[FOM] Ordering a rank

[Joe Shipman]

“Pure ZFC”? That seems to contradict this paragraph from Mancosu’s paper (December 2009 Review of Symbolic Logic):

“But is there a model of such a set of axioms? Yes, there is. The construction consists in taking numerosities to be equivalence classes of nondecreasing functions from the natural numbers into the natural numbers that are equivalent modulo a ‘selective’ (or ‘Ramsey’) ultrafilter. Indeed, the existence of a numerosity function on countable sets is equivalent to the existence of a selective ultrafilter (Benci & Di Nasso, 2003). It is also well known that the existence of a selective ultrafilter is independent of ZFC.”

— JS

Sent from my iPhone

> On Jan 2, 2020, at 11:50 AM, Marco Forti <forti at dma.unipi.it> wrote:
> 
> 
> In fact, the main problem of the "Euclidean" theory of numerosity was  that the "algebraic" total ordering of the ring of numerosities should have the natural set theoretic characterization, namely:
>    (#)             n(A)<n(B) if and only if exists C s.t. A (strictly) included in C, and n(C)=n(B).
> This fact has been established up to now, only for countable sets, by means of selective ultrafilters, so a little beyond ZFC.
> (on the other hand, also the totality of the Cantorian weak cardinal ordering that
>                      |A|<=|B| if and only if exists C s.t. A (weakly) included in C, and |C|=|B|
> had to wait till Zermelo's new axiom of choice!)
> However  a proof in pure ZFC of the general statement (#) for all sets of sufficiently comprehensive families, e.g. V_kappa or H(kappa) for any cardinal kappa, has been found in the last couple of month, and will be posted soon to the arxive.
> Marco Forti
> .
> 
> 
> Marco Forti
> Dipartimento di Matematica
> Università di Pisa
> Largo B. Pontecorvo 5
> 56123 PISA (Italy)
> Tel. +39 050 2213876
> email: marco.forti at unipi.it
> 
> 
>> On Wed, Jan 1, 2020 at 12:35 AM Joe Shipman <joeshipman at aol.com> wrote:
>> 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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