FOM: Cantor'sThm:IssuesNotAddressed
Robert Tragesser
rtragesser at hotmail.com
Tue Feb 13 04:08:07 EST 2001
The responses on fom to my query have
been very useful, especially in helping
frame what is perplexing, which has not
been addressed, and I think there are
substantial and fundamental issues at
stake:
[1] Whether one has a constructive or
intuitionistically acceptable proof of
a theorem depends very much on the
statement of the theorem. I am not so
sure that fomers I've read have been
considering proofs of Cantor's Theorem
as stated.
[CT}"For any set M and P[M], the set of all
subsets of M, _the power of P[M]
is greater than the power of M_."
The crucial detail is the expression
'the power of'. In Cantor, 'the power
of X" refers to a substantial entity
secured by an act of abstraction. But
it would not be contrary to the spirit
of the statement of this theorem to
substitute for "the power of", 'the
cardinal number of'.
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]."
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?
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].
[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.
It needs some argument whether or not
"intuitionistic ZF" is at all either
an intuitionistically acceptable or
constructively acceptable theory of
"sets". That is: Can one make
intuitionistic or constructive good
sense of "intuitionistic ZF"?
[3] Some variation on question [Q}
above has to be the fundamental question.
The matter of whether or not
impredicative constructions are problematic
for intuitionist or constructivist thinking
is not satisfactorily touched by
simply citing some authority - however
great - on the matter, as Bill Tait cited
Godel. The important thing, the only
thing that should interest us, is the
thinking behind the authoritative opinion.
That is, something like question [Q] ought
to be answered, and answered out of the
deepest motivations for intuitionism or
constructivism.
robert tragesser
westbrook, connecticut
rtragesser at hotmail.com
860 399 6305
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