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Does Mathematics Need New Axioms?
 American Mathematical Monthly
, 1999
"... this article I will be looking at the leading question from the point of view of the logician, and for a substantial part of that, from the perspective of one supremely important logician: Kurt Godel. From the time of his stunning incompleteness results in 1931 to the end of his life, Godel called f ..."
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this article I will be looking at the leading question from the point of view of the logician, and for a substantial part of that, from the perspective of one supremely important logician: Kurt Godel. From the time of his stunning incompleteness results in 1931 to the end of his life, Godel called for the pursuit of new axioms to settle undecided arithmetical problems. And from 1947 on, with the publication of his unusual article, "What is Cantor's continuum problem?" [11], he called in addition for the pursuit of new axioms to settle Cantor's famous conjecture about the cardinal number of the continuum. In both cases, he pointed primarily to schemes of higher infinity in set theory as the direction in which to seek these new principles. Logicians have learned a great deal in recent years that is relevant to Godel's program, but there is considerable disagreement about what conclusions to draw from their results. I'm far from unbiased in this respect, and you'll see how I come out on these matters by the end of this essay, but I will try to give you a fair presentation of other positions along the way so you can decide for yourself which you favor.
Does category theory provide a framework for mathematical structuralism?
 PHILOSOPHIA MATHEMATICA
, 2003
"... Category theory and topos theory have been seen as providing a structuralist framework for mathematics autonomous vis à vis set theory. It is argued here that these theories require a background logic of relations and substantive assumptions addressing mathematical existence of categories themselves ..."
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Category theory and topos theory have been seen as providing a structuralist framework for mathematics autonomous vis à vis set theory. It is argued here that these theories require a background logic of relations and substantive assumptions addressing mathematical existence of categories themselves. We propose a synthesis of Bell’s “manytopoi” view and modalstructuralism. Surprisingly, a combination of mereology and plural quantification suffices to describe hypothetical large domains, recovering the Grothendieck method of universes. Both topos theory and set theory can be carried out relative to such domains; puzzles about “large categories ” and “proper classes ” are handled in a
On Founding the Theory of Algorithms
, 1998
"... machines and implementations The first definition of an abstract machine was given by Turing, in the classic [20]. Without repeating here the wellknown definition (e.g., see [6]), 13 we recall that each Turing machine M is equipped with a "semiinfinite tape" which it uses both to comp ..."
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machines and implementations The first definition of an abstract machine was given by Turing, in the classic [20]. Without repeating here the wellknown definition (e.g., see [6]), 13 we recall that each Turing machine M is equipped with a "semiinfinite tape" which it uses both to compute and also to communicate with its environment: To determine the value f(n) (if any) of the partial function 14 f : N * N computed by M , we put n on the tape in some standard way, e.g., by placing n + 1 consecutive 1s at its beginning; we start the machine in some specified, initial, internal state q 0 and looking at the leftmost end of the tape; and we wait until the machine stops (if it does), at which time the value f(n) can be read off the tape, by counting the successive 1s at the left end. Turing argued that the numbertheoretic functions which can (in principle) be computed by any deterministic, physical device are exactly those which can be computed by a Turing machine, and the correspon...
The Mathematical Development Of Set Theory  From Cantor To Cohen
 The Bulletin of Symbolic Logic
, 1996
"... This article is dedicated to Professor Burton Dreben on his coming of age. I owe him particular thanks for his careful reading and numerous suggestions for improvement. My thanks go also to Jose Ruiz and the referee for their helpful comments. Parts of this account were given at the 1995 summer meet ..."
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This article is dedicated to Professor Burton Dreben on his coming of age. I owe him particular thanks for his careful reading and numerous suggestions for improvement. My thanks go also to Jose Ruiz and the referee for their helpful comments. Parts of this account were given at the 1995 summer meeting of the Association for Symbolic Logic at Haifa, in the Massachusetts Institute of Technology logic seminar, and to the Paris Logic Group. The author would like to express his thanks to the various organizers, as well as his gratitude to the Hebrew University of Jerusalem for its hospitality during the preparation of this article in the autumn of 1995.
The Development of Mathematical Logic from Russell to Tarski: 19001935
"... this paper that from the logical point of view set theory is the proper foundation of the mathematical sciences. Thus, he adds, if one wants to give general definitional principles that hold for all of mathematics it is necessary to account for the definitional principles of set theory. First, he b ..."
