Re: Galileo's Paradox

On Wed, 29 Nov 2006 08:30:28 -0500, "Jesse F. Hughes" <jesse@xxxxxxxxxxxxx>
wrote:

Six Letters writes:

On Tue, 28 Nov 2006 15:14:22 +0000 (UTC), stephen@xxxxxxxxxx wrote:

Six wrote:
I accept that. The contradiction comes about if the one notion
suggests equality of size and the other notion suggests inequality. Which
they do, so there is a prima facie paradox.

The problem is that you are using a word 'size' that you have
not defined.

True. I took it that people knew what I meant. And I think
they do.

Pretty standard fare for someone complaining about cardinality as set
size. They assume that "size of a set" has a natural meaning and that
this natural meaning should satisfy that A c B -> A is smaller than B
(I'm using c for "subset").

I am not claiming any originality, and if everything I have said
has been said before and properly dealt with, I would be gratified to know
that.
It is not so much a case of assumption, though, as presenting a
paradox, or dilemma. For me, anyway. I am just trying to work things out in
my own mind.

But, no, people do not know what you mean.

At least some of the force of my claim was the rhetorical point
that I am not making the simple confusion about size that some people seem
to be suggesting.


We are not born with
flawless intuitions about the sizes of infinite sets.

Agreed

And so, in
trying to make precise our notions about set size, we have
extrapolated from the finite case by characterizing what counting
means. And the analysis that is most successful to date is: two sets
have the same size if there is a one-to-one correspondence between
them.

If it's a matter of comparative success or fruitfulness, I have no
complaint. I do not have the mathematics to judge.

It seems like a well-motivated definition.

Of course it does. Otherwise there would be no paradox.

Counting a finite set does
amount to assigning one number to each element of the set. But it
doesn't satisfy the subset condition and that seems to cause some
people great anguish.

Unfortunately, I have never seen any measure of set size such that the
following hold:

(1) If A c B then A is smaller than B.
(2) Every set A and B is comparable.

Cardinality satisfies (2) but not (1). There's an obvious definition
that satisfies (1) but not (2): Say that A is smaller than B iff
A c B. But that's not very interesting. Off-hand, I do not know of
*any* interesting definition of set size that satisfies (1).

Of course, (1) and (2) are mutually satisfiable by cardinality just
so long as there are no infinite sets.

Thanks. That is informative. It's clear to me that I need to learn
a lot more mathematics before I could usefully respond to this.

Much appreciated, Six Letters
.

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