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Section 3 explains the ultrapower construction of the hyperreals it also includes an explanation of the notion of a free ultrafilter. Section 2 focuses on some common ingredients of various axiomatic approaches to NSA, including the star-map and the Transfer principle.
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Section 1 is best-suited for those who are familiar with logic, or who want to get a flavor of model theory. Instead of choosing one option, these notes include three introductions. Abstract NSA can be introduced in multiple ways. The document consists of two parts: sections 1–3 introduce NSA from different perspectives and sections 4–9 discuss applications, with an emphasis on topics that may be of interest to formal epistemologists and to philosophers of mathematics or science. It does not aim to be exhaustive or to be formally precise instead, its goal is to direct the reader to relevant sources in the literature on this fascinating topic. It is set up as an annotated bibliography about hyperreals. This document is prepared as a handout for two tutorial sessions on "Hyperreals and their applications", presented at the Formal Epistemology Workshop 2012 (May 29–June 2) in Munich.
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NSA was developed by Robinson in the 1960's and can be regarded as giving rigorous foundations for intuitions about infinitesimals that go back to Leibniz (at least). Just like standard analysis (or calculus) is the theory of the real numbers, non-standard analysis (NSA) is the theory of the hyperreal numbers. The set of hyperreal numbers is denoted by * R or R * in these notes, I opt for the former notation, as it allows us to read the * -symbol as the prefix 'hyper-'.
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Nonstandard analysis offers great simplicity in defining basic concepts.Overview Hyperreal numbers are an extension of the real numbers, which contain infinitesimals and infinite numbers. Part of the effort was directed towards utilizing some of the new methods and techniques in teaching. Since then, much has been written on the theory of infinitesi- mals and their applications. Nonstandard analysis was created (or perhaps recreated) by Abraham Robinson (4), who reintroduced infinitesimals to mathematics using the formal apparatus of mathematical logic. At this point let me just mention that the theorem excludes the possibility of a simple presentation of some structures, and the number system of nonstandard analysis is one of them. The notion of complexitz7, as used in Tennenbaum's theorem, has precise meaning, and I will say more about it later. A theorem of Stanley Tennenbaum's (5), pub- lished in 1959, states that certain mathematical structures are very complex. A level of abstraction and set theoretic methods are necessary to prove that certain structures even exist. Some mathematical structures, however, cannot be visual- ized easily. The same pictures also seem indispensable in learning mathematics. Rather, mathematicians work with pictures visual representa- tions of mathematical structures. We do not limit ourselves to computations and rigorous proofs. When we "do"mathemat- ics, we rarely think formally. Precision and formal rigor are essential in mathematics, but these qualities apply only to the final products of mathematical activity. With simple images we can illustrate, quite accurately, more abstract concepts. Those structures often do not seem to have direct interpretations in the physical universe we know, but nevertheless we can picture them in the mind's eye. We use our natural intuition of the geometiy of three dimensional space as a starting point for constructions of abstract mathematical structures, such as higher dimensional vector spaces, non- euclidean geometries, and topological spaces of various kinds. In mathematics we have tools that allow us to think visually about objects far beyond direct physical perception. This might seem an obvious observation since infinitesimals are infinitely small, but that is not the point I wish to make here. Consider, for example, the difficulty of seeing a number system that includes infinitesimals. Visualizing mathematical structures is a broad and important subject in mathemat- ics education.