Re: Skolem's Paradox and why is math the way it is?
From: Shmuel (Seymour J.) Metz (spamtrap_at_library.lspace.org.invalid)
Date: 10/05/04
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Date: Tue, 05 Oct 2004 00:55:45 -0300
In <39d6e584.0409291042.75243cea@posting.google.com>, on 09/29/2004
at 11:42 AM, troubled6man@yahoo.com (J.E.) said:
>I'm saying that it does land everywhere.
OK, so you adhere to Many Worlds[1]. It's viable, but it also has
issues.
>the >wave-function splits
Please describe the apparatus that measures the splitting.
>Thank you for telling me your opinion about the ZF axioms.
Actually, I haven't. How do you know that I don't prefer GBN?
>So it is interesting to you that there is no bijection from
>the naturals to the power set of the naturals, right?
Yes.
>Are you concerned if that is bothersome to other people?
No. Are you concerned that SR[2] is bothersome to other people?
>It is bothersome to the people who are not making further axioms to
>add to ZF and instead are merely using ZF as is, because there
>doesn't seem to be an uncountable number of things described by
>ZF, given the faithful countable model by Skolem.
What do you mean by "be"? Are you taking a Platonic position that
every time you prove (Exists x)P(x), that there is a "real" object
described by that x? Platonism is not a necessary part of Mathematics.
>I want to get rid of unobservables from math because I think it
>would *help* to remove them from math first and *then* from physics.
It would make more sense to first remove unobservables from Physics.
>I was trying to say that the fact one can prove theorems with axioms
>doesn't say anything about the axioms themselves.
Why would you want to say anything about the axioms beyond stating the
theorems that can be derived from them?
>I still don't understand these other criteria,
That's the problem; Mathematics is an art form. One of the key terms
is "elegant". The criteria for judging Mathematics are more intuitive
to a musician or a sculptor than to a Physicist.
>I'm willing to consider that, but then I
>want to know why you think the power set is interesting.
Because it generalizes. Because it leads into other results. Because
the proof is so simple.
>I understand talking about the collection of all subsets that exist
>in a model of a particular set,
But you seem to be equating the model with the theory that it is a
model of. They are not the same.
>but to talk about there being more subsets when you
>haven't yet introduced any further axioms is weird.
What further axioms do you need? ZF already includes the requisite
axioms. ZF is not a constructive theory; proving (Exists x)P(x) does
not mean that we have show a construction for such an x, but only that
the statement is a theorem of the system.
>But from physics we want a "finished" mathematical theory
That's impossible, if you want it to be consistent. It will always be
incomplete.
>it's cumbersome to add another mathematical
>axiom and then go do all our physics again,
It's also unnecessary. ZF is perfectly adequate for the tasks to which
physicists have applied it to date.
>I wanted to know why uncountability is desirable,
LIttle things like the MVT.
>What is your operational definition? Show me a power set in the lab
>and say what is in it.
The lab is a notebook. What is in it are marks on a piece of paper,
forming inferences from the axioms in accordance with the rules of
inference.
>The problem is that WHICH elements are in a
>power set depend on OTHER axioms than ZF
No. You can only talk about a set being an element of another set in
the context of a specific theory.
>and you haven't CHOSEN those other axioms YET,
If I chose other axioms then it wouldn't be ZF any more. If you find
it more convenient to work with, e.g., inaccessible cardinals, that
doesn't change the theorems of ZF.
>so you just insist that there ARE more subsets than can be proven
>to exist
No. I note that it is a theorem in ZF that they exist; I don't draw
any metaphysical conclusions from that. If you want a constructive
theory, they're down the hall ---->
>since IF another axiom was made to count the previously existing ones
Are you talking about inaccessible cardinals?
>You toss around the word all in a very UNoperational way in ZF.
Not at all; you simply don't understand what the relevant operations
are. They're rules of inference.
>The THEORY is about things you can measure
That hasn't been true for a century. The theories used in Physics
involve gauge fields that are quite removed from things that we can
measure.
