Nonstandard Labour Values

Nonstandard Gaussian quadrature formulae based on operator values

In this paper, we develop the theory of so-called nonstandard Gaussian quadrature formulae based on operator values for a general family of linear operators, acting of the space of algebraic polynomials, such that the degrees of polynomials are preserved. Also, we propose a stable numerical algorithm for constructing such quadrature formulae. In particular, for some special classes of linear op...

Nonstandard Analysis by Means of Ideal Values of Sequences

A notion of ideal value of N -sequences is axiomatized through elementary properties. The resulting theory ZFC[α] provides nonstandard analysis with a general foundational framework. 1991 MSC. Primary: 26E35, 03E70, 03H05. Secondary: 03C20, 03E35. In this paper, the axiomatic system ZFC[α] is presented. It is a generalization in a set theoretic context of an approach to nonstandard analysis rec...

FOUNDATIONS OF NONSTANDARD ANALYSIS A Gentle Introduction to Nonstandard Extensions

Nonstandard Analysis in Topology: Nonstandard and Standard Compactifications

Let (X,T ) be a topological space, and X a non–standard extension of X . There is a natural “standard” topology T on X generated by G, where G ∈ T . The topological space ( X, T ) will be used to study, in a systematic way, compactifications of (X,T ) . Mathematics Subject Classification: 031105, 54J05, 54D35, 54D60.

Working with Nonstandard Models

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...