16/06/04 05:09:39.74 CtkyGlEO.net
>>339
>「無限公理」
これは、英文だが。>>335 渕野昌先生の記事にも出てくるので
URLﾘﾝｸ(en.wikipedia.org)
Axiom of infinity
(抜粋)
Independence
The axiom of infinity cannot be derived from the rest of the axioms of ZFC, if these other axioms are consistent.
Nor can it be refuted, if all of ZFC is consistent.
Indeed, using the Von Neumann universe, we can make a model of the axioms where the axiom of infinity is replaced by its negation.
It is V ω , the class of hereditarily finite sets, with the inherited element relation.
If allowed, the empty domain also satisfies the axioms of this modified theory, as all of them are universally quantified, and thus trivially satisfied if no set exists.
The cardinality of the set of natural numbers, aleph null ( aleph _{0}), has many of the properties of a large cardinal.
Thus the axiom of infinity is sometimes regarded as the first large cardinal axiom, and conversely large cardinal axioms are sometimes called stronger axioms of infinity.