![]() 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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