Re: Skolem's Paradox and why is math the way it is?
From: J.E. (troubled6man_at_yahoo.com)
Date: 09/19/04
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Date: 19 Sep 2004 10:14:34 -0700
> No, I haven't studied IF-friendly logic. If we used pure logic there
> wouldn't be any structures to talk about.
You start with a language and then define a structure as anything that
is shared in common by all consistent interpretations of the language.
Isn't that a structure?
> > > If you think that moving to second-order logic would solve this
> > > problem, why not just say that our intended model is the unique model
> > > up to isomorphism of such-and-such second-order axioms, and work with
> > > the first-order axioms.
> >
> > I don't think that, I wanted to know if someone else knew if it did.
> > I don't know why the mathematicians at my school don't care about
> > this. It seems like a serious problem to assume the existance of
> > things that can't be seen or have theorems proven about them. Almost
> > religious. So different than the mathematical bravado.
>
> They can have theorems proved about them. You have to start somewhere,
> you have to have some axioms.
I agree that you can't get something for nothing, but WHAT is this
intended model? How can you show that there are REALLY uncountably
many reals in YOUR model? You can only describe countably many, and
the closest I've seen anyone come to say that there are more is to say
that they lack a bijection in THEIR model, which doesn't distinguish
for me the difference between the class of bijections being small and
the range of the hypothetical missing bijection being large. How do I
tell the difference? What are the second-order axioms that allegedly
uniquely define up to isomorphism an uncountable number of real
numbers? If I don't have these second order axioms, how am I supposed
to use them?
> > > > How do we know which model is the "intended" interpretation of set
> > > > theory?
> > >
> > > We can describe it (fragments of it, anyway) by second-order
> > > statements.
> >
> > Only fragments, so why pretend the rest is real if you believe it can
> > NEVER be described.
> >
>
> Well, we can describe the set of all sets of rank less than the first
> inaccessible cardinal by a second-order statement. Lots of people
> would be happy to live there. And we can describe larger fragments of
> the universe, too.
How do you have rules to demonstrate the existance of more numbers
than can be put on a list?
> > > > Why do we care about "uncountably many" real numbers when in reality
> > > > there are only a countable number that we can "prove theorems about"?
> > >
> > > I don't think that's true. We prove theorems about the real numbers,
> > > and there are uncountably many of them. So we prove theorems about
> > > uncountably many real numbers.
> >
> > No, we do NOT because every theorem can be interpreted in the
> > countable model, where there are NOT uncountably many singletons (in
> > the class of all sets). Why isn't the countable model the "intended"
> > model, and why can't we fix the axioms so that this illusion of more
> > members goes away?
>
> The countable model isn't the intended model because there are some
> collections of its elements of rank smaller than some fixed ordinal
> which don't get counted as sets. Why would you want to fix the axioms?
> There's nothing wrong with them.
I was interested in fixing the axioms because I don't see the point of
numbers that we can't talk about. If we had a bijection that said,
these are the sets and the other axioms don't apply to this particular
bijection, then we'd know what the model was confined too, no
mysterious other elements whose properties are based on what order I
introduced the axioms. From INSIDE the countable model it looks the
same as your model, how can you tell from the inside that it is
"defective", and what makes it defective in your opinion.
> > > Well, even if you don't assume Platonism, it's still a theorem that
> > > there are uncountably many real numbers. So surely that's a good
> > > reason to "care" about uncountably many real numbers?
> >
> > I disagree, there is a theorem, that says for any given model, and any
> > injection in that model, there exists another number that is in the
> > reals that has no preimage. It only proves ONE more number, and since
> > the set of all reals in ONE model is actually countable (from the
> > outside) then it seems that the "uncountably many reals" are ghosts in
> > the wind. Seems like fancy talk to say that the set of bijections
> > between the reals is incomplete because our axioms made it so. Why
> > not fix the axioms?
>
> It's not incomplete because the axioms made it so, it's incomplete
> because you chose a model that isn't the intended model. Since you say
> the model is countable, you must acknowledge there are bijections not
> in the model. Why not work with a model that has all the bijections?
> As I say, there's nothing wrong with the axioms so we don't need to
> fix them.
It seems like you made it incomplete to me because you removed took
the natural numbers, reinterpreted them to represent a class of sets
and then PRETNED that the onto function you just performed can't be
done so that you can PRETEND other reals exist that you can't prove
specific theorems about, so that you can PRETEND that some of the
numbers represent infinte sets that are "bigger" than others. Why
create the illusion of more numbers than you can describe anyway? It
is your alleged "intended" model that threw away the bijection from
the class of marks on paper to the class of sets we can describe with
finite marks on paper. Seems silly so that you can pretend your class
contains more numbers, when these numbers can't be described by you
anyway.
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