[FOM] Ordering a rank

[Joe Shipman]

Yes, I should have been clearer. I want the order not only to be total, but also to be compatible with ordered semiring operations as in the theory of “numerosities” that was referenced here recently — a way of measuring size of sets that is finer-grained than cardinality.

Numerosities need not represent a total ordering of V or of some V_alpha because sets incomparable under inclusion can have the same numerosity; however, with choice one can, I think, get an appropriate total ordering from them. But the existence of numerosities even for V_(omega+1) seems to be independent of ZFC if I understand Mancosu correctly (also consistent without needing large cardinals, but I don’t know how far up you can go).

— JS

Sent from my iPhone

> On Dec 31, 2019, at 6:19 PM, Noah Schweber <schweber at berkeley.edu> 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?
> 
> 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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