Re: Skolem's Paradox and why is math the way it is?

From: J.E. (troubled6man_at_yahoo.com)
Date: 10/08/04 

Date: 8 Oct 2004 10:26:02 -0700

daryl@atc-nycorp.com (Daryl McCullough) wrote in message news:<cik66301l4v@drn.newsguy.com>...
> J.E. says...
> >
> >> >> Well, even if you don't assume Platonism, it's still a theorem that
> >> >> there are uncountably many real numbers. So surely that's a good
> >> >> reason to "care" about uncountably many real numbers?
> >> >
> >> > I disagree, there is a theorem, that says for any given model, and any
> >> > injection in that model, there exists another number that is in the
> >> > reals that has no preimage. It only proves ONE more number,
>
> >I've SEEN the diagnal arguement!!!!! I asked a question, I didn't
> >make a statement
>
> Yes, you did. You said you disagreed that there were uncountably
> many real numbers. The paragraph starting with "I disagree..." and
> ending with "It only proves ONE more number" was not a question,
> it was a number of statements.

The proof just demonstrates the lack of a bijection. It doesn't
demonstrate that many elements exist in the set of reals. I haven't
seen any proof of that, and my question is where people get that idea,
since it's not what the theorem says.

> >I'm asking WHY mathematicians DEFINE real numbers so broadly as to
> >include numbers that can be neither described NOR constructed.
>
> Because it is the most straight-forward way to define the real
> numbers: It is the smallest set containing the rational numbers
> which is closed, in the sense that every Cauchy sequence converges.
> That isn't true for the definable reals or the constructible
> reals. That is, if you have an infinite convergent sequence of
> definable reals
>
> r_0, r_1, ...
>
> the limit isn't necessarily definable.

You may, in reality, be correct, but your statement wasn't very
convincing (or detailed or complete). Can you prove (in ZF) that the
sequence exists? Can you prove (in ZF) that the sequence is cauchy?
If so, then can't you define the limit to be the equivalence class of
cauchy sequences that have the same limit? What is undefinable about
that? In fact if you care about completeness, I thought that that WAS
the definition of real numbers you use. Since there are only a
countable class of sequences that you can PROVE are cauchy, there is
NO proof that there are MORE real numbers required to be limits of
cauchy sequences. How do you get around that, your example didn't
show very many details.
 
> To work exclusively with constructible, or definable reals is much
> more difficult, with no actual benefit for physics or science in
> general. You say that you are motivated by physics and science, but
> I certainly don't see any motivation from *physics* to restrict our
> attention to constructible reals, or to assume that all reals are
> definable.

How is it more difficult? How can you even work with things you do
NOT define? THAT seems hard.

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