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Let $\Omega$ be an uncountable set. Without assuming the axiom of choice, does there exist a subset $S\subset \Omega$ such that neither $S$ nor $\Omega\setminus S$ is countable?

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Impossible. It is consistent with ZF (i.e. Zermelo-Frankel set theory without axiom of choice) that there is an infinite set $X$ such that if $A$ and $B$ divides $X$ (i.e. $A\cap B = \varnothing$ and $A\cup B = X$) then either $A$ or $B$ must be finite. Such set is called amorphous set. Of course, such set does not exist when assuming choice.

If $X$ is amorphous and $S\subset X$, then one of $S$ or $X-S$ should be finite, so it is impossible that both $S$ and $X-S$ are uncountable. Moreover, countable set should not be amorphous, since we can divide countable sets into two countable sets (e.g. divide $\Bbb{N}$ into odds and evens) so any amourphos set is uncountable. (In fact, every amorphous set can not contain countably infinite subsets.)

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    $\begingroup$ If you allow me, I would suggest you to include in your answer that an amorphous set can NOT be countable. $\endgroup$ – Ramiro Oct 1 '15 at 4:28
  • $\begingroup$ @Ramiro Thanks for your comment. I add it. $\endgroup$ – Hanul Jeon Oct 1 '15 at 4:55
  • $\begingroup$ I see: even the existence of a countable subset of an uncountable set relies on the axiom of choice. $\endgroup$ – pre-kidney Oct 1 '15 at 6:22
  • $\begingroup$ @pre-kidney: You can actually construct $\aleph_1$-amorphous set, where every subset is countable or co-countable. These sets can carry quite a lot more structure than just amorphous sets. $\endgroup$ – Asaf Karagila Oct 1 '15 at 6:34

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