Proof that an infinite set can be written as the countably infinite union of pairwise disjoint infinite subsets without using the axiom of choice.

Question

I have the following theorem

Any infinite set can be written as the countably infinite union of pairwise disjoint infinite subsets

I have found a couple of proofs for this but I was wondering if there is some proof that doesn't use the axiom of choice (and it's equivalent statements). Does there exist such a proof?

It is easy to construct such a partition explcitely for $\Bbb N$, say. Now if only you could prove that for your infinite set $A$, there exists an injective map $\Bbb N\to A$ ... — Hagen von Eitzen, Nov 17 '20 at 19:02

score 4 · Answer 1 · answered Nov 17 '20 at 19:00

4

No. An amorphous set is infinite but cannot be written even as the union of two disjoint infinite sets, and it is consistent with $\mathsf{ZF}$ without choice that amorphous sets exist.

answered Nov 17 '20 at 19:00

Brian M. Scott

616,228

Proof that an infinite set can be written as the countably infinite union of pairwise disjoint infinite subsets without using the axiom of choice.

1 Answers1