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

From: J.E. (troubled6man_at_yahoo.com)
Date: 11/05/04 

Date: 5 Nov 2004 09:09:25 -0800

kramsay@aol.com (KRamsay) wrote in message news:<20041104185237.21658.00000013@mb-m14.aol.com>...
> In article <39d6e584.0410301037.7c9b415@posting.google.com>,
> troubled6man@yahoo.com (J.E.) writes:
> |I have taken classes on set theory, my professors lied to me. I own
> |set theory books, and they use circular logic.
>
> You know, I've seen you make this claim of "circularity" a number
> of times, and I don't remember you ever saying that any specific
> thing is circular, giving the cycle of dependency you think there is.
> I think you're mistakenly perceiving a dependency in their presentation
> that isn't there. A lot of the things you say come off to me as if you
> are trying to put the cart before the horse, making the meaning of a
> concept depend on the axioms used to explore it and so on. I think
> it would make things much clearer if you expressed this kind of
> complaint specifically.

I wasn't trying to critize all teachers or all books, just the ones
I've had. Someone here suggested "Set theory, Logic, and thier
limitations", but that books assumes set theory (and induction) in the
metalanguage to define set theory. I think it's just that the
teachers I had before weren't clear, it seems like something has to
come first. We can assume induction in the langauge and then latter
show there there is an induction INSIDE the theory as well, so that we
don't have to use induction outside the theory, but that's very very
different than proving induction without proving induction. That's
proving induction in a theory using induction outside the theory. Set
theory can't do it's own model theory and have every existentionally
possible set be in the model, but that doesn't mean there isn't a
strong theory that *can* do it's own model theory that has set theory
(and hence everything based on it) as a component. That's what I'm
looking for now, and I think the excluded middle is the only thing in
the way really. There is a subsection of the universe where the
excluded middle holds, and that's what we call set theory, but it's
intended models (if it has any) live outside that subsection.

Of course, no one really wants an entirely new system, so I'll look
for a new system (theory A) without excluded middle from within set
theory, and then reconstruct the full set theory within that. That
should show Theory A <--equiconsistent--> Set Theory, which should
make everyone happy, but then we could take theory A as the real
theory, and it will have models of itself inside of itself that aren't
incomplete. I'm pretty sure the models will be countable (but not
ZF-countable, for ZF's made inside the model), and in fact that there
will be a universe and that there will be invertible functions from
every infinite set (but not ZF-invertible ZF-functions between every
ZF-infinite ZF-set), so it seems a lot like Ross Finlayson's system
(which I don't really know) except it has axioms and no excluded
middle, whereas Ross Finlayson's system (if I understoof correctly)
has no axiom and an excluded middle.

If the system can describe it's own satisfaction (or truth) criteria
and is equiconsistent with ZF set theory, then that's good enough for
me. Then there isn't infinite regress about satisfaction or truth and
yet there are still models of the axioms.

> And now you claim someone lied to you, without saying what they
> said that you can somehow tell they knew was incorrect. Again,
> what specific claim was a lie? I consider it very rude indeed to go
> around accusing people of lying without backing it up, even if you're
> not accusing them by name.

The math classes I took assumed that for every sentance T(x) such that
for every set X such that for all x (x in X) => ((T(x) is true ) or
(T(x) is false)) there exists a set Y such that for all y (y in Y) <=>
((x in X) and T(x)), where set and sentance were both undefined", then
the standard interpretation of set and sentance were "assumed" outside
the theory, but maybe the set part is fine, but they should have been
more honest about what was a valid sentance, some professors actually
wrote "the sky is blue" as an example sentance, with the proviso that
it is true. But this sneaks a truth predicate into set theory that
isn't supposed to be there, and I know they were smart enough to know
better, so I'm left to conclude that they did it on purpose. And I
shouldn't have to wait for weeks to get a book that defines formulas
without assuming set theory first, it's a bit sad that so many people
do this in a non-rigorous way.

Then there is the whole colloquialness of truth, theorem, theory,
model, proof, that people use. I don't think they were trying to be
dishonest there, but it's very very very difficult for students to
learn when people are using the words different ways.

> Mathematicians seem generally, even the ones who are not formalists,
> to treat the job of deducing consequences from axioms as playing a
> special role in doing mathematics. It is supposed to be what we can
> all agree on. I certainly hope that there is no circularity in your
> set theory books in that part! Your set theory books should contain
> many theorems that follow definitely from one of the usual sets of
> axioms for set theory.

If the axioms aren't described clearly enough, it's not much an
exercise in anything.
 
