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A simple proof of Parsons' theorem
"... Let I# 1 be the fragment of elementary Peano Arithmetic in which induction is restricted to #1formulas. More than three decades ago, Charles Parsons showed that the provably total functions of I# 1 are exactly the primitive recursive functions. In this paper, we observe that Parsons' resul ..."
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Let I# 1 be the fragment of elementary Peano Arithmetic in which induction is restricted to #1formulas. More than three decades ago, Charles Parsons showed that the provably total functions of I# 1 are exactly the primitive recursive functions. In this paper, we observe that Parsons' result is a consequence of Herbrand's theorem concerning the of universal theories. We give a selfcontained proof requiring only basic knowledge of mathematical logic.
P is not equal to NP
, 2009
"... SAT ̸ ∈ P is true, and provable in a simply consistent extension B ′ of a first order theory B of computing, with a single finite axiom B characterizing a universal Turing machine. Therefore P ̸ = N P is true, and provable in a simply consistent extension B ′ ′ of B. ..."
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SAT ̸ ∈ P is true, and provable in a simply consistent extension B ′ of a first order theory B of computing, with a single finite axiom B characterizing a universal Turing machine. Therefore P ̸ = N P is true, and provable in a simply consistent extension B ′ ′ of B.
unknown title
, 2000
"... Neil Leslie asserts his moral right to be identified as the author of this work. cfl Neil Leslie Abstract We explain how to program with continuations in MartinL "of's theory of types (MLTT). MLTT is a theory designed to formalize constructive mathematics. By appealing to the Curry ..."
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Neil Leslie asserts his moral right to be identified as the author of this work. cfl Neil Leslie Abstract We explain how to program with continuations in MartinL &quot;of's theory of types (MLTT). MLTT is a theory designed to formalize constructive mathematics. By appealing to the CurryHoward `propositions as types ' analogy, and to the BrouwerHeytingKolmogorov interpretation of intuitionistic logic we can treat MLTT as a framework for the specification and derivation of correct functional programs. However, programming in MLTT has two weaknesses: ffl we are limited in the functions that we can naturally express; ffl the functions that we do write naturally are often inefficient. Programming with continuations allows us partially to address these problems. The continuationpassing programming style is known to offer a number of advantages to the functional programmer. We can also observe a relationship between continuation passing and type lifting in categorial grammar. We present computation rules which allow us to use continuations with inductivelydefined types, and with types not presented inductively. We justify the new elimination rules using the usual prooftheoretic semantics. We show that the new rules preserve the consistency of the theory. We show how to use wellorderings to encode continuationpassing operators for inductively defined types. Acknowledgements An earlier version of some of the material in Chapter 6 appeared as [70]. I would like to thank: ffl Peter Kay of Massey University's Albany campus, and Steve Reeves of Waikato University, for providing invaluable support and guidance; ffl Ross Renner, and the School of Mathematical and Computing Sciences at Victoria University of Wellington, for indulging me with time and money to visit Waikato University to talk with Steve; ffl Mhairi for caring (again) that I should finish a thesis; ffl Keir and Ailidh for not caring about theses at all.
Construction and Schemata in Mathematics
"... scandal in philosophy is the problem of free will ” [17, p. 205]. I very much agree with Suppes that the problem of the free will is a major puzzle, which we should try to get a better understanding of by examining the deeper issues connected with the free will. This essay, however, does not treat t ..."
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scandal in philosophy is the problem of free will ” [17, p. 205]. I very much agree with Suppes that the problem of the free will is a major puzzle, which we should try to get a better understanding of by examining the deeper issues connected with the free will. This essay, however, does not treat the problem of the free will. It concerns the problems of the ontology and epistemology of mathematics. In genereal, the problems of the philosophy of mathematics are just as old and—if it makes sense to talk about solvability of such problems— perhaps just as unsolved as the problem of the free will. Mathematics is a very important ingredient of knowledge. In its most simple form mathematics plays a necessary role in our understanding of the surrounding world and is necessary for solving simple problems of ordinary life. At the other end of the simplicityscale we find the mathematics as used in science. Also here mathematics has a necessary role in our descriptions of nature and the way in which we are involved with it and each other. It truly amazes me that there seems to be very little consensus with respect to the ontology and epistemology
Contents
, 2014
"... The purpose of this short article is to build on the work of Ghirardi, Marinatto ..."
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The purpose of this short article is to build on the work of Ghirardi, Marinatto
Double Exponential Inseparability Of Robinson Subsystem Q+  From The Unsatisfiable Sentences In The Language Of Addition
, 1992
"... this report is organized in the following way. As shown by a careful analysis in the first sections, a small set of properties of addition and successor are sufficient to define an exp(3; k)representation ..."
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this report is organized in the following way. As shown by a careful analysis in the first sections, a small set of properties of addition and successor are sufficient to define an exp(3; k)representation
1. Mathematical structuralism and the BurgessKeränen objection According to mathematical structuralism, mathematics is the science of
"... A structure, as the notion is understood in contemporary mathematics, is a set (or possibly a proper class) with distin1 ..."
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A structure, as the notion is understood in contemporary mathematics, is a set (or possibly a proper class) with distin1