Re: Galileo's Paradox

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

Six wrote:
On Mon, 27 Nov 2006 02:21:33 +0000 (UTC), stephen@xxxxxxxxxx wrote:

Six wrote:
On Fri, 24 Nov 2006 18:26:37 +0000 (UTC), stephen@xxxxxxxxxx wrote:

Six wrote:
On Fri, 24 Nov 2006 16:04:12 +0000 (UTC), stephen@xxxxxxxxxx wrote:

Six wrote:

<snip>

I want to suggest there are only two sensible ways to resolve the
paradox:

1) So- called denumerable sets may be of different size.

2) It makes no sense to compare infinite sets for size, neither to say one
is bigger than the other, nor to say one is the same size as another. The
infinite is just infinite.


My line of thought is that the 1:1C is a sacred cow. That there is
no extension from the finite case.

What do you mean by that? The one-to-one correspondence works
perfectly in the finite case. That is the entire idea behind
counting. Given any two finite sets, such as { q, x, z, r} and
{ #, %, * @ }, there exists a one-to-one correspondence between
them if and only if they have the same number of elements.
This is the idea that let humans count sheep using rocks long
before they had names for the numbers.

I love this quaint, homely picture of the origin of arithmetic. I
am sure that evolutionary arithmetic will soon be taught in universities,
if it is not already. Disregarding the anthropology, however, you have said
absolutely nothing about whether !:!C is adequate for the infinite case.

I was addressing your claim that there was "no extension from the
finite case". In the finite case, two sets have the same number
of elements if and only if there exists a one to one correspondence
between them. This very simple idea has been extended to the
infinite case.

OK. The idea of a 1:1 corresondence is indeed a simple idea. The idea of
infinity is not.

That depends on what 'idea of infinity' of you are talking about.
The mathematical definition of 'infinite' is as simple as the
idea of a 1:1 correspondence.

The mathematical definition of infinity may be simple, but is it
unproblematic? It seems to me that infinity is a sublte and difficult
concept.

What concept of infinity? Note, I said 'infinite', not 'infinity'.
You have been talking about Cantor and one-to-one correspondences,
so you have been talking about set theory. The word 'infinity'
is generally not used in set theory. It has no formal definition.
'infinite' is used to describe sets, and it has a very simple
definition.

I'm talking about mathematical meaning. Specifically I'm talking
about "How many?", more or less etc..

"How many" is not a technical term. Cardinality corresponds to our
notion of "how many" in the finite case, and that is likely what people
will think of when you ask "how many". I know that later on you complain
about the term "cardinality", but I will respond to that later.


And that we are entitled to ask how well the simple mathematical
defintion captures what we mean by it, not necessarily in all its wilder
philosphical nuances, but what we mean by it mathematically, or if you
like, proto- mathematically.

A set is infinite if there exists a bijection between the set and
a proper subset of itself. That is what mathematicians mean when
they say a set is infinite. There are other equivalent definitions.

I know already.

So what are you asking? That is the definition of 'infinite set'.
It means mathematically exactly what it says.

There is no point in dragging
philosophical baggage into a mathematical discussion.

In my opinion the philsosopy is already there, and it impoverishes
mathematics to pretend otherwise.

Do you have the same problem with prime numbers? Or even numbers?
The words 'prime' and 'even' have meanings outside of mathematics.
Do you feel obligated to drag those meanings into a discussion
of prime or even numbers?

See above

I do not see an answer to the question above.

<snip>


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.

No. I do not know what it means when applied to a set. Does
it mean "cardinality"? If so then we would not be having this discussion.
If it does not mean "cardinality", what does it mean? Can you give
me a mathematical definition of "size"?

<snip>

Noone is doing anything because 'Cantor says so'. Childish comments
like that are a sure way to make this thread degenerate.

Certainly I write things in the heat of the moment which I later
regret. But this wasn't meant as a cheap jibe. I've already conceded that
following Cantor might in some deep way be right, if it comes down to
following productive branches and forsaking dead ends.

Look at what you've written. It consists of repeating things I
already know (definitions etc.) coupled with the suggestion that I'm mixing
up different notions of size. Saying that people are confusing two
different notions of X is a classic manoeuvre of 20th century philosophy in
the moribund analytic movement, and in every case, I'd venture to say, it
sells the argument short. As if anybody that disagreed with your point of
view was a complete idiot.

You seem to be taking this all far too personally. You have not provided
a definition of 'size'. You are using a vaguely defined word, which
is always going to get you into trouble in mathematics.

There is an intuition that there are less squares (even numbers,
primes, whatever) than naturals. We are talking here precisely of
intuitions about infinite sets. It is not good enough to say: You're
getting mixed up with finite sets, or: You can't rely on common sense
intuitions in maths.

So if there are less squares than naturals, then since they have
the same cardinality, how can cardinality have anything to do with size
(how many)? Why not just say there's a bijection and forget about
cardinality.

Why not just say 'having no factors other than itself and one' instead of
'prime'? Whe not just say 'divisible by 2' instead of even? Cardinality
has a very precise definition. Yes, we could replace the word 'cardinality'
with its definition. It would not change anything.

Again, your problem is insisting that cardinality match some vague notion of 'how many'
that you have not defined. Until you come up with a precise definition of 'how many',
any questions about 'how many' elements are in a set simply cannot be answered.

You suggested I conduct my argument without using the term
'infinity'. I am quite happy to do that. I suggest you conduct the rest of
your argument without using the term 'cardinality'.

Why? Cardinality has a definition in set theory. 'infinity' does not have
a definition. Do you really think that the two words are on an equal footing?

Stephen

.

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