Re: Han's startling new set theory.

Han.deBruijn@xxxxxxxxxxxxxx wrote:
> But there IS a proof that Euclidian Geometry is consistent. Right?

As far as I know, Hilbert did provide an axiom system for geometry and
proved it's consistency. That means by Godel's theorem that euclidean
geometry as stated by Hilbert is not powerful enough do whole number
arithmetic. But I'm not familiar with Hilbert's proof of this.

> And for the rest: if Nature can prove its Consistency, so can we.

What the hell are you babbling about? How did nature "prove" it's
consistency? For all we know nature may not follow any logical system
at all. We will NEVER know this, we just suspect that it does, and
from observations it seems that our suspicion is correct. But
empirical evidence is NOT a mathematical proof.

Even if there was a consistent model of all physics someone came up
with (there is none) it would still not prove that the actual world was
consistent with this. All you get is empirical evidence. And by
empirical evidence, ZFC is incredibly consistent, a lot more consistent
then physics. ZFC has been emprically consistent since the 30's
(people have tried and failed to show inconsistency for 70 years,
that's pretty good empirical record). On the other hand physics
evolves all the time, and still all the main theories are inconsistent
with each other.

> A little bit of Physics would be NO Idleness in Mathematics.
>
> I keep repeating this mantra, in the somewhat desperate hope that it
> will sink into your stubborn brains within a century or so. Physics
> has _everything_ to do with a _provable consistency_ of mathematics.

Witty remarks are remembered if famous people say them. Obviously you
want to be famous, but nobody gets famous by just providing witty
remarks. You have to actually do something interesting and THEN have
witty remarks. It doesn't work in reverse. If Einstein have said that
maybe people would remember it. If you say it, especially if you say
it in messages where you display utter lack of knowledge about
mathematics, people will not remember it.

Secondly, physics has nothing to do with any provable consistency. It
is a THEOREM that any sufficiently powerful axiom system is unable to
prove its own consistency.

> Yes, you need arguments _from outside_ to prove consistency, because
> most theories cannot prove their consistency from within, as you have
> said. Why don't you see then that my mantra offers a key to success?

Please read a book on mathematical foundations. You make yourself into
a total moron if you argue about something without knowing anything
about it.

> > THERE ARE NO CLASSES IN ZF! THERE ARE NO CLASSES IN ZF! THERE ARE NO
> > CLASSES IN ZF! Nobody is escaping anything. Learn about ZF before
> > talking about it. You are sinking deeper and deeper into the total
> > crank category, or you are not reading what people have written.
>
> I really appriciate your attempts to convince me, Jiri. I really do.
> And I *hate* the fact that I *have* to disagree with people like you.
> But what is the reason that *you* didn't submit a far better article
> to Wikipedia than the one I had to rely on:
>
> http://en.wikipedia.org/wiki/Mathematical_class
>
> I mean, do you really think that you can be convincing by sending us
> from one book to another? Dutch: "Van het kastje naar de muur"?

There is NOTHING wrong with that article. It is only an overview and
doesn't give the specifics. It is only an introduction article for
people who know nothing about classes. I suppose you did not read the
sentence that said:

"A proper class cannot be an element of a set or a class and is not
subject to the Zermelo-Fraenkel axioms of set theory"

You confuse everything together and then whine that it doesn't make
sense. Just because you don't understand something doesn't mean that
it's inconsistent. This is the kind of arrogant ignorance that pretty
much defines a "crank".

> > No. It is possible to find an inconsistency. If you find it, ZF will
> > be dead and people will have to fix it by fixing the axioms or changing
> > to a different axiom system.
>
> No. The interests involved are of a non-mathematical nature. Which
> makes it virtually impossible for mainstream mathematics to give up
> on Set Theory as its foundation par excellence, despite of whatever
> evidence of the contrary. Google("The Political Economy of Sets").

I don't know where you get your conspiracy ideas. There is NO EVIDENCE
to the contrary. There is evidence that naive set theory is bad, but
that was sorted out in the beginning of the 20th century. Just because
you don't understand the theory (you still have not shown how Russell's
paradox applies to ZF) doesn't mean that it is inconsistent.

Mainstream mathematics will not give up ZF(C) or NBG until it is shown
to be inconsistent.

> > You are obviously starting with a preconcieved notion that
> > set theory is all wrong.
>
> It's wrong to use it as a foundation for the whole of mathematics.

But your belief in that is not enough as an argument. On the other
hand, the fact that no inconsistency has been found yet, and the fact
that the theory has been used to derive incredibly useful results which
have then been used successfully in physics, engineering, etc... is
what keeps people using set theory. There is no other rigorous
alternative. You have not presented one. Perhaps one day there will
be. Until then most mathematicians will think of ZFC as their
foundation.

> > You take a problem with NAIVE SET THEORY (as practiced by early
> > set theorists) and generalize it to a theory with does not have
> > this problem (namely ZF).
>
> Denied. ZF has the same problems as naive set theory, in disguise.
> Hiding them behind a far more complicated formalism will not help
> in the end, though.

Huh? And that is true because? If you make a statement like that you
have to provide a reason for this. It is nice that you think so, but
provide reason. This is why mainstream mathematics WILL NOT listen to
your whining about set theory. Mathematicians need actual reasons and
not just "well, HdB says ZF is bad, so we'll just have to give it up."
Provide a proof that ZF has the same problems as naive set theory and
then people will listen. Given that until a few days ago you didn't
see a problem with "for all x, x = {x}" I would very much doubt you
have any grasp at all of what ZF(C) even says.

Jiri

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