純粋・応用数学（含むガロア理論）6

716 ：１３２人目の素数さん：2021/04/04(日) 09:26:22.08 ID:J+JfVsHB.net

>>715
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https://ja.wikipedia.org/wiki/%E9%9B%86%E5%90%88%E8%AB%96
集合論
素朴集合論と公理的集合論
パラドックスを解消すべく建設された公理的集合論では集合や帰属関係の概念はそれらの性質を取り出した記号論理学的な公理系によって間接的に定義される。この捉え方においては集合と帰属関係はユークリッド幾何学の点や線のような根源的な概念で、それ自体は他のものを用いて定義されることはない。 実際には数学を行う上では、集合を素朴集合論の立場で理解しておけば十分なことが多い。

https://en.wikipedia.org/wiki/Set_theory
Set theory
Applications
Set theory as a foundation for mathematical analysis, topology, abstract algebra, and discrete mathematics is likewise uncontroversial; mathematicians accept (in principle) that theorems in these areas can be derived from the relevant definitions and the axioms of set theory. However, it remains that few full derivations of complex mathematical theorems from set theory have been formally verified, since such formal derivations are often much longer than the natural language proofs mathematicians commonly present.

https://en.wikipedia.org/wiki/Naive_set_theory
Naive set theory

Naive set theory is any of several theories of sets used in the discussion of the foundations of mathematics.[1] Unlike axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language. It describes the aspects of mathematical sets familiar in discrete mathematics (for example Venn diagrams and symbolic reasoning about their Boolean algebra), and suffices for the everyday use of set theory concepts in contemporary mathematics.[2]

Sets are of great importance in mathematics; in modern formal treatments, most mathematical objects (numbers, relations, functions, etc.) are defined in terms of sets. Naive set theory suffices for many purposes, while also serving as a stepping-stone towards more formal treatments.

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