On 7/6/2013 10:49 PM, Shmuel (Seymour J.) Metz wrote: > In <Rt-dna0-mbsy7ErMnZ2dnUVZ_rmdnZ2d@giganews.com>, on 07/05/2013 > at 08:23 PM, fom <fomJUNK@nyms.net> said: > >> modern mathematics differentiates ordinal numbers from cardinal >> numbers. > > Modern Mathematics treats cardinals as special cases of ordinals. >
Coming from one who slipped in a few references to New Foundations in other threads, this is surprising. New Foundations uses Fregean number classes, does it not?
The source of that assertion in the sense I used it can be found in William Tait's "Frege vs. Cantor and Dedekind" reprinted in a Blackwell anthology (2002) edited by Jacquette. ISBN: 0-631-21869-6
I posted once that my foundational studies had led me to mathematical systems in which the distributivity laws did not necessarily hold or required modification.
I remarked in that post, that I believed this arose from the use of canonical representatives within classes of terms. This mathematical practice is not "represented" in logic.
Also, I directed attention to Lawvere's discussion of the category of pointed sets in which the distributive laws do not hold.
Fregean number classes cannot form sets in the usual set theories to which your remarks seem to relate. One way to interpret von Neumann cardinals, however, is as a canonical representative from its given Fregean number class. It is in this way that a cardinal number is representable as a set.
So, if one asks "What is a number?" does the question of identity include consideration of pointed classes? Or, relative to a model, pointed sets?