# [FOM] obscure technical question concerning ZF without choice

Arnold W Miller miller at math.wisc.edu
Fri Feb 6 11:49:06 EST 2009

I am not sure if this question has already been answered.
In
Gitik, Moti; All uncountable cardinals can be singular. Israel J.
Math. 35 (1980), no. 1-2, 61--88.
it is shown that there may be no such ordinal \alpha.
I recently wrote a paper on long Borel hierarchies which
has some related results.
see www.math.wisc.edu/~miller/res/longbor.pdf

Arnold W. Miller

On Sun, 25 Jan 2009, Thomas Forster wrote:

>
> If we drop AC$_\omega$ then we can no longer prove that a union of
> countably many countable sets is countable.  Let
>
> 	$C_0$ be the class of countable sets;
> 	$C_{\alpha + 1}$ be the class of countable unions of things
> 		in $C_\alpha$;
> 	$C_\lambda$ is the union of earlier $C_\alpha$.
>
> (Use Scott's trick if you are worried about classes: this isn't a question
>
> Must there be an $\alpha$ such that $C_\alpha$ = C_{\alpha +1}\$?