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this paper that from the logical point of view set theory is the proper foundation of the mathematical sciences. Thus, he adds, if one wants to give general definitional principles that hold for all of mathematics it is necessary to account for the definitional principles of set theory. First, he begins his definitional analysis with geometry. Relying on Pieri's work on the foundations of geometry he starts with two relations, x y and E(x, y , z). E(x, y , z) means that y and z are equidistant from x. Then he adds that all definitions in Pieri's geometry can be obtained by closing the basic relationships under five principles: 1. Permutation of variables: if A(x,y , z) is a ternary relation so is A(x,z,y)
The emergence of firstorder logic
 University of Minnesota Press, Minneapolis
, 1988
"... To most mathematical logicians working in the 1980s, firstorder logic is the proper and natural framework for mathematics. Yet it was not always so. In 1923, when a young Norwegian mathematician named Thoralf Skolem argued that set theory should be based on firstorder logic, it was ..."
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To most mathematical logicians working in the 1980s, firstorder logic is the proper and natural framework for mathematics. Yet it was not always so. In 1923, when a young Norwegian mathematician named Thoralf Skolem argued that set theory should be based on firstorder logic, it was
Models Of SecondOrder Zermelo Set Theory
, 1999
"... The paper discusses models of secondorder versions of Zermelo set theory that are not given by certain initial segments of the cumulative hierarchy. These models show that common versions of infinity do not, absent replacement, guarantee the existence of the first transfinite stage of the cumulativ ..."
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The paper discusses models of secondorder versions of Zermelo set theory that are not given by certain initial segments of the cumulative hierarchy. These models show that common versions of infinity do not, absent replacement, guarantee the existence of the first transfinite stage of the cumulative hierarchy. Another construction shows that a version of secondorder Zermelo set theory that results when infinity is strengthened to deliver the existence of that stage is satisfied in nonwellfounded models. A variant of secondorder Zermelo set theory is considered all of whose models are given by certain initial segments of the hierarchy.
Pluralism and the Foundations of Mathematics
, 2006
"... A plurality of approaches to foundational aspects of mathematics is a fact of life. Two loci of this are discussed here, the classicism/constructivism controversy over standards of proof, and the plurality of universes of discourse for mathematics arising in set theory and in category theory, whose ..."
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A plurality of approaches to foundational aspects of mathematics is a fact of life. Two loci of this are discussed here, the classicism/constructivism controversy over standards of proof, and the plurality of universes of discourse for mathematics arising in set theory and in category theory, whose problematic relationship is discussed. The first case illustrates the hypothesis that asufficiently rich subject matter may require a multiplicity of approaches. The second case, while in some respects special to mathematics, raises issues of ontological multiplicity and relativity encountered in the natural sciences as well. 1
Towards a unified treatment of induction, I: the general recursion theorem, unfinished draft manuscript
, 1996
"... The recursive construction of a function f: A → Θ consists, paradigmatically, of finding a functor T and maps α: A → TA and θ: TΘ → Θ such that f = α; Tf; θ. The role of the functor T is to marshall the recursive subarguments, and apply the function f to them in parallel. This equation is called pa ..."
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The recursive construction of a function f: A → Θ consists, paradigmatically, of finding a functor T and maps α: A → TA and θ: TΘ → Θ such that f = α; Tf; θ. The role of the functor T is to marshall the recursive subarguments, and apply the function f to them in parallel. This equation is called partial correctness of the recursive program, because we have also to show that it terminates, i.e. that the recursion (coded by α) is well founded. This may be done by finding another map g: A → N, called a loop variant, where N is some standard well founded srtucture such as the natural numbers or ordinals. In set theory the functor T is the covariant powerset; in the study of the free algebra for a free theory Ω (such as in proof theory) it is the polynomial Σr∈Ω(−)ar(r), and it is often something very crude. We identify the properties of the category of sets needed to prove the general recursion theorem, that these data suffice to define f uniquely. For any pullbackpreserving functor T, a structure similar to the von Neumann hierarchy is developed which analyses the free Talgebra if it exists, or deputises for it otherwise. There is considerable latitude in the choice of ambient category, the functor T and the class of predicates admissible in the induction scheme. Free algebras, set theory, the familiar ordinals and novel forms of them which have arisen in theoretical computer science are treated in a uniform fashion. The central idea in the paper is a categorical definition of well founded coalgebra α: A. TA, namely that any pullback diagram of the form