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

From: J.E. (troubled6man_at_yahoo.com)
Date: 10/29/04 

Date: 29 Oct 2004 14:18:36 -0700

tchow@lsa.umich.edu wrote in message news:<41825198$0$577$b45e6eb0@senator-bedfellow.mit.edu>...
> In article <41811eb8$0$576$b45e6eb0@senator-bedfellow.mit.edu>, I wrote:
[snip]
> As I understand your view, sets are mysterious, amorphous things. Only when
> I write down a syntactic formula that defines a set does it leap into focus.
> And to have a good physical theory, all the sets we might want to refer to
> must be in sharp focus. You're worried that there might be some shadowy,
> unfocused sets that physics really needs but that ZFC can't focus. (I keep
> wondering how you manage to talk fearfully about those shadowy sets if your
> language is allegedly incapable of referring to them, but let's gloss over
> that point since you don't seem to understand what I'm driving at.)

C={xe{0,1}: (CH and x=1) or (~CH and x=0)} already seems pretty
shadowy to me. In ZF "proof is everything" to most mathematicians,
and given an a number n I can't tell if neC or not. In a model I
could tell you, but every model of set theory is wrong in that things
that should be true "in all models" are not true.

> Thus, you want to write down a bunch of formulas. Each one of these is
> supposed to define a set. And now you're worried because
>
> 1. the existence of different models seems to mean that even a syntactic
> formula can't pin down a set properly; and

Sorry, I didn't realize you
 
> 2. there are only countably many formulas and there are uncountably many
> sets, so we must be missing some.

Are there really countably many formulas? What about the arguement?

B={S(x): S(x) is a one place predicate}

If B is countable, then let f(n) be an onto function from N to B, and
let fn(x) be true iff (S(x) and f(n)=S).

Then consider any predicate Q(n), An (neN)=>(Q(n) <=> ~fn(n))

Which is very very strange, the only result I could figure is the
claim "EB (B={S(x): S(x) is a one place predicate}) and (B is
countable)" was false.

How about R={xeN:Q(n)} does Re(2^N)? Separation doesn't tell us that
it does (or does it?). But powers wants to say it's there (under
those weird conditions some mathematicians take "as real" that start
from making premises that are later proven contradictory), in fact it
the counterexample to the claim 2^N = {{xeN: S(x)}: All (countable)
predicates}, so the standard interpretation wants to say R was always
there.

> These worries make you feel that ZFC is weak and incomplete in some sense,
> and since there's some theorem of Goedel that goes by the name of the
> "incompleteness theorem," you assume that this theorem must have something
> to do with your vague feeling that ZFC is somehow incomplete.

Isn't it the (standard) interpretation of ZFC that is wrong? Godel's
incompleteness is about true results that can't be proved with fixed
formal methods. Hintikka has true results that are false in every
model of set theory. I don't quite grasp Hintikkas result. But with
a Godel numbering of predicates and a truth predicate in the language,
it's not hard to imagine that set theory is just plain lacking the
"sets" that the standard interpretations claims are there.
 
> O.K., so now for my assessment. The key difficulty is the notion of a
> "definable set." Naively, it seems that there's no problem. I have a
> formula, and it defines a set---the set of all things that satisfy the
> formula. Right? "x is a real" and "x is a natural" define the set of
> reals and the set of naturals respectively, right? And the axiom of
> separation has something to do with this, right? So why can't we say:
>
> Definition. A set X is "definable" if X = {x : P(x)} for some formula P(x)?
[snip]
> The real problem is Berry's paradox. If you're not careful, you'll find
> yourself arguing that there are only countably many definable sets, and
> so there must be a subset of integers that is not definable. But now
> we've just defined an undefinable subset of integers. How did we manage
> to do the impossible?

I'd hardly call invoking some version of choice (from outside the
theory even) as "defining" a set. So where is the alleged paradox?
If you considered M={{xeN: Q(x)}: All FOL predicates Q}, then all you
prove about it, is that if M were a set (which it isn't) then it's not
equal to 2^N. But all we are *really* proving is that M isn't a set,
not that there is something else in 2^N. But M *should* be a set
*and* there should be more sets in 2^N, specifically, (M e 2^N) should
be true. That's what the standard interpretation is all about,
"having *every* set" instead of "having the sets exposing your
ignorance about having every set being conviently left out" (which is
the non-standard interpretation). You want AN [AS ((seS)=>(seN)) <=>
(Se(2^N))] to be true, but with only one relation (e) then the
assumption that S is a set is buried inside, so we can't tell if
Me(2^N) or not. We know that meM=>meN, but we don't know if the
sentance "EX X=M" is true, because the definition of M is "outside ZF"
(because it talks about all predicates).

And that's probably what you are trying to say about language and
definability is about predicates, that if you could talk about a
completed infinity of all predicates, then there'd be another
predicate you could define from the set of predicates. But why are we
using weak languages with that property? That's something I don't
see. FOL has that property, and I think IF-logic does too, we need a
stronger class of predicates, something that can really handle
infinitary combinatorics.

We've got a weird enough situation anyway, like with, AS
(Se[2^N])=>(EA A={xeN: xeS}), it's clearly a theorem, (A=S works), but
"these" (xeS) are those "uncountable" predicates, they are impredicate
predicates slipped in in disguise. It's possible that that was not
real a statement of ZF and someone will troll out another caveat in
the definition of the separation schema. But it should be true anyway
(since A=S satisfies it). I can see the very very strong (and
appealing) slip to just say that there are more subsets out there, but
it's just a bit circular to me. You could reason about the "any
subset", but then you are begging the question about what was an what
wasn't a subset. It's a bit hard to take seriously that FO set theory
has "all sets" when it doesn't even contain the skolem functions of
true statements, and skolem functions are very very special functions,
it wouldn't have been hard to have them all.

> Definability in a language (such as the first-order language of set theory)
> cannot be (consistently) expressed in the language itself.
[snip]
> Moving to some other logic or axiom set isn't going to get around this
> fundamental problem.

I've never seen a theorem as strong as the one you stated, about
definability. "All languages" is a big claim. IF-logic has it's own
truth predicate, lot's of people want to say that that is impossible.
But science isn't a democracy, so saying so doesn't make it so. So
what's the basis for your claim about definability? Isn't there some
problem about "defining definability" in the language you use to prove
your result *before* proving that the thing you defined (definability)
doesn't exist? It seems like the theorem would apply to itself, and
claim that such a theorem can be stated, so how is it proved? Maybe I
just missed something.

> But more to the point, it's not really a problem for *physics*. There's
> no reason to judge the adequacy of ZFC according to the criterion of
> whether it can define all the sets needed for physics. As I've said
> before, you should judge it based on whether it successfully mimics all
> statements and deductions that you want to make.

I can't tell what ZFC is "mimicing" because I can't tell what is is
really "doing" itself. That makes it pretty useless as a "surface
large enough to contain a square that looks like a mirror of the
universe". I want a theory where if two physicists make the same
predictions from the same model. To do that they have to know what
they are doing well enough to copy each other. Mathematiciams can
shrug off "non standard models" all they want, but we don't want to
reject potentially good physical models because someone somewhere
makes a (to the universe) "non standard" interpretation and then
falsifies the *interpretation* and we think he falsified the *model*.
That is catestrophically bad bad bad.

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