[FOM] Ordering a rank (Joe Shipman)

[Marco Forti]

David, I'm not sure you have seen my response to Joe Shipman, so I copy it
here,
in case it might be of interest to you:

two remarks are in order:
1) the property I quoted was
  (#)             n(A)<n(B) if and only if exists C s.t. A (strictly)
included in C, and n(C)=n(B),
and not the similar
  ($)             n(A)<n(B) if and only if exists C s.t. C(strictly)
included in B and n(C)=n(A).
I do not claim that they are equivalent, nor that they are not.
What is actually proved in [FM] is that (#) is equivalent to the symmetric
property
  (@)             n(A)<n(B) if and only if exists C,D s.t. D (strictly)
included in C, and n(C)=n(B), n(D)=n(A).
I don't know wether the same is true for ($), only that (#) seems easier to
obtain in our uncountable models.
2) You are right in your last assertion about stronger requirements, in
fact the only articles of mine where equivalence with special ultrafilters
is claimed (and proved) are:
[QSU] A. Blass, M. Di Nasso, M. Forti - Quasi-selective ultra lters and
asymptotic numerosities, Adv. Math. 231 (2012),
where the equivalence is established between "asymptotic" numerosities of
subsets of N that generalize the usual asymptotic density,
and a class of ultafilters, introduced n ad hoc and intermediate between
selective and P-point, so a little beyond ZFC.
[FM] M. Forti, G. Morana Roccasalvo - Natural numerosities of sets of
tuples, Trans. Amer. Math. Soc. 367 (2015),
where the same result is extended to numerosities explicitly qualified as
equivalent to the asymptotic ones, on the one hand;
on the other hand, a new kind of ultrafilters, called "gauge", is
introduced and proved equivalent to the existence of numerosities
satisfying (#)
(but again only on countable sets): actually I conjectured that also those
ultrafilters were beyond ZFC, but now I think that
I have proved that they exist in ZFC, even on sets of arbitrary cardinality.

Best wishes, MF
.


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 Sat, Jan 4, 2020 at 4:07 AM David Ross <ross at math.hawaii.edu> wrote:

>
> 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
>
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