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

From: KRamsay (kramsay_at_aol.com)
Date: 11/09/04 

Date: 09 Nov 2004 10:12:31 GMT

In article <39d6e584.0411041450.1066c969@posting.google.com>,
troubled6man@yahoo.com (J.E.) writes:
[...]
|It's like DnD,
|where if you have armor-making tools, and armor-making skill, then you
|can make armor, but heaven forbid the player askes who makes the
|armor-making tools. Someone could have armor-making-tools-making
|tools and armor-making-tools-making skill, and then make armor-making
|tools, but HE needs a guy with the skill and tools to make his tools.

It's like Douglas Adams wrote, the secret is you HIT THE ROCKS
TOGETHER. Everything boostraps its way up from something very
primitive that doesn't require special skills or tools.

|I thought this was silly when I heard the rules, but set theory seems
|to be the same way, the standard interpretation is to have every
|exestentionally possible set, but the tools are always inadequate to
|get them all, no matter what tools you have.

I think you should distinguish carefully between "getting" objects
in the sense that you can define them individually, and "having"
them in the sense of being able to talk about their properties.
Nobody thinks that they can "get" each real number by being able
to define it. But that doesn't mean that we can't talk about the
properties that they have. We can say "real number" and have it
mean "real number", even if we can't define each real number
individually. All we need is a definition of "real number".

[...]
|> The only way to keep it from being circular is to base your acceptance
|> of the first domain of sets on something not requiring any set theory.
|> Now you can go either down the realist road, treating them like any other
|> entities of which we have theories, or down the nonrealist road, where
|> you explain them away as merely hypothetical or fictional. But you can't
|> start with a model that depends for its definition on an already existing
|> set theoretical domain.
|
|All other axiom systems in mathematics are about proving a theorem of
|an axiom system so that it's a truth in all models.

No, this is a misconception. This is one of the biggest confusions
that all this talk about "models" gets people into. When people
axiomatized arithmetic, it was not because they thought "models of
my axiom system" were interesting. It was because they thought
arithmetic was interesting.

[...]
|Either ZFC is BAD and we need new axioms, or ZFC is FINE (for physics)
|in which case a countable model is OK (for physics). I think it is my
|realist opponents that want it both ways, not me.

I have a somewhat different point of view from some of your other
critics.

Your inference from "ZFC is fine for physics" to "a countable model
is okay for physics" is unclear, and I think it's faulty, but for
different reasons than some.

What would it mean to say that you are "using" a countable model
for physics? It would mean adding substantial complexity to your
theory of physics, with no predictable consequences. Oh, I agree,
you could do it in such a way as also not to ruin any predictions
either (assuming that the initial conditions are in the model,
which is not necessarily the case, but we can pretend is so for
the sake of inferring predictions).

The countability of the model provides you with no benefits, unless
you step outside of it, and then you're not simply "using" it; you're
using it as it lives in a larger set theoretical universe. Why bother?

Keith Ramsay

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