[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.
thanks
tf
--
URL: www.dpmms.cam.ac.uk/~tf;
DPMMS ph: +44-1223-337981;
UEA ph: +44-1603-592719
mobile in UK +44-7887-701-562;
mobile in US: +1-412-818-1316;
mobile in NZ +64-210580093.
More information about the FOM
mailing list