[FOM] Ordering a rank

[Noah Schweber]

> For which ordinals alpha can there exist a total ordering < on V_alpha
such that A<B whenever A is a proper subset of B?

Unless I'm misunderstanding the question, the answer is "all of them" since
every partial order can be extended to a linear order (this is Szpilrajn's
Extension Theorem). Of course, this requires choice; without choice, I
believe it is consistent with ZF that already $V_{\omega+1}$ cannot be so
ordered (I think Cohen's original model of the failure of choice witnesses
this).



 - Noah

On Sun, Dec 29, 2019 at 2:48 PM Joe Shipman <joeshipman at aol.com> wrote:

> For which ordinals alpha can there exist a total ordering < on V_alpha
> such that A<B whenever A is a proper subset of B?
>
> — JS
>
> Sent from my iPhone
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