Re: Cantor's circular "proof" that evens = integers



"G. Frege" wrote:

On 8 May 2007 01:11:57 -0700, "R. Srinivasan" <sradhakr@xxxxxxxxxx>
wrote:

[

george wrote:
Phil wrote:

Well, I am pointing out that that is true ONLY when that
infinite set is combined with the proper axioms.

It is NOT POSSIBLE for it NOT to be so combined.
"that infinite set" DOES NOT HAVE *ANY* EXISTENCE
apart from these axioms! THAT infinite set was PRODUCED BY
*the axiom of infinity*!

]

Classically the position you take may not be tenable. Existence of
infinite sets will have to be "prior" to axioms. I doubt if your
colleagues in this NG will agree with your stand.

Of course, most of us _will_ agree.*) Claiming that "existence of
infinite sets will have to be 'prior' to axioms" seems to embrace a
"realistic" (->mathematical Platonism) point of view, imho. While
George formulated a "formalistic" (or "fictionalistic") position.
This means, (form this point of view) we can justly claim that
there is a an empty set in ZFC BECAUSE we can derive the statement

ExAy~y e x

in this theory, but NOT because there "really" is an empty set.


To be sure, we have comments such as Aatu Koskensilta's

" ... According to any ordinary mathematical
understanding the naturals and the reals are fixed objects, which must
thus be reflected in any formalization of our ordinary mathematical
reasoning. The idea that a formal theory could "vindicate" or "justify"
mathematical reasoning is wrong-headed; rather, we must judge the formal
theories on basis of how well they capture the relevant aspects of our
mathematical knowledge, conceptions, reasoning and so on ..."

although it must be added that Aatu rejects this as expressing any
sort of "realism", or indeed as any sort of philosophical position:

"Talking about our mathematical knowledge, conceptions,
reasoning, and the extent some of their features are captured in this or
that formal theory is not in any obvious sense related to any questions
about realism or anti-realism ..."

and

"that the naturals are a 'fixed object' is not intended as any
philosophical or metaphysical claim ..."

This all does seem to suggest that there is something '"prior" to
axioms', though exactly what that may be is not especially clear,
and possibly not especially relevant to mathematical practice.

To what extent Aatu (and others of this NG) disagree with George's
formalist or "fictionalist" position I would not hazard to guess.

Whether "THAT infinite set was PRODUCED BY *the axiom of infinity*!"
or had some existence "prior" to the axioms, my personal opinion
is that, assuming consistency of the axioms of ZFC, that infinite
set is a logically possible entity, whatever its status as an
actually existent entity, or as a conditionally existent entity.

Whether this position qualifies as formalism or not, I'm not sure.

I also imagine that George might say that his remark was heavily
context related, and must be so understood.

--
hz

P.S. -- I hope George won't regard this post as stepping all
over his conversation with Phil.

--
Posted via a free Usenet account from http://www.teranews.com

.

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