19/11/27 22:09:56.09 qnEhNItW.net
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＞基礎付け問題
これは、下記が、元記事だな（＾＾
URLﾘﾝｸ(en.wikipedia.org)
Finite set
(抜粋)
Contents
1 Definition and terminology
2 Basic properties
3 Necessary and sufficient conditions for finiteness
4 Foundational issues
5 Set-theoretic definitions of finiteness
5.1 Other concepts of finiteness
Foundational issues
Georg Cantor initiated his theory of sets in order to provide a mathematical treatment of infinite sets. Thus the distinction between the finite and the infinite lies at the core of set theory.
Certain foundationalists, the strict finitists, reject the existence of infinite sets and thus recommend a mathematics based solely on finite sets.
Mainstream mathematicians consider strict finitism too confining, but acknowledge its relative consistency: the universe of hereditarily finite sets constitutes a model of Zermelo?Fraenkel set theory with the axiom of infinity replaced by its negation.
Even for those mathematicians who embrace infinite sets, in certain important contexts, the formal distinction between the finite and the infinite can remain a delicate matter.
The difficulty stems from Godel's incompleteness theorems. One can interpret the theory of hereditarily finite sets within Peano arithmetic (and certainly also vice versa), so the incompleteness of the theory of Peano arithmetic implies that of the theory of hereditarily finite sets.
In particular, there exists a plethora of so-called non-standard models of both theories. A seeming paradox is that there are non-standard models of the theory of hereditarily finite sets which contain infinite sets, but these infinite sets look finite from within the model.
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