Re: Platonism

tchow_at_lsa.umich.edu
Date: 12/06/04 

Date: 05 Dec 2004 23:46:44 GMT

In article <covbi6$2bb$1@charm.magnus.acs.ohio-state.edu>,
Nath Rao <nXatTHEhraCAPSo@yahoo.com> wrote:
>The problem is that "the equivalence class of all well-ordered set with
>2 elements" is unmanageable at best, and undefinable at best. So we pick
>a consistently definable representative.
>
>It may be the case that the above is not mentioned in the set theory
>treatment you use.

No argument from me, except that I think you're missing the context of the
discussion. The issue is not how to present the concepts of finite ordinal
and finite cardinal in the cleanest and most logically rigorous way, but
whether there is a conceptual distinction between the ordinal 2 and the
cardinal 2 that is lost when one formalizes them as *identical* sets.
I think there is.

For example, in some context where you're studying ordered sets, it may
be good to define an ordered set as a set equipped with a relation "<"
satisfying certain conditions. The ordinal 2 would be a special case of
this, and so wouldn't be the von Neumann ordinal but a set with 2 elements
equipped with a total ordering. 

-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences