FOM: Cantor's subsets
Dennis E. Hamilton
dennis.hamilton at acm.org
Sat Jun 22 17:59:34 EDT 2002
Thanks for stepping out under the spot light!
I wanted to comment on one thing that is more consequential than you say.
There is a pitfall in speaking of "all of the N's", because it leaves open a
fallacy with regard to the principle of induction. And we need to be
careful when we say all numbers are numbers, because we tend to think that
all operations with numbers produce numbers. I have been engaged in an
inquiry into the problems that arise when one asserts that the product of
*all* the non-zero natural numbers is a natural number. (That is, it would
be in N) and whether such an assumption can be supported by application of
Peano's axiom for induction. I am unwilling to accept that interpretation,
but my partner in this discussion is very much committed to it and also to
it being a demonstration of the inconsistency of Peano's arithmetic theory.
(I think this last bit is an attribution error, in that it is the
interpretation that is invalid, and it cannot be projected onto Peano, but
let's not get into that here.)
So I think mathematicians cannot ignore this, and do sometimes get it on
their shoes. The reason for the retreat to symbols -- that is, the safe
ground of an axiomatization, is to protect us and give us places to be
cautious, and to give us some grip on the complexity of the situation.
Also, it supports us in noticing where we may be reading into something that
which is simply not there. There's no guarantee of course. It seems that
we (including mathematicians) are mostly not "reasoning" in the symbol
system, but about the symbol system or what we interpret it to mean, ....
... and here we are.
Thank you, by the way, for your questions. I have failed in all of these
years to dig into Cantor and it was a rewarding experience. I even found
something that may provide some insight into the business of equivalence of
unbounded subsets of N. Cantor uses equivalence, not equinumerous wherever I
have looked. This is the same as equipollence (and Suppes uses it that
way), though I haven't found that precise term used in Cantor.
From: owner-fom at math.psu.edu [mailto:owner-fom at math.psu.edu]On Behalf Of
Sent: Friday, June 21, 2002 10:16
To: fom at math.psu.edu
Subject: RE: FOM: Cantor's subsets
[ ... ]
(d) that we can speak of "the" A's, as in the definite noun phrase "the
natural numbers". If this is meant to signify "all" natural numbers, it
fails. Since, for any collection of numbers (B's), there can be found other
numbers (A's). None of our arbitrarily chosen numbers are the A's (the
other numbers), yet, of course, the other numbers are numbers - all numbers
are numbers. There is nothing that satisfies the expression "all the
natural numbers". This is of course is a merely philosophical
consideration, concerning the failure of reference of a purportedly
referring expression of the form "all the F's". Mathematicians can
therefore ignore it as concern about useof English words, semantics or
whatever, and go back to their symbols!
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