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Truth Definitions, Skolem Functions And Axiomatic Set Theory
 Bulletin of Symbolic Logic
, 1998
"... this paper, it will turn out logicians have universally missed the true, exceedingly simple feature of ordinary firstorder logic that makes it incapable of accommodating its own truth predicate. (See Section 4 below.) This defect will also be shown to be easy to overcome without transcending the fi ..."
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this paper, it will turn out logicians have universally missed the true, exceedingly simple feature of ordinary firstorder logic that makes it incapable of accommodating its own truth predicate. (See Section 4 below.) This defect will also be shown to be easy to overcome without transcending the firstorder level. This eliminates once and for all the need of set theory for the purposes of a metatheory of logic.
History of Constructivism in the 20th Century
"... notions, such as `constructive proof', `arbitrary numbertheoretic function ' are rejected. Statements involving quantifiers are finitistically interpreted in terms of quantifierfree statements. Thus an existential statement 9xAx is regarded as a partial communication, to be supplemented ..."
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notions, such as `constructive proof', `arbitrary numbertheoretic function ' are rejected. Statements involving quantifiers are finitistically interpreted in terms of quantifierfree statements. Thus an existential statement 9xAx is regarded as a partial communication, to be supplemented by providing an x which satisfies A. Establishing :8xAx finitistically means: providing a particular x such that Ax is false. In this century, T. Skolem 4 was the first to contribute substantially to finitist 4 Thoralf Skolem 18871963 History of constructivism in the 20th century 3 mathematics; he showed that a fair part of arithmetic could be developed in a calculus without bound variables, and with induction over quantifierfree expressions only. Introduction of functions by primitive recursion is freely allowed (Skolem 1923). Skolem does not present his results in a formal context, nor does he try to delimit precisely the extent of finitist reasoning. Since the idea of finitist reasoning ...
The Logic of Brouwer and Heyting
, 2007
"... Intuitionistic logic consists of the principles of reasoning which were used informally by ..."
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Intuitionistic logic consists of the principles of reasoning which were used informally by
Continuous truth II: reflections
 In Workshop on Logic, Language, Information and Computation
, 2013
"... Abstract. In the late 1960s, Dana Scott first showed how the StoneTarski topological interpretation of Heyting’s calculus could be extended to model intuitionistic analysis; in particular Brouwer’s continuity principle. In the early ’80s we and others outlined a general treatment of nonconstructi ..."
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Abstract. In the late 1960s, Dana Scott first showed how the StoneTarski topological interpretation of Heyting’s calculus could be extended to model intuitionistic analysis; in particular Brouwer’s continuity principle. In the early ’80s we and others outlined a general treatment of nonconstructive objects, using sheaf models—constructions from topos theory—to model not only Brouwer’s nonclassical conclusions, but also his creation of “new mathematical entities”. These categorical models are intimately related to, but more general than Scott’s topological model. The primary goal of this paper is to consider the question of iterated extensions. Can we derive new insights by repeating the second act? In Continuous Truth I, presented at Logic Colloquium ’82 in Florence, we showed that general principles of continuity, local choice and local compactness hold in the gros topos of sheaves over the category of separable locales equipped with the open cover topology. We touched on the question of iteration. Here we develop a more general analysis of iterated categorical extensions, that leads to a reflection schema for statements of predicative analysis. We also take the opportunity to revisit some aspects of both Continuous Truth I and Formal Spaces (Fourman & Grayson 1982), and correct two longstanding errors therein.
Cohesiveness
 INTELLECTICA, 2009/1, 51, PP.
, 2009
"... It is characteristic of a continuum that it be “all of one piece”, in the sense of being inseparable into two (or more) disjoint nonempty parts. By taking “part ” to mean open (or closed) subset of the space, one obtains the usual topological concept of connectedness. Thus a space S is defined to b ..."
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It is characteristic of a continuum that it be “all of one piece”, in the sense of being inseparable into two (or more) disjoint nonempty parts. By taking “part ” to mean open (or closed) subset of the space, one obtains the usual topological concept of connectedness. Thus a space S is defined to be connected if it cannot be partitioned into two disjoint nonempty open (or closed) subsets – or equivalently, given any partition of S into two open (or closed) subsets, one of the members of the partition must be empty. This holds, for example, for the space R of real numbers and for all of its open or closed intervals. Now a truly radical condition results from taking the idea of being “all of one piece ” literally, that is, if it is taken to mean inseparability into any disjoint nonempty parts, or subsets, whatsoever. A space S satisfying this condition is called cohesive or indecomposable. While the law of excluded middle of classical logic reduces indecomposable spaces to the trivial empty space and onepoint spaces, the use of intuitionistic logic makes it possible not only for nontrivial cohesive spaces to exist, but for every connected space to be cohesive.In this paper I describe the philosophical background to cohesiveness as well as some of the ways in
Mass problems and intuitionistic higherorder logic
"... In this paper we study a model of intuitionistic higherorder logic which we call the Muchnik topos. The Muchnik topos may be defined briefly as the category of sheaves of sets over the topological space consisting of the Turing degrees, where the Turing cones form a base for the topology. We note t ..."
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In this paper we study a model of intuitionistic higherorder logic which we call the Muchnik topos. The Muchnik topos may be defined briefly as the category of sheaves of sets over the topological space consisting of the Turing degrees, where the Turing cones form a base for the topology. We note that our Muchnik topos interpretation of intuitionistic mathematics is an extension of the well known Kolmogorov/Muchnik interpretation of intuitionistic propositional calculus via Muchnik degrees, i.e., mass problems under weak reducibility. We introduce a new sheaf representation of the intuitionistic real numbers, the Muchnik reals, which are different from the Cauchy reals and the Dedekind reals. Within the Muchnik topos we obtain a choice principle (∀x∃y A(x, y)) ⇒ ∃w ∀xA(x,wx) and a bounding principle (∀x∃y A(x, y)) ⇒ ∃z ∀x∃y (y ≤T (x, z) ∧ A(x, y)) where x, y, z range over Muchnik reals, w ranges over functions from Muchnik reals to Muchnik reals, and A(x, y) is a formula not containing w or z. For the convenience of the reader, we explain all of the essential background material on intuitionism, sheaf theory, intuitionistic higherorder
History of constructivism in the 20th century
"... In this survey of the history of constructivism, more space has been devoted to early developments (up till ca 1965) than to the work of the last few decades. Not only because most of the concepts and general insights have emerged before 1965, ..."
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In this survey of the history of constructivism, more space has been devoted to early developments (up till ca 1965) than to the work of the last few decades. Not only because most of the concepts and general insights have emerged before 1965,