超实数的来历
严格说来,超实数就是柯西序列(超精密细分)等价类的数学同构物。
那么,什么是超积?简单地说,序偶集合就是最最简单的“超积”。
坦言之,如果没有数学“超积”,什么无穷小微积分都是一句空话。
我们认为,中国数学教育实现现代化,引入基于模型论无穷小微积分是必由之路。
袁萌 陈启清 9月12日
附:Ultraproduct(超积
An
ultraproduct is a mathematical construction that appears mainly in
abstract algebra and in model theory, a branch of mathematical
logic.
Ultraproduct is a quotient of the direct product of a family of
structures. All factors need to have the same
signature. The ultrapower is the special case of this construction
in which all factors are equal.
For
example, ultrapowers can be used to construct new fields from given
ones. The hyperreal numbers(超实数), an
ultrapower of the real numbers, are a special case of
this.
Some
striking applications of ultraproducts include very elegant proofs
of the compactness theorem and the completeness theorem, Keisler's
ultrapower theorem, which gives an algebraic characterization of
the semantic notion of elementary equivalence,
and the Robinson-Zakon presentation of the use of superstructures
and their monomorphisms to construct nonstandard models of
analysis, leading to the growth of the area of non-standard
analysis, which was pioneered (as an application of the compactness
theorem) by Abraham Robinson.(鲁宾逊)
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