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

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

Date: 10 Oct 2004 12:27:59 -0700

Neil W Rickert <rickert+nn@cs.niu.edu> wrote in message news:<ciut6u$sl3$1@usenet.cso.niu.edu>...
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> troubled6man@yahoo.com (J.E.) writes:
>
> >> >WHY can't I use a countable model while doing physics?
>
> >> Why do you need to use any model at all?
>
> >To me we you language, and the axioms of set theory I introduce as
> >definitions of my terms.
>
> But isn't that where your mistaken thinking begins?
>
> Surely you should be starting with terms that come from physics.
> - From that, one can invent the mathematics needed to deal with those
> terms.
>
> The standard mathematical conception of the real numbers really
> arises as an idealization of the measurements used by physicists.
> The axioms of set theory are a late comer to the scene.
> Mathematicians were using real numbers long before they studied set
> theory.
>
> For the mathematician, the role of the axioms is clear. It provides
> a consistent logical framework for discussing mathematical objects,
> and proving theorems about them. There relevance to physics is less
> obvious.
>
> Mathematical foundations gives the impression that the real numbers
> are constructed from set theory. But this is mostly a clever
> pretense. Mathematicians went to a lot of trouble to make sure that
> what they could appear to construct from set theory, was really the
> old familiar real numbers that arose out of the needs of physics.
>
> If you jump to a non-standard model (such as countable model of set
> theory) you lose all of that. Maybe you will feel more comfortable
> with the countable model. But you will be dealing with a model whose
> real numbers have at most a dubious connection with physics.
>
> Take an example. Suppose that you settle on the constructable
> reals. Then you know that every mathematical operation you could do
> would finish up with a constructable number.
>
> But now let's suppose that a physical event is observed. You want to
> say that the event occurred at time t. Is reality such that the time
> t is constrained to be a constructable real? If so, how would you
> prove that?
>
> Or are you going to say that if the time t is not constructible, then
> the event didn't actually occur, since t is not in your model of the
> reals?
>
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In reality the accuracy of our time measurements are limited by the
energy available to us to do experiments. Rationals are good enough
for time measurements in the lab. I agree that ZF(C) is good enough
for experimentalists that just take measurements and use the existing
theories that are already made in ZF(C). It is theorists who must
make better theories that predict more phenomina where we need a
powerful math engine that contains enough stuff and not too much
stuff. Because a real world symmetry can use infinite precision
cancellation to generate real world effects, and so theory needs more
than the finite precision needed in the lab to take measurements. We
need a mathematical language that allows the appropriate real world
symmetries to exist and to have the right operations. Do you
understand anything I am saying? ZFC appears to be "good" through an
eloborate social illusion that it contains ALL subsets, when there is
no truth behind the claims that it does, as the countable model
EXPOSES to mathematical observation.

J.E.

Relevant Pages

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