Re: Han's startling new set theory.

Han.deBruijn@xxxxxxxxxxxxxx wrote:
> That *is* true in a _consistent_ theory. Suppose that I give you
> a geometrical shape and call it a triangle. Suppose that you can
> prove, though, that the sum of the angles in that shape is not 180
> degrees, but 360 degrees. Then you may safely conclude that this
> shape is not a triangle. All within Euclidian Geometry, of course.
>
> But the situation in ZF is different. Nobody has ever proven that
> ZF is consistent. The occurrence of a paradox therefore does _not_
> necessarily mean then that your assumption has been wrong. It can
> also mean that you have evidence that ZF itself is _inconsistent_.

Firstly, it is a proof that any sufficiently complicated system is
unable to produce a proof of it's own consistency. You will NEVER have
a ZF proof that ZF is consistent. You will NEVER produce a set of
axioms that allows the use of arithmetic with natural numbers that will
be able to prove itself consistent.

Secondly, there is NO PARADOX in ZF. Russells paradox is in naive set
theory. It is like you claiming that euclidean geometry is
inconsistent because the triangle in spherical geometry doesn't have
angles sum to 180 degrees. The paradox is in a different theory.

> Congratulations! I have been wrong. Such a Set Theory IS consistent.
> But something much worse has happened:
>
> THIS CONSISTENCY IS COMPETELY WORTHLESS
>
> Why? Because the introduction of "proper classes" simply renders it
> _impossible_ to ever find a contradiction. Just by _vicious circle
> reasoning_! It's quite simple. As soon as a paradox is threatening
> ZF then change the name "set" into "class" to avoid the paradoxes.
> (Or, as you write, by a more sophisticated but equivalent of this)

NO! There is NO PARADOX in ZF nor ZFC, nor NBG. The paradox is in
NAIVE SET THEORY!

Secondly, of course if you ever find an inconsistency, people will try
to correct it by correcting the axiom set. Thus ZF was born.
Russell's paradox made it utterly obvious that you cannot just form
sets willy nilly.

If a contradiction will be found in ZF and must be fixed, it will NOT
be ZF anymore. You will have to MODIFY the axioms, not just change
wording! You are totally off target with what you think people are
doing with "proper classes." It is most definately NOT changing the
name of some sets. It is introducing NEW objects that DON'T EXIST IN
ZF(C). It is like when you extend the naturals to include the negative
integers. You have not CHANGED how the naturals work. You have just
added some new objects to work with.

What you are (perhaps rightly) protesting is naive set theory. But
where you are totally off is that you are confusing this with ZF(C).

> > Uh huh. Sorry, what was the inconsistency of ZF? A nice proof
> > in ZF of P & ~P for some formula P would be swell.
>
> Duuhh, NOT with an Escape Sequence like those "classes" built in!

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 breathlessly await.
>
> _Everybody_ has to give up. The structure of ZF itself renders it
> _impossible_ to ever find an inconsistency. Because mathematicians
> have been so "smart" that they have just _killed_ the opportunity
> to prove any inconsistency. And how did they do it? By cheating !!
> They _use_ paradoxes to _define_ consistency.

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. You cannot just call things differently,
it doesn't work that way.

> Did I say:
>
> Set Theory _still is_ inconsistent to the bone.
>
> I'm sorry. That should be:
>
> Set Theory is _rotten_ to the bone.
>
> Oh Lord, tell me that this is only a nightmare and that mathematics
> has not sunk so deep.
>
> Han de Bruijn

Tell me that you are NOT as ignorant as you seem to be. You are
obviously starting with a preconcieved notion that set theory is all
wrong. 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). Do you in physics also take a problem with an
older theory to say that the new theory is inconsistent? Should I
argue that Newtonian mechanics is inconsistent becase Ptolemy's system
of planetary motions has problems? I hope this is not the way you
argue in physics.

Jiri

.

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