Re: Skolem's Paradox and why is math the way it is?
From: Ross A. Finlayson (raf_at_tiki-lounge.com)
Date: 09/23/04
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Date: Thu, 23 Sep 2004 01:47:13 -0700
"J.E." wrote:
> > > Why do we care about "uncountably many" real numbers when in reality
> > > there are only a countable number that we can "prove theorems about"?
> >
> > Actually, that's not true. Although there can be only a countable
> > number of theorems, they can still address an uncountable number of
> > reals. As a simple example, there are an uncountable number of reals
> > in the Cantor set, and a simple procedure by which to determine given
> > any real if it is a member (namely, can its decimal expansion, when
> > convereted to base 3, be written without using the digit 1).
> >
> > Jonathan Hoyle
>
> Thank you for your polite response. I do think that I need to ask my
> question more clearly. There are many sets, we agree on that. Some
> sets have bijections with the set of naturals, others don't, we agree
> on that.
I don't. You can apply a canonical ordering operator onto any set that is
not ordering-sensitive.
> ... But each time you use an axiom to prove a theorem, the axiom
> produces, at most, a finite number of sets, and you can use each axiom
> only a finite number of times in a single proof. Doesn't it seem that
> each proof generates only a finite number of sets, even though some of
> these sets might be considered "large" themselves? And if the number
> of theorems is "somewhat countable", then doesn't it seem like there
> is only a "somewhat countable" number of sets that we can prove
> theorems about?
>
I eschew axioms, but if you call infinity an axiom then it generates
infinitely many sets and all the proofs about those sets.
>
> You, for instance, cited a theorem about ONE set (the cantor set). So
> there is at least one set in the universe. But you want be to believe
> that you just proved a theorem about many sets. However, all you
> proved is that there is a certain subset of another set, namely the
> cantor set is a subset of the reals. That is a theorem about ONE set.
> How am I supposed to tell if there are as MANY sets in the universe
> as you seem to want me to believe there are? Thank you again.
Easily, put them in a line.
Are second order logic and "countable" models of the "uncountable" not
slippery slopes? (They are.)
It's a theory about one set: all of them.
Ross
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