Re: Zenkin's paper on Cantor (reply of Dr. Zenkin)

From: Eray Ozkural exa (examachine_at_gmail.com)
Date: 11/22/04 

Date: 22 Nov 2004 02:13:32 -0800

I'm kind of in a hurry, so I will give quick answers, from which you
can infer the rest.

stephen@nomail.com wrote in message news:<cnqnrn$1pe9$2@msunews.cl.msu.edu>...
> In sci.math Eray Ozkural exa <examachine@gmail.com> wrote:
> : stephen@nomail.com wrote in message news:<cnp65n$ibp$1@msunews.cl.msu.edu>...
> You have to define "size of set". When talking about infinity
> size is not all that well defined. For a finite set "size"
> can be determined by counting, which is the same thing as
> putting the elements in a bijection with a subset of the natural
> numbers. "Cardinality" has been defined. What you think
> about the matter is not all that relevant, especially if you
> think "cardinality" should mean something it does not.

I think "size of a set" is a very well defined term. If cardinality of
an infinite set is not a *complete* explanation for its size, then
this can only cast doubt on the bijection account.

> : I'm saying that there are two approaches for measuring the size of
> : sets, and in general the bijection account does not seem to be
> : satisfactory to reason about the size of infinite sets. Do you accept
> : that there is a paradox of the infinitely big? In other words, do you
> : agree with the received view in philosophy of mathematics, or not?
>
> What other way is there of measuring the size of a set? Are
> you claiming that the fact that |E| = |N| where E is the even
> natural numbers is somehow paradoxical? Even using a naive
> idea of size and infinity this seems intuitive. How many
> even numbers are there? Infinity, so |E|=infinity. There
> are twice as many natural numbers as even natural numbers,
> so |N|=2*|N|=2*infinity. What is 2*infinity? 2*infinity=infinity.
> I am not claiming that this is at a rigourous, but just a
> naive and intuitive way at looking at the problem.

No, your above arithmetic is indeed the naive way, but it's a step.

> In general subsets seem particular useless as a measure of size
> because often the domains of the sets are disjoint.

Right. So, the subset account does not seem satisfactory, either.
Hence, we have another paradox of the infinitely big.

> I suppose
> you could recast everything into sets of sets, but that seems
> cumbersome.

That would be the wrong approach.

> A very classic problem where the bijection definition
> of cardinality has immediate consequences is the fact the cardinality
> of the set of all Turing machines is less than the cardinality of the
> set of all functions f : N -> { 0, 1 }. Can you make this argument
> using subsets or some other definition of size? The fact that
> unsolvable problems exist shows us that in a very real way
> there are not as many TM's as functions.

Making that argument, exactly in that fashion, is not necessary. But
it is easily done using alternative definitions of size. (If it is
indeed real!)

> I have yet to see any paradoxes involving the infinitely big.

You have yet to read any serious textbook in philosophy of
mathematics.

Again. Your post is challenging the received view. But defending some
current, and changeable state of mathematics, at the expense of
assailing philosophical wisdom is not a good idea. Not a good idea at
all. With what you write above you'd get A if you took a class in
math. dept, but get a nice and permanent F in a phil. dept, if you
continue the above way of argumentation.

Philosophers don't have to see Cantor's theory as the ultimate
solution to every problem of infinity. There are complete books on the
riddles of infinity! Current set theory hardly is an answer.

To think that it does in fact give a satisfactory solution to all the
antinomies would be naive. (Much more naive than writing 2*infinity)

Regards, 

--
Eray Ozkural

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