Paradoxical sets exist as a consequence of the Axiom of Infinity. Admitting infinite classes as sets is sufficient to allow paradoxical sets.
An example of a paradoxical set is the natural numbers. They are paradoxical with respect to the group of functions generated by the natural function :
Split the natural numbers into the odds and the evens. The function maps boths sets onto the whole of . Since only finitely many functions were needed, the naturals are -paradoxical.
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