🚀 go-pugleaf

On Mar 14, 9:34 am, James Burns <burns...@osu.edu> wrote:
> Transfer Principle wrote:
> > We need something less trivial then simply redefining
> > "reals" to mean "naturals" to satisfy Herc's desideratum,
> > since Herc refers to reals that aren't naturals in some
> > of his posts.
> And also because it is very unlikely (unlikely to
> point that we should actually ASK HERC!) that he
> thinks the real numbers and the natural numbers are
> the same set. Isn't that right, Transfer?

OK. So let's ask Herc right now:

Herc, you _don't_ consider R and N to be the same set, right?

> However, as I see you mention below and as
> Aatu Koskensilta also tells us, we can prove the
> reals are uncountable in NFU.
> And, anyway, we don't think Herc is using either
> NFU or zuhair's ND, do we?

We can eliminate NFU based on Aatu's statement. So far, I
don't know whether to eliminate ND or not. We must wait to
see whether Herc posts something refuted by ND.

> It seems to me that NFU has not so much eliminated
> the perplexing relationships called paradoxes as
> moved the perplexity to other relationships.
> I don't imagine Herc being any
> happier by trading one paradox for another.
> I wonder if there is some law of meta-logic that
> conserves perplexity.

Maybe so. I've said it once, and I'll say it again -- I
doubt that there is _any_ theory that avoids _every_
single counterintuitive result.

But some posters might be willing to, as you put it,
trade one counterintuitive result for another. For
example, you mention to RE that ultrafinitism likely
leads to counterintuitive results (when you discuss
what happens when you take a large set and try to find
its powerset). But to an ultrafinitist, these paradoxes
involving powersets might be a _small_price_to_pay_ to
avoid having infinite or arbitrarily large finite sets,
and so on. Of course, to a poster on the standard side,
that _isn't_ a small price to pay just to avoid the (to
them, harmless) counterintuitive results regarding
infinite sets.

> > Now zuhair would be much better than I to prove that R is
> > countable in his theory, so you might want to ask zuhair
> > for the full proof in the revelant thread.
> Perhaps you could do that. You were the one suggesting
> that that R is countable in zuhair's system, I believe.

Actually, the proof should follow, in a single line, from
that Axiom of Countability:

Every set is countable. Therefore R is countable. QED

What I'm curious about -- and perhaps I might ask this in
the zuhair thread itself -- is whether a given real number,
say pi (since today is, after all, Pi Day) is a set (and not
a proper class) in his theory. I suspect that based on his
Set Separation Schema, any _definable_ real is a set, which
would include pi.

> A question for myself (and, PLEASE, anyone who might
> know the answer) is whether there is a proof of
> "If there exists a complete ordered field (that is, the
> real numbers) then there is no complete list of the
> reals." a conjecture I made up thread.

I'm curious about that proof as well. I suspect that one
might be able to use Cantor's _First_ Uncountability proof
to show it, but we'll wait and see.