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

From: J.E. (troubled6man_at_yahoo.com)
Date: 09/26/04 

Date: 26 Sep 2004 06:39:05 -0700

> >http://www.mrao.cam.ac.uk/~clifford/publications/abstracts/astrop.html
> >and finally, in this paper they show how this theory is EASIER than
> >manifold theory because one has the FREEDOM to choose a gauge that
> >simplifies the equations.
>
> Gauge? Sounds like you're still using manifolds.

I'd have to know your definition of manifold. I'm used to a manifold
being a set that is an ordered triple, this set has a vector space.
The difference that I note is that Penrose proved some criteria for a
manifold having a singularity, and the cosmological models in the
manifold theory of gravitation met the criteria and people mentioned
explanding spacetime and a big bang. The cambridge group doesn't have
the entire universe compress down to an infinite-
density in a finite amount of time in the past, so I assumed that the
theory didn't meet Penrose's definition of a manifold. But I could be
wrong about that, because the logical possibility remains that the
Cambridge group uses a "different" cosmological model, and that's why
Penrose's proof doesn't apply. I would be happy to not call the
manifold theory by that name, I worry that "classical gravitation" is
a bad name since people might think I'm talking about Newtonian
gravity when I'm not. I can call it General Relativity. I worry that
that gives too much credit to the Cambridge group because I think
their model is just an implementation of Einstein's principles, but if
it makes you happy I can use those words.

> >I agree that getting rid of unobservable phases is difficult, but
> >one can have them and realize that a LOCAL change is just a change
> >in an arbitray gauge (EM from QFT) and a GLOBAL change is completely
> >physically meaningless
>
> But still in your theory.

I don't get your point. The physical model can talk about relative
phases, and if I use absolute phases because the math is easier,
that's fine as long as I don't interpret that absolute phase as
meaning something, since it wasn't in the physical model.

> >But YOU brought this up and FAILED to explain to ME
>
> It is enough that I explained it. I can't explain anything to a man
> that doesn't wish to look at the facts. The facts are that Physicists
> do exactly what you are complaining of in Mathematics; their theories
> have unobservable entities.

I'm not trying to do mathematics. I've said that all along. What I'm
doing is asking mathematicians about what the axioms THEY use mean to
THEM so that I can decide if I was to use them in my PHYSICAL models.
I have to make a model, so I'm interested in how it will or should
compare to the models that mathematicians make. If you say that it
isn't MATHEMATICIANS that make models of ZF, then tell me who does and
I'll ask them instead. Should I take this to some other usenet group?
Philosophy? Logic? Is there a model theory section? I came here
because I thought it was the right place, but if somehow people think
I'm trying to not look at facts and this misunderstanding is based on
me being in the wrong place, I'm happy to move.

> >WHY can't I use a countable model while doing physics?
>
> Feel free; either the Universe will cooperate with you or it won't.
> The methods currently used by Physicists are based on the Real Line,
> and you have not presented a viable theory using an alternative. If
> you are asking why the World is as it is, that is not a question for
> Mathematics.

I'm not asking about the world. Other people brought up various
physical models. I wanted a small model to avoid making mistakes
about things that I don't know anything about. I thought that was
reasonable. I didn't want to make my model too small, so I was asking
about how you tell which models are good, and which axioms explain the
various things people talk about. I thought this was a question that
mathematicians would know something about. I agree that it isn't a
theorem about ZF, it's about HOW people choose axioms and about which
axioms do what and what do they not do. I thought this was related at
least to the PRACTISE of mathematics, if not the SUBJECT of
mathematics.

> >It doesn't change any theorems about anything we can actually talk
> >about without introducing NEW axioms.
>
> Of course it does; the fact that you don't like the talk doesn't mean
> that it doesn't exist. You're in the position of a tune-deaf man
> demanding that the local orchestra stop using stringed instruments.

I didn't try to say it didn't exist because I didn't like it. I
thought I had a basis for saying it didn't change anything. If I'm
wrong I'd be happy for you to explain how I was wrong. You could do
that by explaining how it does matter, or explaining how/where I made
an error. I don't care if mathematicians continue to use ZF axioms or
some other axioms system to do mathematics. That's their business. I
just wanted to decide which axioms to use to teach physics.
 
> >In fact, the NON issue about it is *why* I wonder why mathematicians
> >like to pretend
>
> I'm nopt a psychiatrist and can't explain your delusions. The fact
> that you "wonder why" about something that is in fact false is beyond
> the realm of Mathematics.

I'm trying to ask why mathematicians like the axioms they do. Were
you trying to say that a psychiatrist might know that? I have some
books on the physcology of mathematical discovery and I got the
impression that most psychiatrists didn't care, they were interested
in drugs and such. I understand that there is a theorem about a like
of a bijection in ZF, but I was under the impression that that was
different that actually exhibiting more elements. I used the
countable model as an example where there is a lack of a bijection,
but that there were the same number of elements. I agree that with
the definition of "number" IN the model, that there were different
"numbers" of reals and naturals. But all I was trying to do was
distinguish between the idea of lacking a bijection and having
different elements. I thought the countbale model made that
distinction clear. Did I misunderstand the countable model? Did I
make an error already? I'm trying to ask if there is some OTHER
reason (other than the diagonal lemma) that people like big models. I
was under the impression that all the models were faithful and so I
didn't understand the preference. I thought this prefence might be
based on a penchant for pretending that there are many real numbers,
and that's why I asked? I based that hypothesis on how people were
talking here on sci.math. But you are saying that the hypothesis is
based on a false fact. That'd be great if you explained how it was
based on a false fact, because then I could conclude since
mathematicians are sensible people, that THAT was not the reason they
didn't like the countable model. I still woulnd't know why, but I'd
have one reason ruled out at least.

> >The diagnonalization proof says that given a SET that is a bijection
> >from the naturals to the reals, the set is NOT onto. It does NOT
> >say that given a "logical correspondance" from the naturals to the
> >reals, that the correspondance is not invertable.
>
> What do you mean by a "logical correspondence"? If it's not a term in
> ZF and you can't define it in terms of ZF then it has noting to do
> with ZF.

I meant the statement I provided about the bijection in the base
model. It's a set in the base model, but there is no corresponding
set in the model.

> >Skolem says that in ZF one can take the naturals and use them a
> >the domain of a model of set theory, that is a FAITHFUL model of
> >the ZF axioms.
>
> Which is not what you attributed to him.

Then I apologize, because that was what I was TRYING to say all along.

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