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To formalize the concepts of countably and uncountably infinite sets, we need the [[set theory]] concept of [[cardinality]].  Using this concept it can be shown that there are infinitely many distinct infinite cardinalities.

To formalize the concepts of countably and uncountably infinite sets, we need the [[set theory]] concept of [[cardinality]].  Using this concept it can be shown that there are infinitely many distinct infinite cardinalities.

In [[Zermelo-Fraenkel]] set theory, there is the [[Axiom of Infinity]], asserting the existence of an infinite set.  The set that it creates is essentially the same as the integers.  Some constructive mathematicians work without this axiom, and determine which results may be proved without assuming it.  

In [[Zermelo-Fraenkel]] set theory, there is the [[Axiom of Infinity]], asserting the existence of an infinite set.  The set that it creates is essentially the same as the integers.  Some "constructive" mathematicians work without this axiom, and determine which results may be proved without assuming it.  

==References==

==References==