Re: Han's startling new set theory.

Jesse F. Hughes wrote:

> Han.deBruijn@xxxxxxxxxxxxxx writes:
>
> > Jesse F. Hughes wrote:
> >> Consider the formulas of ZF (with one
> >> free variable) and quotient out by the equivalence relation:
> >>
> >> Phi R Psi <=> |- (A x)(Phi(x) <-> Psi(x)).
> >>
> >> Call the elements of that quotient the *classes* of ZF.
> >
> > Huh? Quotient? I know that 1/2 is a quotient, but where is the
> > quotient in the above? And where are the "elements" in, if there
> > is no set to contain them? But ah, your set is a "class" now ...
>
> Different sense of quotient, but utterly standard.
>
> An equivalence relation ~ on a set S defines a set of partitions
> denoted S/~ and called the *quotient* of S by ~.

Hmm, now I remember again. It has been long ago: < 1973.
Quotient Groups for example.

> Assuming that there is a set satisfying a particular definition and
> then deriving a contradiction does indeed prove that no such set
> exists.

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_.

> Anyway, I'm happy that Wikipedia provided a better resource than my
> off-the-cuff remarks.
>
> > Plenty of rationale to maintain HdB's view that:
> >
> > Set Theory _still is_ inconsistent to the bone.
>
> Utter nonsense. Show the inconsistency.

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)

> 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!

> 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.

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

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