Re: Zenkin's paper on Cantor (reply of Dr. Zenkin)
From: Chas Brown (cbrown_at_cbrownsystems.com)
Date: 11/29/04
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Date: 28 Nov 2004 16:13:49 -0800
examachine@gmail.com (Eray Ozkural exa) wrote in message news:<320e992a.0411231630.18c36d7c@posting.google.com>...
> cbrown@cbrownsystems.com (Chas Brown) wrote in message news:<ba7dcfd7.0411230310.337a3ac9@posting.google.com>...
> > Mathematics doesn't always erase confusion in our intuition by
> > conforming to it; it also does it by _informing_ it.
> >
> > It seems unintuitive (at least to me) at first that, even in
> > principle, you can't trisect every angle using the usual straight edge
> > and compass; but it is something you can prove; and eventually it
> > becomes "intuitive".
>
> I disagree with this formalist interpretation of mathematics.
>
That is your perogative. But to return to the topic of the thread, it
should have no bearing on the correctness or incorrectness of Zenkin's
argument - his argument fails because it doesn't follow logically; not
because he espouses a formalist or other philosophy.
> Your intuition was simply wrong at first.
That was my point. Your intuition regarding "size of a set" appears to
be "wrong" as well - at least when you use the standard definition of
"size of a set". If you want a definititon of "size of a set" which
conforms to your intuitions, you'll need to be more specific about
what your intuitions require of the term, "size of a set".
> It's muddled to think that
> you just get used to some formal relation. If you don't understand it,
> it's just syntax.
Until you understand it, it's "just" syntax. When you understand it,
it becomes incorporated in your new, _correct_ intuition about the
result in question. Or don't you think one can acquire new intuitions
which extend and correct one's old ones?
> (So, I don't like that statement of Von Neumann at
> all...) That's another argument, though, and I don't think it's
> necessary for me to make it.
>
> > Why should the _name_ of the things
> > we define have any significance in the logical systems we construct?
>
> I never said the names matter. What matters is that they have
> reference. I think that should be obvious to anybody who took a logic
> class.
>
You have claimed (without particular proof) that "cardinality" is
antimonius - i.e., leads to a _logical_ contradiction.
Then if we define the term "chasanality" identically to "cardinality",
and list its properties, then it should create a _logical_
contradiciton iff "cardinality" creates a _logical_ contradicition. I
have seen no evidence of such a contradicition.
It seems the "antimony" you are claiming is not a _logical_
contradiction, but instead offends your preconception of what "size of
a set" _should_ mean.
The onus is then on you to resolve this for yourself via a different
definititon of "size of a set". The standard one seems to work well
for most mathematicians; in particular, as others have pointed out, it
is consistent with one's intuition regarding finite sets, unlike the
alternate definition you mentioned.
> Regards,
Cheers - Chas
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