> There are always other parts of the story, however, that are less
> formal. Platonists and most constructivists have a kind of set theory
> that they consider to mean something more than just abstract deductions
> from given axioms-- we consider the statements about sets to have
> actual meaning. We will often have a discussion of what kind of thing
> we are talking about when we talk about sets, and a more or less lengthy
> process of investigating them informally before we state some of the
> most key claims about them as axioms. Then there follows essentially
> the same process of deduction of consequences from the axioms as
> formalists have.
>
> A theorem can be considered two ways. There is a more "formal" side
> (which may be done informally, however) where we are only interested
> in whether the theorem follows from the axioms and deductive principles.
> Then there may also be an intended interpretation, where we say that
> the theorem is not only a consequence of the axioms, but true in some
> way. To a formalist, imagining that one has this kind of interpretation
> probably seems like a kind of illusion. To Platonists, most constructivists
> and some others, the fact that this intended meaning holds true for
> theorems proven from the axioms rests upon the fact that the axioms are
> true (in whatever sense is intended). Since the truth of the axioms can't
> be established formally, it's inevitably an informal process by which we
> argue that they are correct.

Lost you on the definitions again. Is a theorem a truth of all models
or a provable statement of a language (assuming a fixed standard of
proof)? If the latter, then what does it MEAN to be "interested in
whether a theorem follows from the axioms", since they all do?

> Informally established Selection of
> Intuition--------------> facts about mathematical---->axioms that are
> objects regarded as true
> |
> v
> Theorems considered
> true based on formal
> proofs from intuitively
> justified axioms
> ^
> |
> Axioms considered as ----------------------------> Deductive consequences
> arbitrary givens (formal theorems)
> (no justification needed)
>
> Since you don't say what specific circularity you see, there's no way
> of knowing, but it appears as though you think you see a dependency
> that I would describe as going from something in the formal side back
> to the informal side.

The dependancy was in defining the axioms. What I think you call
formal (what I'm used to called pure,a s opposed to applied)
mathematics is about propositional relations, like x is a y, where you
don't say (or know) what x is or y is or even "is a" is or means, and
the statement "x is a y" is obviously netihre true or false, it's
meaningless. But what you do is assume that certain relations BETWEEN
propositional relations hold, like "for all x, for all y, (x is a y)
or (y is a x)", then you can consider what other propositional
relations must ALSO hold that hold INDEPENDANT of any meaning ascribed
to x, y, or "is a". Then later if a model exists, that means someone
can make an interpretation where the x's, y's, and "is a"
propositional relations are interpreted to be mean something, and the
model is faithful is the axioms (as propositional relations) hold true
in the model (as meaningful statements), and a theorem of the axiom
system is a statement in the language of the thoery that is true in
all faithful models of the axioms. That's how it works for group
theory, field theory, geometry, etc. But current set theory seems to
be different. And it's totally possible that I missed some math 101
conversation that was supposed to define everything differently so
that everything in math makes sense and everything I said is nonsense.
 But I'm a bit frustrated that people tell me to go get a book or go
take a class, as if I haven't done that before.

> A formalist considers everything above the bottom line to be just a
> kind of rhetorical flourish. A Platonist will tend to regard the formal
> side as being just another technique for refining informal reasoning.
> Not many mathematicians are very much interested in either refining
> our explanation of what the undefined terms like "set" mean, or justifying
> the truth of the axioms in those terms, however. Whether a given
> mathematician believes the axiom of choice tends to be treated as a
> matter of personal belief.

That's VERY annoying. I took a class in functional analysis where the
professor actually changed whether the axiom of choice was true
halfway through the semester, I basically had to go redo everything.
Hintikka gives a justification about why mathematicians like the axiom
of choice because it translates the standard interpretation second
order formulas into equivalent first order formulas, but then he shows
that that doesn't work in general. I think that's because he was
holding onto an "excluded middle" for atomic sentances when he didn't
need to, but that's his problem, there is no reason I can't assume no
excluded middle.

> It's tempting to think that the informal reasoning used to try to justify
> the axioms should be replaced with something better, perhaps by formalizing
> it. But doing that would amount to developing a kind of proto-set-theory
> underneath set theory. It would shift the boundary between where the
> informal theory ends and the formal theory starts. There still would be
> an informal development underlying it.
>
> Keith Ramsay

If the proto-set-theory is equiconsistent with regular set theory and
can do it's own truth predicate and model theory, then I'm all for it.
 Maybe we can even get all the A=>Con(A) things from an
equiconsistency arguement switching back and forth between the two. If
A => Con(B) and B=>A and A=>B and B => Con(A) and so on, then we might
have a strong strong system

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