Part 3 presents applications of Alpha-Theory, such as (1) alternative proofs for standard results, (2) the notion of gauge spaces and quotients, and (3) a simplified treatment of stochastic differential equations. A more thorough discussion of the foundational aspects of this principle is postponed to Part 4. Curiously, this principle turns out to be independent of Alpha-Theory. This principle states that each positive infinitesimal is the Alpha-limit of some decreasing sequence with a corresponding standard limit equal to zero. This chapter also presents the Cauchy Infinitesimal Principle, which formalizes an idea due to Augustin-Louis Cauchy (or at least one possible interpretation of what he wrote). The standard topology on hypersets is defined and the relation between Alpha-limits and this standard topology is investigated. Earlier in this part, we also encounter some notions familiar from other approaches to non-standard analysis, such as over- and under-spill, as well as (countable) saturation. The final chapter of Part 2 includes a consistency proof for Alpha-Theory (in Section 6.7), using the notion of superstructures (which is introduced beforehand) and that of Alpha-morphisms, which are introduced in this section. In fact, the book includes an alternative axiomatization of Alpha-Theory with just four axioms that include transfer as well. (For instance, while both the Archimedean property and the completeness of the standard field of real numbers can be expressed as elementary formulas, their transferred counterparts do not express the Archimedean property or the completeness of the hyperreals, which lack both.) Some axiomatic approaches to non-standard analysis include a form of transfer as an axiom. They offer some examples and include a warning of how to avoid erroneous conclusions. The authors consider this theorem to be the fundamental result of Alpha-Theory. This formalism is needed to state the transfer principle precisely, which crucially applies only to elementary formulas, and to prove that it applies to Alpha-limits. While the book keeps the promise of its blurb that technical notions from logic are not required from the beginning, this part requires the introduction of some notions from first-order logic. Starting from real closed fields, which are fields with the same first-order properties as |$\mathbb^\ast)^\ast$|. Infinitely large numbers are numbers larger than |$n$| for all natural numbers |$n$| non-zero infinitesimals are their multiplicative inverses. Robinson developed an alternative framework for differential and integral calculus based on infinitely large numbers and infinitesimals. Non-standard analysis was first developed by Abraham Robinson in the 1960s. This book gives a systematic exposition of these developments, spanning nearly two decades. They collaborated on both topics with logician Marco Forti and, in recent years, with Emanuele Bottazzi and occasionally other mathematicians. In a series of papers starting in 2003, the authors developed their own approach to non-standard analysis, which they call Alpha-Theory, as well as a new approach to measuring the size of labelled countable sets, which they call Numerosity Theory. In addition, the authors have a long-standing collaboration on non-standard analysis. Benci specializes in partial differential equations and Di Nasso in mathematical logic. Vieri Benci and Mauro Di Nasso are two mathematicians affiliated with the University of Pisa.
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