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A weak theory nonstandard analysis, with types at all finite levels over both the integers and hyperintegers, is developed as a possible framework for reverse mathematics. In this weak theory, we investigate the strength of standard part principles and saturation principles which are often used in practice along with first order reasoning about the hyperintegers to obtain second order conclusions about the integers.

It is known that IST (internal set theory) is a conservative extension of ZFC (Zermelo-Fraenkel set theory with the axiom of choice); see for example the appendix to [] for a proof using ultrapowers and ultralimits. But these semantic constructions leave one wondering what actually makes the theory work—what are the inner mechanisms of Abraham Robinson’s new logic. Let us examine the question syntactically.

The thesis underlying this text is that the various approaches of mathematics, both the conventional or the diverse non standard approaches, pure or applied, are characterized primarily by their mode of referring and in particular by the more or less important use of the reference, the reference to the sets and collections, that I will distinguish from the reference, the reference to the world of the facts in a broad sense. The direct reference, in traditional mathematics as well as in non standard mathematics (for the main part) is ritually performed in the classical form of modelling, consisting in confronting the facts to a small paradise (a set) correctly structured. So the discourse on the model acts like a metaphor of the modelized reality.

It is now over forty years since Abraham Robinson realized that “” (Robinson [], Introduction, p. 2). The magnitude of Robinson’s achievement cannot be overstated. Not only does his framework allow rigorous paraphrases of many arguments of Leibniz, Euler and other mathematicians from the classical period of calculus; it has enabled the development of entirely new, important mathematical techniques and constructs not anticipated by the classics. Researchers working with the methods of nonstandard analysis have discovered new significant results in diverse areas of pure and applied mathematics, from number theory to mathematical physics and economics.

Elementary Recursive Nonstandard Analysis, in short ERNA, is a constructive system of nonstandard analysis proposed around 1995 by Chuaqui, Suppes and Sommer. It has been shown to be consistent and, without standard part function or continuum, it allows major parts of analysis to be developed in an applicable form. We briefly discuss ERNA’s foundations and use them to prove a supremum principle and provide a square root function, both up to infinitesimals.

Nonstandard Analysis suggests several ways in which the standard theories of distributions and other generalised functions could be reformulated. This paper reviews the contributions of José Sousa Pinto to this area up to his untimely death four years ago. Following the original presentation of nonstandard models for the Sebastião e Silva axiomatic treatment of distributions and ultradistributions he worked on a nonstandard theory of Sato hyperfunctions, using a simple ultrapower model of the hyperreals. (This in particular allows nonstandard representations for generalised distributions, such as those of Roumieu, Beurling, and so on.) He also considered a nonstandard theory for the generalised functions of Colombeau, and finally turned his attention to the hyperfinite representation of generalised functions, following the work of Kinoshita.

Neutrices are convex subgroups of the nonstandard real number system, most of them are external sets. They may also be viewed as modules over the external set of all limited numbers, as such non-noetherian. Because of the convexity and the invariance under some translations and multiplications, the external neutrices are appropriate models of orders of magnitude of numbers. Using their strong algebraic structure a calculus of has been developped. which includes solving of equations, and even an analysis, for the structure of external numbers has a property of completeness. This paper contains a further step, towards linear algebra and geometry. We show that in ℝ every neutrix is the direct sum of two neutrices of ℝ. The components may be chosen orthogonal.

In this article my research on the subject described in the title is summarized. I am not the only person who has worked on this subject. For example, several interesting articles by Steve Leth [], [], [] were published around 1988. I would like to apologize to the reader that no efforts have been made by the author to include other people’s research.

A major open question in combinatorial number theory is the Erdős-Turán conjecture which states that if = 〈〉 is a sequence of natural numbers with the property that ∑ 1/ diverges then contains arbitrarily long arithmetic progressions []. The difficulty of this problem is underscored by the fact that a positive answer would generalize Szcmerédi’s theorem which says that if a sequence A⊂ ℕ has positive upper Banach Density then A contains arbitrarily long arithmetic progressions. Szemerédi’s theorem itself has been the object of intense interest, since first, conjectured, also by Erdős and Turán, in 1936. First proved by Szemerédi in 1974 [], the theorem has been re-proved using completely different approaches by Furstenberg in 1977 []

Let () be an exponential family in ℝ. After establishing nonstandard results about large deviations of the sample mean , this paper defines the nonstandard likelihood ratio test of the null hypothesis : θ ∈ hal(), where is a standard subset of Θ and hal() its halo. If is the level of the test, depending on whether ln/ is infinitesimal or not we obtain different rejection criteria. We calculate risks of the first and second kinds (external probabilities) and prove that this test is more powerful than any “regular” nonstandard test based on .