>If I don't PUT IN absolute phases,
What do you mean by that? Theories involving local gauge invariance
require absolute phases, but the dynamics are invariant under gauge
transformations.
>so approximate solutions are fine if they are accurate enough.
Then why do you care if the axiom systems aren't accurate in some
vague metaphysical sense if the results are accurate enough?
>For instance, I know I guy in England who is spending his career
>looking for closed timelike curves that are solutions to Einstein's
>field equations, and he can't seen to prove that they either exist
>or can't exist.
Huh? Wasn't that solved a long time ago? Or do you mean that he is
looking for solutions subject to a constraint that he is interested
in?
>It would be nice to have a decidable model
Kurt Gödel demolished that hope a long time ago. And you're confusing
model with theory. Models aren't decidable or undecidable; only
theories are.
>If a small enough model now would avoid things like that being ambiguis,
A small *theory* would leave more statements undecidable than a larger
theory.
>but some results are independant of teh ZF axioms,
What are you trying to say? The results are inferences from the
axioms; they can't be independent of them. The best that you can say
is that certain results exist in multiple theories. The results that
appear to be relevant to Physics exist in both ZF and GBN.
>I don't want an infinite regress based on a mathemtaically undecidable
>proposition. It may be that no matter which axioms I choose that will
>happen anyway, but I can't know unless I try the axioms out.
You know that the theory will be incomplete. You can't tell whether
that incompleteness is relevant to a physical theory until you have
the physical theory. And if it is relevant, the solution is to add
axioms, not to take them away.
>You describe your inial conditions as a wavefunction, and your
>observations as sections of a wavefunction.
What do you mean by "sections of a wavefunction"? And is isn't a wave
function rather unphysical?
>Huh? You do your calculations, no one walks in and forces you to
>pretend things commute when they don't.
The fact that they don't commute has some observable properties. In
fact, some physicists start with the commutation relations.
>I'm not an idiot,
Perhaps, but when you write things like "Who cares about
commutations?" and "You need an infinite space because you have an
infinite number of potential particles" then it's not obvious,
especially when you contradict yourself.
>What experiment are you performing,
I'm measuring the position and momentum of an electron.
>what is the tolerances and the precisions.
Arbitrary. Except that I can't, BECAUSE OF THOSE COMMUTATION
RELATIONS.
>Why are you computing something like that?
Maybe I want to know the physical limits on what I can measure. There
are observable consequences.
>The results you need/want are the shape (location) of the wavefunction.
How do I measure the wave funtion?
>I'm REALLY upset with people pretending that there is a complete
>basis all the time.
It may be a crooked game, but it's the only game in town. QFT is a
mess, but we don't yet know how to clean it up, and meanwhile we get
some very accurate predictions.
>If we only wanted to calculate the ones we make in
>the lab, then it's OK, but then we don't need an infinite model,
>like you claim.
You need an infinite state space even for a single particle system.
What do you think the wave functions are? There is no finite basis for
them.
>Especially because you don't like models,
There you go again. If I tell you that I don't want raspberries in my
chili, that doesn't mean that I dislike raspberries. It just means
that they don't belong in chili, possibly because it's a waste of
perfectly fine raspberries. What I don't like is confusing models with
things that aren't models.
>I understand that isomorphism, you are the one claiming we have to
>use ONE particular model with an infinite algebra even though it
>makes distinctions between events and states that we can't
>distinguish between.
No, I am the one that realizes that we've known for decades that the
commutation relations can not be satisfied with a finite dimensional
representation. If you're not aware of it, it's not my job to remedy
the lack. That has nothing to do with the fact that we use
approximations in our calculations.
[1] Sort of.
[2] For that matter, Newtonian Mechanics is bothersome to some
people, who believe in an Aristotelian world.
-- Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel> Unsolicited bulk E-mail subject to legal action. I reserve the right to publicly post or ridicule any abusive E-mail. Reply to domain Patriot dot net user shmuel+news to contact me. Do not reply to spamtrap@library.lspace.org
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