Nonstandard Analysis, Axiomatically
Nonstandard Analysis, Axiomatically
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非标准分析，公理化
DOI：
10.1007/978-3-662-08998-9
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发表时间：
2004
期刊：
影响因子：
--
通讯作者：
M. Reeken
中科院分区：
文献类型：
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作者：
V. Kanovei;M. Reeken
在数学基础的发现之后，令人惊讶的是，对整个数学的影响微乎其微。如果我们看一下不同数学学科的标准教科书，特别是那些更接近于所谓的应用数学的教科书，几乎没有关于数理逻辑和模型论之外的发展的痕迹。但似乎可以公平地说，人们普遍认为，Zermelo-Fraenkel选择理论（ZFC）中体现的原则是对数学集合论基础的正确描述。在上面提到的大多数教科书中，当然没有讨论这些问题，集合论被非正式地假定，尽管更高级的原则，如选择或有时替代经常被明确地提到。这隐含地确定了一个与基础结果不一致的物理宇宙的观点。例如，大多数数学家仍然想当然地认为，真实的数系是唯一确定的，直到同构，这是一个正确的观点，只要一个人不接受看”不自然”的解释的成员关系。
In the aftermath of the discoveries in foundations of mathematiC's there was surprisingly little effect on mathematics as a whole. If one looks at stan dard textbooks in different mathematical disciplines, especially those closer to what is referred to as applied mathematics, there is little trace of those developments outside of mathematical logic and model theory. But it seems fair to say that there is a widespread conviction that the principles embodied in the Zermelo-Fraenkel theory with Choice (ZFC) are a correct description of the set theoretic underpinnings of mathematics. In most textbooks of the kind referred to above, there is, of course, no discussion of these matters, and set theory is assumed informally, although more advanced principles like Choice or sometimes Replacement are often mentioned explicitly. This implicitly fixes a point of view of the mathemat ical universe which is at odds with the results in foundations. For example most mathematicians still take it for granted that the real number system is uniquely determined up to isomorphism, which is a correct point of view as long as one does not accept to look at" unnatural" interpretations of the membership relation.