Nonstandard analysis is often used to prove that certain objects exist, i.e., that certain sets are not empty. In the literature one can find many existence theorems whose only known proofs use nonstandard analysis; see, for example, [AFHL]. This article will survey a new method for existence proofs, based on the concept of a neometric space. We shall state definitions and results (usually with...

The sheaves over the category of lters, with a joint cover topology, serve as a universe of sets where nonstandard analysis can be developed along constructive principles. In this paper we show that all functions between internal sets in this model are internal. We also illustrate how this model can be used by giving a constructive nonstandard proof of the Implicit Function Theorem.

The paper establishes equivalence of two axiomatizations of the relative set theory GRIST, examines dependence of nonstandard concepts on the choice of the level, and proves that several nonstandard definitions of Lebesgue measure are equivalent. The last section extends GRIST to a theory of external sets. 2000 Mathematics Subject Classification 26E35 (primary); 03E70, 03H05, 28E05 (secondary)

A general method of interpreting weak higher-type theories of nonstandard arithmetic in their standard counterparts is presented. In particular, this provides natural nonstandard conservative extensions of primitive recursive arithmetic, elementary recursive arithmetic, and polynomial-time computable arithmetic. A means of formalizing basic real analysis in such theories is sketched. §

We prove that a nonstandard extension of arithmetic is eeectively conservative over Peano arithmetic by using an internal version of a deenable ultrapower. By the same method we show that a certain extension of the nonstandard theory with a saturation principle has the same proof-theoretic strength as second order arithmetic, where comprehension is restricted to arithmetical formulas.

Cohesive sets play an important role in computability theory. Here we use cohesive sets to build nonstandard versions of the rationals. We use Koenigsmann’s work on Hilbert’s Tenth Problem to establish that these nonstandard fields are rigid. As a consequence we obtain results about automorphisms of the lattices of computably enumerable vector spaces arising in the context of Ash’s conjecture.

Most of the research in foundations of mathematics that I do in some way or another involves the use of nonstandard models. I will give a few examples, and indicate what is involved. 1. General algebra and measurable cardinals. An unexpectedly direct connection. 2. Borel selection and higher set theory. A descriptive set theoretic context extensively pursued by some functional analysts. 3. Equa...

This paper presents analysis of data aggregated from three experiments, investigating whether there are social differences in the processing of sociolinguistic variation. Three self-paced reading experiments tested the processing of variable subject-verb agreement in English. Sentences appeared in one of three agreement conditions: standard [singular+doesn’t] or [plural+don’t], nonstandard [sin...

Intuitively, it is tempting to view a cube in R as a discrete lattice with infinitesimal edges. A similar infinitesimal picture suggests itself with other fractal spaces such as the Sierpinski gasket or the Sierpinski sponge (see [13], [5].) The purpose of this paper is to show certain classes of metric spaces characterized by volume growth properties of balls ([4],[2]) can viewed as graphs wit...

Subject to some conditions, the input data for the Drinfeld-Sokolov construction of KdV type hierarchies is a quadruplet (A,Λ, d1, d0), where the di are Z-gradations of a loop algebra A and Λ ∈ A is a semisimple element of nonzero d1-grade. A new sufficient condition on the quadruplet under which the construction works is proposed and examples are presented. The proposal relies on splitting the...