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Without the axiom of choice, these may not be equivalent. Sets that are neither finite nor Dedekind infinite are called '''amorphous sets'''.
Within [[ZFC]], the existence of infinite sets is guaranteed by the [[axiom of infinity]], which implies the existence of the natural numbers, which can be proven infinite. Without this axiom, infinite sets can't be proven to exist. A model of this theory is provided by the [[hereditarily finite set]]s.
Important kinds of infinite sets include [[countable]] and [[uncountable]] sets.
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