19/10/06 07:57:27.47 d8OQiN+r.net
>>95 追加
>Infinity
>This final axiom asserts the existence of an infinitely large set which contains the empty set, and for each set a that it contains, also contains the set {a}.
> (Thus, this infinite set must contain Φ, {Φ}, {{Φ}}, ….)
で、N＝{Φ, {Φ}, {{Φ}}, …}で、自然数の集合Nができるけど
無限公理で最初は、Nよりも大きな集合ができるんですよね、確か（下記wiki）
それを、最小の無限集合に絞って小さくする操作が必要です
最小の無限集合に絞った結果、Nには有限の元ｎしか含まれないものができる
なので、無限公理でできた最小に絞る前の無限集合には、
自然数を表現する以上の
つまり、真に無限の{・・・{Φ}・・・}なる無限多重カッコ{}の集合が
含まれていることは
明白ですね
QED
（参考）
URLﾘﾝｸ(en.wikipedia.org)
Axiom of infinity
(抜粋)
In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo?Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing the natural numbers. It was first published by Ernst Zermelo as part of his set theory in 1908.[1]
Thus the essence of the axiom is:
There is a set, I, that includes all the natural numbers.
Extracting the natural numbers from the infinite set
The infinite set I is a superset of the natural numbers. To show that the natural numbers themselves constitute a set, the axiom schema of specification can be applied to remove unwanted elements, leaving the set N of all natural numbers. This set is unique by the axiom of extensionality.