[FOM] obscure technical question concerning ZF without choice

Thomas Forster T.Forster at dpmms.cam.ac.uk
Sun Jan 25 09:57:26 EST 2009

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 
about classes)

Must there be an $\alpha$ such that $C_\alpha$ = C_{\alpha +1}$?

Is this known?  (What i am hoping is that as soon as i have lifted my head 
above the parapet in posting this i will discover a simple solution: it 
often works.)  I do remember many years ago being told that someone called 
`Morris' proved (i think i've got this right) that there could be models 
of ZF-minus-choice in which one could find arbitrarily large things that 
were the size of a power set of a countable union of countable sets -- 
which has the same sort of flavour.  I would be grateful to anyone telling 
me more about either of these.



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