Re: Cantor's paradise of fools

Ross A. Finlayson wrote:
> david petry wrote:
> > daz wrote:
> >
> > > Based on a conversation I had with Paul Cohen around 1993 about what he
> > > thought is the truth status of the Continuum Hypothesis, he showed
> > > every sign of believing that there *is* a truth status.
> >
> > Did he indicate what he think it *means* to say there must be a truth
> > status to CH? Did he say that he thinks humans will someday have
> > access to that truth status? Why should we believe you, and not
> > Zeilberger, who claimed that Cohen knew it was all a game?
> >
> >
> > > Let's suppose what could easily be the case: that the Twin Prime
> > > Conjecture cannot be proved or disproved with standard axioms of
> > > arithmetic.
> > >
> > > Then, what would you believe? That it is neither true nor false
> > > (because verifying either case would involve infinitely many
> > > operations) ???
> >
> > You've picked a poor example. We have very strong heuristic
> > (probabilistic) arguments in favor of the twin prime conjecture being
> > true. We can say right now that we have every reason for believing it
> > to be true, without worrying whether we can prove it formally.
> >
> >
> > > But that is an entirely different thing from asserting that what humans
> > > cannot access must therefore be meaningless. Such an assertion seems
> > > to reveal the hubris of presuming that if humans can't know something,
> > > it makes no sense to talk of it.
> > >
> > > (E.g., suppose there is a separate universe from ours, which for some
> > > reason never was and never will be connectable to ours via any form of
> > > information transfer. Therefore
> > > such a universe cannot exist or be meaningful? Try telling that to the
> > > citizens of the other universe who, at this very moment, are laughing
> > > raucously about the hubris of humans!)
> >
> > Your thought experiment is terribly silly. You are requiring us to
> > imagine "Try telling that to the citizens of the other universe who
> > ... never will be connectable to ours via any form of information
> > transfer"
> >
> >
> > >I find [Zeilberger's] anti-infinity harangue to be more than a little silly.
> >
> > Your criticisms of it seem to be even sillier.
>
> Zeilberger, who is a respected combinatorialist and well-known for his
> opinion pages there, and his freely available text with H. Wilf, and
> apparently hypergeometric infinite series summation methods, does have
> there basically an tongue-in-cheek "anti-infinity harangue".
>
> http://www.cis.upenn.edu/~wilf/AeqB.html
>
> Finite combinatorics, graph and matroid systems, the discrete and
> umbral calculus or calculus of finite differences, those are widely
> useful in such notions as CLP, databases, container, sort, and search
> algorithms, etcetera.
>
> While that's so, he acknowledges systems that via induction have not
> finitely many elements. It seems he is trying to reestablish the power
> of concrete mathematics.
>
> We might ask him his opinion, Dr. Zeilberger, what do you mean by that?
>
> It seems he suggests that the efforts spent on consideration of the
> transfinite are better spent in analytical methods that can provide
> solutions to real-world problems. That's a view shared by many, that
> to a large extent transfinite cardinals as a subfield of foundations of
> mathematics, and to some arguably foundationless subfield, draws too
> much attention, where to actually produce something useful to humanity
> has not been shown to be a fruit of the inimitable succession of the
> transfinite, basically re-addressing the infinite.
>
> There are a lot of ways that mathematics uses the infinite.
>

Hi,

This conversation is extended over at the FOM mailing list.

http://www.cs.nyu.edu/pipermail/fom/2005-December/date.html

Mentioned is Friedman's consideration that some true statements about
the natural integers are requiring cardinals larger than ever before.
Yeah!

One way to intepret that is as has been discussed with "nonstandard
countable" "models" of the natural integers, Paris and Kirby, where
there is the notion that there needs be an infinite element of that set
to prove certain soi-disant facts about that set.

Another is to consider that those sets have a maximal element. That is
one of the assumptions of this "model theory", that assigns an
arbitrary ordinal to be the maximal element.

That was a focal point of recent discussions on sci.logic about the
applicability of a form of Goedel's incompleteness, and a form of
reasoning with acknowledgment of a maximal element that allows the
rejection of the notion that no consistently strong theory is complete.

The universe is infinite. Infinite sets are equivalent.

There's only one theory with no axioms.

Ross

.

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