[FOM] Ordering a rank (Joe Shipman)

[David Ross]

Marco can answer for himself, but I suspect he means that # can be proved for some families in ZFC provided there is no assumption that the 
numerosities are "Euclidean" (ie, if A is a strict subset of B then n(A)<n(B)).

I'm a little puzzled at the formulation of (#); usually these principles are stated in terms of finding subsets instead of extending to 
supersets.  (For example the "Zermelian" principle in the numerosity theory is that n(A)<=n(B) iff n(A)=n(C) for some subset C of B.)  I 
just did a quick flip-through of Benci and di Nasso's recent book, can't find any formulations in terms of extensions.


> ?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
>> Universit? di Pisa
>> Largo B. Pontecorvo 5
>> 56123 PISA (Italy)
>> Tel. +39 050 2213876
>> email: marco.forti at unipi.it


--
David A. Ross
Dept. of Mathematics, University of Hawaii
808-956-4673       ross at math.hawaii.edu