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An intuitionistic theory of types
"... An earlier, not yet conclusive, attempt at formulating a theory of this kind was made by Scott 1970. Also related, although less closely, are the type and logic free theories of constructions of Kreisel 1962 and 1965 and Goodman 1970. In its first version, the present theory was based on the strongl ..."
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An earlier, not yet conclusive, attempt at formulating a theory of this kind was made by Scott 1970. Also related, although less closely, are the type and logic free theories of constructions of Kreisel 1962 and 1965 and Goodman 1970. In its first version, the present theory was based on the strongly impredicative axiom that there is a type of all types whatsoever, which is at the same time a type and an object of that type. This axiom had to be abandoned, however, after it was shown to lead to a contradiction by Jean Yves Girard. I am very grateful to him for showing me his paradox. The change that it necessitated is so drastic that my theory no longer contains intuitionistic simple type theory as it originally did. Instead, its proof theoretic strength should be close to that of predicative analysis.
The Witness Function Method and Provably Recursive Functions of Peano Arithmetic
"... This paper presents a new proof of the characterization of the provably recursive functions of the fragments I\Sigma n of Peano arithmetic. The proof method also characterizes the \Sigma kdefinable functions of I\Sigma n and of theories axiomatized by transfinite induction on ordinals. The proofs ..."
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This paper presents a new proof of the characterization of the provably recursive functions of the fragments I\Sigma n of Peano arithmetic. The proof method also characterizes the \Sigma kdefinable functions of I\Sigma n and of theories axiomatized by transfinite induction on ordinals. The proofs are completely prooftheoretic and use the method of witness functions and witness oracles. Similar methods also yield a new proof of Parson's theorem on the conservativity of the \Sigma n+1induction rule over the \Sigma ninduction axioms. A new proof of the conservativity of B\Sigma n+1 over I\Sigma n is given. The proof methods provide new analogies between Peano arithmetic and bounded arithmetic.
Preliminary investigations on induction over real numbers
"... In arithmetic, the induction principle permits to give more direct and more intuitive proofs than alternative principles such as the existence of a minimum to all nonempty sets of natural numbers or the existence of a maximum to all bounded nonempty sets of natural numbers, that usually require de ..."
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In arithmetic, the induction principle permits to give more direct and more intuitive proofs than alternative principles such as the existence of a minimum to all nonempty sets of natural numbers or the existence of a maximum to all bounded nonempty sets of natural numbers, that usually require detours in proofs. Moreover, when we prove a general property by induction and use it on a particular natural number, we can eliminate the invocation of the induction principle and get a more elementary proof [1]. The idea of induction is that when a property holds for some natural number a and is hereditary, then it holds for all numbers greater than or equal to a. The fact that the property is hereditary means that when it holds for some number n then it propagates &quot;at least a little bit &quot; to greater numbers. In various formulations of the scheme, this takes the form P (n) ) P (S(n)) or (8p! n (P p)) ) (P n). We present in this note a similar principle for real numbers. The fact that some property is hereditary is expressed by the fact that if P holds for a real number c, then its holds on an interval [c:::c + &quot;]. Of course, this is not enough to prove that if P holds for a real number a, then it holds for all real numbers greater than a, because unlike a bounded set of natural numbers, a bounded set of real numbers need not have a maximum. However, a closed bounded set of real numbers does. Thus, we shall restrict our induction principle to closed properties P. In this note, we state the induction principle, we prove it and we give several examples applications. All these examples come from [3] where we have formalized and proved results in elementary calculus and kinematics to studying the motion of aircraft and where this real induction principle is implicit. We also briefly discuss which axioms would be replaced by this scheme in an axiomatization of analysis, how this this induction scheme is related to ordinal induction and how, in some cases, the invocation of this scheme can be eliminated when we apply a general theorem to a particular real number.
ae Primitive Recursion on the Partial Continuous Functionals
"... We investigate G"odel's notion of a primitive recursive functional of higher type [5] in the context of partial continuous functionals as introduced by Kreisel in [7] and developed mainly by Scott (see [13], [3]). To make this paper readable for people not familiar with the theory of p ..."
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We investigate G&quot;odel's notion of a primitive recursive functional of higher type [5] in the context of partial continuous functionals as introduced by Kreisel in [7] and developed mainly by Scott (see [13], [3]). To make this paper readable for people not familiar with the theory of partial continuous functionals we have included a short exposition of the basic material, in a form convenient for our later arguments.
ABSTRACT THE LOGIC OF RESOURCES AND TASKS
"... Acknowledgments I would like to express my sincere gratitude to my advisor, Scott Weinstein, for the support, encouragement and advice that he has given me over the past four years, and for creating an incredibly friendly atmosphere that made my work and life in Philadelphia productive, enjoyable an ..."
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Acknowledgments I would like to express my sincere gratitude to my advisor, Scott Weinstein, for the support, encouragement and advice that he has given me over the past four years, and for creating an incredibly friendly atmosphere that made my work and life in Philadelphia productive, enjoyable and easy. I would also like to thank the members of my committee, Sergei Artemov, Peter Buneman, Jean Gallier and Val Tannen, for taking the time to comment on this thesis and give me useful suggestions for improved presentation. Special thanks go to Sergei who has been the prime inspirer of my work in logic since 1985. I owe my survival as a scientist the privilege that not all of my colleagues were lucky to have after the collapse of the USSR to my deseased teacher and friend, George Boolos, as well as Sergei Artemov, Dick de Jongh, Michael Detlefsen, Scott Weistein and Val Tannen. Their efforts helped me to escape the economic devastation in my country and peacefully continue my work in the West, instead of possibly ending up vending cigarettes and beer in the streets of Moscow. I am truly endebted to my Georgian teachers Leri Mchedlishvili, Michael Bezhanishvili and Leo Esakia for introducing the world of logic to me back in the early 1980s. I have had the good fortune to be supported by the National Science Foundation, on grants CCR9403447 and CCR9057570, while working on this thesis. ii