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Unifying Constructive and Nonstandard Analysis
 Bull. Symbolic Logic
, 1999
"... This paper is partly a survey of this development. In Section 2 we review the construction of the nonstandard universe N . Section 3 discusses the internal sets and functions of N . Here we present two new results for N : the @ 1 saturation principle and a characterisation of internal functions bet ..."
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This paper is partly a survey of this development. In Section 2 we review the construction of the nonstandard universe N . Section 3 discusses the internal sets and functions of N . Here we present two new results for N : the @ 1 saturation principle and a characterisation of internal functions between nonstandard versions of standard sets. We also briefly indicate how to make the Loeb measure construction over hyperfinite sets. Section 4 discusses the relation between nonstandard real numbers and the canonical real numbers of N . In the final section we exemplify the use of the model to prove results in the calculus of several variables, e.g. the Implicit Function Theorem.
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 has a pla ..."
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Cited by 6 (0 self)
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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.
Constructive Nonstandard Representations of Generalized Functions
 Indagationes Mathematicae
, 1998
"... Using techniques of nonstandard analysis Abraham Robinson showed that it is possible to represent each Schwartz distribution T as an integral T (OE) = R f OE, where f is some nonstandard smooth function. We show that the theory based on this representation can be developed within a constructive se ..."
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Using techniques of nonstandard analysis Abraham Robinson showed that it is possible to represent each Schwartz distribution T as an integral T (OE) = R f OE, where f is some nonstandard smooth function. We show that the theory based on this representation can be developed within a constructive setting. Mathematics Subject Classification: 03F60, 03H05, 26E35, 46F10. Keywords: Constructive analysis, nonstandard analysis, generalized functions. 1 Introduction Robinson (1966) demonstrated that Schwartz' theory of distributions could be given a natural formulation using techniques of nonstandard analysis, so that distributions become certain nonstandard smooth functions. In particular, Dirac's deltafunction may then be taken to be the rational function ffi(x) = 1 ß " " 2 + x 2 where " is a positive infinitesimal. As is wellknown, the classical nonstandard analysis is based on strongly nonconstructive assumptions. In this paper we present a constructive version of Robinson's the...
Ultrapowers as Sheaves on a Category of Ultrafilters
, 2001
"... In 1993 I. Moerdijk presented a new model of nonstandard arithmetic in the topos of sheaves on a category of filters, Sh($\mathbb{F}$). This was later extended by E. Palmgren to a model of nonstandard analysis. The model in particular makes use of the sheaves ${}^*S$, which at any filter $\mathcal{F ..."
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In 1993 I. Moerdijk presented a new model of nonstandard arithmetic in the topos of sheaves on a category of filters, Sh($\mathbb{F}$). This was later extended by E. Palmgren to a model of nonstandard analysis. The model in particular makes use of the sheaves ${}^*S$, which at any filter $\mathcal{F}$ is the reduced power of the set $S$ over $\mathcal{F}$, ${}^*S(\mathcal{F})$. The details of this will be given in section 1.3. Before this, in section 1.1, we will give a short background to the subject of sheaves and logic and, in section 1.2, some preliminaries. In this paper we focus our attention on the sheaves on the subcategory of ultrafilters, Sh($\mathbb{U}$). The category $\mathbb{U}$ will be discussed in section 2. The sheaves of the form ${}^*S$ now, at an ultrafilter $\mathcal{U}$, represents the ultrapower of $S$ over $\mathcal{U}$, ${}^*S(\mathcal{U})$. More details on the sheaves over $\mathbb{U}$ can be found in section 3. In section 4 we study the internal logic in the topos of sheaves, which is classic since Sh($\mathbb{U}$) is an atomic topos. We prove that this logic does not coincide with the logic in any of the ultrapowers ${}^*S(\mathcal{U})$. The category of ultrafilters has a strong connection with ultrafilters under the RudinKeisler ordering, for instance we have $\mathcal{U} \leq \mathcal{V}$ if and only if $\textup{Hom}_{\mathbb{U}}(\mathcal{V}, \mathcal{U}) ot = \emptyset$. In the paper we define the RudinKeisler ordering on Sh($\mathbb{U}$) and study the consequences of it in our setting. In the paper we investigate the properties of Sh($\mathbb{U}$). We establish two transfer principles: external transfer, which is corresponding to {\L}o{\'s} theorem, and an internal transfer principle. We show that the topos theoretic axiom of choice does not hold in Sh($\mathbb{U}$) but establish some weak form of it and also prove some other properties similar to results proved by Palmgren about Sh($\mathbb{F}$). In section 5 we show that the topos can be used to model Nelson's internal set theory (IST). IST is an axiomatic approach to nonstandard analysis, which adds to ZFC a undefined unary predicate St($x$), for the standard sets, and axioms relating the standard and nonstandard sets.
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 with ..."
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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 ...
UPPSALA DISSERTATIONS IN MATHEMATICS 30
"... In the classical theory of ultrapowers, you start with an ultrafilter (I,U) and, given a structure S, you construct the ultrapower SI /U. The fundamental result is ̷Lo´s’s theorem for ultrapowers giving the connection between what formulas are satisfied in the ultrapower and in the original structur ..."
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In the classical theory of ultrapowers, you start with an ultrafilter (I,U) and, given a structure S, you construct the ultrapower SI /U. The fundamental result is ̷Lo´s’s theorem for ultrapowers giving the connection between what formulas are satisfied in the ultrapower and in the original structure S. In this thesis we instead
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
Realizability with a Local Operator of A.M. Pitts
, 2013
"... We study a notion of realizability with a local operator J which was first considered by A.M. Pitts in his thesis [7]. Using the SuslinKleene theorem, we show that the representable functions for this realizability are exactly the hyperarithmetical ( ∆ 1 1) functions. We show that there is a realiz ..."
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We study a notion of realizability with a local operator J which was first considered by A.M. Pitts in his thesis [7]. Using the SuslinKleene theorem, we show that the representable functions for this realizability are exactly the hyperarithmetical ( ∆ 1 1) functions. We show that there is a realizability interpretation of nonstandard arithmetic, which, despite its classical character, lives in a very nonclassical universe, where the Uniformity Principle holds and König’s Lemma fails. We conjecture that the local operator gives a useful indexing of the hyperarithmetical functions.
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 Elsevier’s archiving and manuscript policies are encouraged to visit: