Definition talk:Dedekind-Infinite

Is there another kind of infinite? --prime mover 01:10, 5 April 2012 (EDT)

What's your point? --abcxyz 01:21, 5 April 2012 (EDT)

If there's no difference between "infinite" and "Dedekind-Infinite", what's the point of defining the latter? --GFauxPas 01:26, 5 April 2012 (EDT)

Apparently the two are not equivalent in Zermelo-Fraenkel set theory. The equivalence can be proven using the axiom of countable choice, if I'm not mistaken. See Equivalent Conditions for Dedekind-Infinite Set (unfinished), as well as [1]. --abcxyz 01:29, 5 April 2012 (EDT)

I've googled it. There's an entry in PlanetMath which suggests that if the ACC specifically does not hold, then there exists an infinite set that is not Dedekind-infinite. However, I have not delved any deeper to find what the PlanetMath definition of infinite is - I've left a comment on the discussion board that they might want to provide a link. --prime mover 01:40, 5 April 2012 (EDT)

The PlanetMath definition of "infinite" appears to be the same as ProofWiki's definition of infinite. --abcxyz 01:45, 5 April 2012 (EDT)

Oh yes so it is. --prime mover 01:54, 5 April 2012 (EDT)