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超实数发现的历史真相

(2019-09-19 13:16:49)

超实数发现的历史真相

    根据最新历史研究表明:1931年,哥德尔不完全性定理导出模型论紧致性定理,据定理此,Skolem1934年证明非标准算术的存在性。

1960年,鲁宾逊

天才地领悟到,利用Skolem证明思路创立非标准分析.由此成功地引入了超实数系统。

本文附件文章,参阅60余历史资料没证明了上述论断。      请读者参阅。

袁萌  陈启清  919

附件:2012 6

Hyperreals and Their Applications

Sylvia Wenmackers

Formal Epistemology Project Faculty of Philosophy University of Groningen Oude Boteringestraat 52 9712 GL Groningen The Netherlands E-mail: s.wenmackers@rug.nl URL: http://www.sylviawenmackers.be/

Overview

Hyperreal numbers are an extension of the real numbers, which contain innitesimals and innite 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 prex ‘hyper-’. Just like standard analysis (or calculus) is the theory of the real numbers, non-standard analysis (NSA) is the theory of the hyperreal numbers. NSA was developed by Robinson in the 1960’s and can be regarded as giving rigorous foundations for intuitions about innitesimals that go back to Leibniz (at least). 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. It is set up as an annotated bibliography about hyperreals. 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. The document consists of two parts: sections 1–3 introduce NSA from dierent 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.

1

Part 1: Introducing the hyperreals

Abstract

NSA can be introduced in multiple ways. Instead of choosing one option, these notes include three introductions. Section 1 is best-suited for those who are familiar with logic, or who want to get a avor of model theory. Section 2 focuses on some common ingredients of various axiomatic approaches to NSA, including the star-map and the Transfer principle. Section 3 explains the ultrapower construction of the hyperreals; it also includes an explanation of the notion of a free ultralter.

1 Existence proofs of non-standard models

1.1 Non-standard models of arithmetic

The second-order axioms for arithmetic are categoric: all models

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