A nonstandard set theory with many levels of standardness was first proposed by Yves Péraire [28]. The theory GRIST formulated here is an extension of Péraire’s RIST. We prove that it is conservative and categorical over ZFC, and universal among theories of its kind. Technically, the paper carries out detailed analysis of the construction of internally iterated ultrapowers introduced in the aut...

In this paper, we propose a new approach to nonstandard analysis without using the ultrafilters. This method is very simple in practice. Moreover, we construct explicitly the total order relation in the new field of the infinitesimal numbers. To illustrate the importance of this work, we suggest comparing  a few applications of this approach with the former methods.

In this paper, we present Internal Set Theory (IST) as a means to more intuitively develop mathematics while still maintaining complete rigor. We begin by stating the axioms of IST and demonstrating several of their central consequences. We then use these results to rigorously introduce the notions of infinitesimal numbers and infinite closeness. Lastly, we develop the calculus in a nonstandard...

1. Internal set theory. We present here a new approach to Abraham Robinson's nonstandard analysis [10] with the aim of making these powerful methods readily available to the working mathematician. This approach to nonstandard analysis is based on a theory which we call internal set theory (1ST). We start with axiomatic set theory, say ZFC (Zermelo-Fraenkel set theory with the axiom of choice [1...

We present a solution to the problem of defining a counterpart in Algebraic Set Theory of the construction of internal sheaves in Topos Theory. Our approach is general in that we consider sheaves as determined by Lawvere-Tierney coverages, rather than by Grothendieck coverages, and assume only a weakening of the axioms for small maps originally introduced by Joyal and Moerdijk, thus subsuming t...