, 43, 28, 29, 44, 22], and it is structured according to the requirements for such theses. I made only minor changes in the original text and added a few further references. Since Godel's main achievement concerns the problem of consistency and some of the problems that I am going to describe had been considered by him, I think that it is appropriate to publish this article in Godel Society. 1 Historical remarks The question that we are going to consider in is interesting per se and is related to some more practical questions, especially in complexity theory, but the original motivation for it comes from foundational studies. Among the variety of streams in foundations of mathematics, the one which had the biggest influence and which very much determined later development of mathematical logic was Hilbert's<F1
Abstract Bounded Arithmetic, Proof Complexity and Two Papers of Parikh
"... This article surveys R. Parikh’s work on feasibility, bounded arithmetic and the complexity of proofs. We discuss in depth two of Parikh’s papers on these subjects and some of the subsequent progress in the areas of feasible arithmetic and lengths of proofs. 1 ..."
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This article surveys R. Parikh’s work on feasibility, bounded arithmetic and the complexity of proofs. We discuss in depth two of Parikh’s papers on these subjects and some of the subsequent progress in the areas of feasible arithmetic and lengths of proofs. 1
Bounded Arithmetic, Proof Complexity and Two Papers of Parikh
, 2002
"... This article surveys R. Parikh's work on feasibility, bounded arithmetic and the complexity of proofs. We discuss in depth two of Parikh's papers on these subjects and some of the subsequent progress in the areas of feasible arithmetic and lengths of proofs. ..."
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This article surveys R. Parikh's work on feasibility, bounded arithmetic and the complexity of proofs. We discuss in depth two of Parikh's papers on these subjects and some of the subsequent progress in the areas of feasible arithmetic and lengths of proofs.
JACQUES DUBUCS & MATHIEU MARION RADICAL ANTIREALISM AND SUBSTRUCTURAL LOGICS
"... (le texte qui suit n’est pas la version définitive et autorisée de cet article, laquelle sera prochainement publiée dans les Proceedings of the International Congress for Logic, Methodology and Philosophy of Science (LMS’99, Krakow) ..."
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(le texte qui suit n’est pas la version définitive et autorisée de cet article, laquelle sera prochainement publiée dans les Proceedings of the International Congress for Logic, Methodology and Philosophy of Science (LMS’99, Krakow)
Alternative Mathematics without Actual Infinity ∗
, 2012
"... An alternative mathematics based on qualitative plurality of finiteness is developed to make nonstandard mathematics independent of infinite set theory. The vague concept “accessibility ” is used coherently within finite set theory whose separation axiom is restricted to definite objective condit ..."
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An alternative mathematics based on qualitative plurality of finiteness is developed to make nonstandard mathematics independent of infinite set theory. The vague concept “accessibility ” is used coherently within finite set theory whose separation axiom is restricted to definite objective conditions. The weak equivalence relations are defined as binary relations with sorites phenomena. Continua are collection with weak equivalence relations called indistinguishability. The points of continua are the proper classes of mutually indistinguishable elements and have identities with sorites paradox. Four continua formed by huge binary words are examined as a new type of continua. AscoliArzela type theorem is given as an example indicating the feasibility of treating function spaces. The real numbers are defined to be points on linear continuum and have indefiniteness. Exponentiation is introduced by the Euler style and basic properties are established. Basic calculus is developed and the differentiability is captured by the behavior on a point. Main tools of Lebesgue measure theory is obtained in a similar way as Loeb measure. Differences from the current mathematics are examined, such as the indefiniteness of natural numbers, qualitative plurality of finiteness, mathematical usage of vague concepts, the continuum as a primary inexhaustible entity and the hitherto disregarded aspect of “internal measurement ” in mathematics. ∗Thanks to Ritsumeikan University for the sabbathical leave which allowed the author to concentrate on doing research on this theme. 1 ar