# A question about Dedekind-infinite sets and Peano natural integers.

I've doubt about Dedekind-infinite sets, sets which are in bijection with a proper part, in the ZF axiomatic framework, without Axiom of Choice.

Assume a Dedekind-infinite set X exists.

Then it can be proved X contains a Dedekind-infinite N set which satisfy the Peano Axioms.

This set N can be well ordered using traditional arguments from Peano Axioms.

It can also be proved its initial chains, in this well order, I_n = { m < n }, are Dedekind-finite.

Can be proved they are also finite using the ZF definition of "finite set", without any further assumption?

• You may find useful my blog posting on defining finite or infinite sets at dcproof.com/Infinity.html There, I start by developing a non-numeric definition of a finite set based on a walk through a finite village. This in contrast to the usual approach using Hilbert's Infinite Hotel to develop the notion of an infinite set. – Dan Christensen Mar 17 at 16:31

Yes. The definition of finite is being equipotent with a proper initial segment of $$\omega$$, that is the ordinal corresponding to $$\Bbb N$$.
Once you have established a bijection, and in fact an order isomorphism, between your "copy of $$\Bbb N$$" and $$\omega$$, and you have by virtue of it being a model of $$\sf PA_2$$, then the initial segments correspond exactly to the actual finite cardinals.