Dedekind finite naturally means that every injective self-map is also surjective.
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In ZF, Kuratowski finite implies Dedekind finite, but not vice versa.
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Another way to phrase the pigeonhole principle for finite sets is similar to the principle that finite sets are Dedekind finite : Let and be finite sets.
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That would make the set of such socks Dedekind finite : there can be no infinite sequence of socks, because such a sequence would allow a choice of one sock for infinitely many pairs by choosing the first sock in the sequence.
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Whether the ACCP holds on the right or left side of " R ", it implies that " R " has no infinite set of nonzero orthogonal idempotents, and that " R " is a Dedekind finite ring.
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