FOM: Cantor'sThm:IssuesNotAddressed
Neil Tennant
neilt at mercutio.cohums.ohio-state.edu
Tue Feb 13 12:24:20 EST 2001
On Tue, 13 Feb 2001, Robert Tragesser wrote:
> I do not think fomer's have in mind
> a proof of this theorem as stated;
> I think that they are thinking of
> a different propostion, perhaps:
>
> [RevCT] "There exists a 1-1 maqpping of
> M into, but not onto, P[M]."
Speaking only for myself, this is not the proposition that I had in mind,
because it does not logically that THERE IS NO 1-1 mapping of M onto P[M].
> ASK THIS QUESTION:
> [Q] What extra
> "information" should a constructive or
> intuitionistic proof of [CT] provide
> that wouldn't be demanded from a
> classically acceptable proof?
Why ask this question> Why not ask, instead,
[Q'] Given a constructive or intuitionistic proof of [CT], what insight
might it afford that would be obscured by, or unavailable from, a
strictly classical (non-constructive) proof?
> Shouldn't one say:
> Given in hand
> the definite name of the power or
> cardinal number of the set M, the
> proof provides a determinate procedure
> for arriving at the definite name
> of the power or cardinal number of
> P[M].
What is meant here by "the definite name"? Let's take omega for M. Here's
one candidate for a definite name for the power or cardinal number of
P[omega]:
"the cardinal number of P[omega]".
That's as good as "The Morning Star" (albeit for a different kind of
object). But, whereas we learned some time ago the truth-value of
the identity-statement
"The Morning Star = The Evening Star"
we are still grappling with that of
"the cardinal number of P[omega] = the first limit ordinal after
omega that cannot be put into a 1-1 correspondence with omega"
(a.k.a. CH).
> [2] When Neil Tennant points out
> that the proof of Cantor's
> Theorem is realizable in "intuitionistic
> ZF" and concludes that therefore
> it is an intuitionistic or constructive
> proof, THIS IS AN ABUSE OF LANGUAGE.
First, a belated correction and strengthening of the claim in question:
Cantor's Theorem is a theorem of intuitionistic Z (as I pointed out in an
earlier posting of Mon, 12 Feb 2001 15:20:38 -0500 (EST)).
Secondly, in reply to the charge of linguistic abuse, let me point out
that I have only ever made claims about the intuitionistic status of the
logical reasoning within a mathematical deduction.
Tragesser is correct, of course, to point out (implicitly) that an alleged
theorem will be acceptable to an intuitionist (if and) only if both (i)
the reasoning from first principles is intuitionistic AND (ii) the first
principles themselves are intuitionistically acceptable.
Interestingly, in addressing the concern (ii), a good case can be made
that all the "first principles" about sets that appear as undischarged
assumptions in the (intuitionistic) proof of Cantor's Theorem within Z are
indeed acceptable to the intuitionist.
Neil Tennant
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