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.

Abstract. 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 Hilbert-style 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 model-theoretic arguments.?1. Introduction. In 1963, Paul Cohen introduced the method of forcing to prove the independence of both the axiom of choice and the continuum hypothesis from Zermelo-Fraenkel set theory. It was not long before Saul Kripke noted a connection between forcing and his semantics for modal and

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 delta-function 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...

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 Rudin-Keisler 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 Rudin-Keisler 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.

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

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