[過去ﾛｸﾞ] 現代数学の系譜　カントル　超限集合論 (1002ﾚｽ)
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504(2): 現代数学の系譜 雑談 ◆e.a0E5TtKE 2019/11/27(水)22:09 ID:qnEhNItW(9/12) AAS
>>491
＞基礎付け問題

これは、下記が、元記事だな（＾＾

外部ﾘﾝｸ:en.wikipedia.org
Finite set
(抜粋)
Contents
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505(1): 現代数学の系譜 雑談 ◆e.a0E5TtKE 2019/11/27(水)22:11 ID:qnEhNItW(10/12) AAS
>>504

つづき

(This can happen when the model lacks the sets or functions necessary to witness the infinitude of these sets.)
On account of the incompleteness theorems, no first-order predicate, nor even any recursive scheme of first-order predicates, can characterize the standard part of all such models. So, at least from the point of view of first-order logic, one can only hope to describe finiteness approximately.

More generally, informal notions like set, and particularly finite set, may receive interpretations across a range of formal systems varying in their axiomatics and logical apparatus. The best known axiomatic set theories include Zermelo-Fraenkel set theory (ZF), Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC),
Von Neumann?Bernays?Godel set theory (NBG), Non-well-founded set theory, Bertrand Russell's Type theory and all the theories of their various models. One may also choose among classical first-order logic, various higher-order logics and intuitionistic logic.

A formalist might see the meaning[citation needed] of set varying from system to system. Some kinds of Platonists might view particular formal systems as approximating an underlying reality.
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508(3): 現代数学の系譜 雑談 ◆e.a0E5TtKE 2019/11/28(木)00:22 ID:QdpmOFrx(1/7) AAS
>>504 追加

外部ﾘﾝｸ:en.wikipedia.org
Finite set
(抜粋)

Necessary and sufficient conditions for finiteness
In Zermelo?Fraenkel set theory without the axiom of choice (ZF), the following conditions are all equivalent:[citation needed]

2.(Kazimierz Kuratowski) S has all properties which can be proved by mathematical induction beginning with the empty set and adding one new element at a time. (See below for the set-theoretical formulation of Kuratowski finiteness.)
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