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Forcing in Proof Theory
 BULL SYMB LOGIC
, 2004
"... Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also ..."
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Paul Cohen's method of forcing, together with Saul Kripke's related semantics for modal and intuitionistic logic, has had profound effects on a number of branches of mathematical logic, from set theory and model theory to constructive and categorical logic. Here, I argue that forcing also has a place in traditional Hilbertstyle proof theory, where the goal is to formalize portions of ordinary mathematics in restricted axiomatic theories, and study those theories in constructive or syntactic terms. I will discuss the aspects of forcing that are useful in this respect, and some sample applications. The latter include ways of obtaining conservation results for classical and intuitionistic theories, interpreting classical theories in constructive ones, and constructivizing modeltheoretic arguments.
Classifying Toposes for First Order Theories
 Annals of Pure and Applied Logic
, 1997
"... By a classifying topos for a firstorder theory T, we mean a topos E such that, for any topos F , models of T in F correspond exactly to open geometric morphisms F ! E . We show that not every (infinitary) firstorder theory has a classifying topos in this sense, but we characterize those which ..."
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By a classifying topos for a firstorder theory T, we mean a topos E such that, for any topos F , models of T in F correspond exactly to open geometric morphisms F ! E . We show that not every (infinitary) firstorder theory has a classifying topos in this sense, but we characterize those which do by an appropriate `smallness condition', and we show that every Grothendieck topos arises as the classifying topos of such a theory. We also show that every firstorder theory has a conservative extension to one which possesses a classifying topos, and we obtain a Heytingvalued completeness theorem for infinitary firstorder logic.
Formal Topologies on the Set of FirstOrder Formulae
 Journal of Symbolic Logic
, 1998
"... this paper that the question has a simple negative answer. This raised further natural questions on what can be said about the points of these two topologies; we give some answers. The observation that topological models for firstorder theories can expressed in the framework of locales appears, for ..."
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this paper that the question has a simple negative answer. This raised further natural questions on what can be said about the points of these two topologies; we give some answers. The observation that topological models for firstorder theories can expressed in the framework of locales appears, for instance, in Fourman and Grayson [6], where the analogy between points of a locale and models of a theory is emphasised; the identification of formal points with Henkin sets, gives a precise form to this analogy. We replace the use of locales by formal topology, which can be expressed in a predicative framework such as MartinLof's type theory. Prooftheoretic issues are also considered by Dragalin [4], who presents a topological completeness proof using only finitary inductive definitions. Palmgren and Moerdijk [10] is also concerned with constructions of models: using sheaf semantics, they obtain a stronger conservativity result than the one in [3]. We will first investigate the difference between the DedekindMacNeille cover and the inductive cover. It easy to see that \Delta DM is stronger than \Delta I , that is, OE \Delta I U implies OE \Delta DM U , but the converse does not hold in general. The notion of point is not primitive in formal topology and therefore it is natural to require that a formal topology has some notion of positivity defined on the basic neighbourhoods; that a neighbourhood is positive then corresponds to, in ordinary point based topology, that it is inhabited by some point. We will show several negative results on positivity, both for the inductive topology and the DedekindMacNeille topology. The points of an inductive topology correspond to Henkin sets, but the DedekindMacNeille topology has, in general, no points. Our reasoning is constructi...
The Filter Construction Revisited
"... The lter construction, as an endofunctor on the category of small coherent categories, was used extensively by A. Pitts in a series of papers in the 80's to prove completeness and interpolation results. Later I. Moerdijk rediscovered the construction on Heyting categories and used it, together ..."
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The lter construction, as an endofunctor on the category of small coherent categories, was used extensively by A. Pitts in a series of papers in the 80's to prove completeness and interpolation results. Later I. Moerdijk rediscovered the construction on Heyting categories and used it, together with E. Palmgren, to construct nonstandard models of Heyting arithmetic. In this paper we describe lter construction as a leftadjoint: applied to a leftexact category it is simply the completion of subobject semilattices under ltered (and thus all) meets. We study ltered coherent logic which is coherent logic extended to arbitrary meets and the rules that existential quantication and binary disjunction distribute over ltered meets. This logic is sound and complete for interpretations in Pitts' ltered coherent categories, and conservative over coherent logic. Restricting further to rstorder logic we show that the minimal models of Heyting arithmetic described by Moerdijk and Palmgren...
An Effective Conservation Result for Nonstandard Arithmetic
, 1999
"... We prove that a nonstandard extension of arithmetic is effectively conservative over Peano arithmetic by using an internal version of a definable ultrapower. By the same method we show that a certain extension of the nonstandard theory with a saturation principle has the same prooftheoretic strengt ..."
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We prove that a nonstandard extension of arithmetic is effectively conservative over Peano arithmetic by using an internal version of a definable ultrapower. By the same method we show that a certain extension of the nonstandard theory with a saturation principle has the same prooftheoretic strength as second order arithmetic, where comprehension is restricted to arithmetical formulas. Mathematics Subject Classification: 03F30, 03H15. Keywords: Nonstandard arithmetic, prooftheoretic strength, bounded ultrapowers. 1 Introduction While it is trivially true that nonstandard methods used inside ZFC will not increase the set of theorems that ZFC proves, the addition of axioms corresponding to transfer and saturation principles to weaker theories may, or may not, increase the strength of the theory. A conservation result for a higher order theory weaker than ZFC was obtained by Kreisel [7]. The prooftheoretic strength of various axioms for nonstandard arithmetic, especially saturation ...
Annals of Pure and Applied Logic
"... This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or sel ..."
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This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevierâ€™s archiving and manuscript policies are encouraged to visit:
Basic Subtoposes of the Effective Topos
, 2012
"... A fundamental concept in Topos Theory is the notion of subtopos: a subtopos of a topos E is a full subcategory which is closed under finite limits in E, and such that the inclusion functor has a left adjoint which preserves finite limits. It then follows that this subcategory is itself a topos, and ..."
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A fundamental concept in Topos Theory is the notion of subtopos: a subtopos of a topos E is a full subcategory which is closed under finite limits in E, and such that the inclusion functor has a left adjoint which preserves finite limits. It then follows that this subcategory is itself a topos, and its internal logic has