Re: Which axiom prohibits this kind of construction?

On Oct 9, 12:47 pm, MoeBlee <jazzm...@xxxxxxxxxxx> wrote:
On Oct 9, 11:22 am, Starblade Enkai <StarbladeEn...@xxxxxxxxxx> wrote:
if you go on with the
empty set, 0, then the set of the empty set, [0], and keep on going,
you have to have a finite number of brackets. If you try to go to
infinity, you'd have an infinite number of brackets, and hence a
contradiction.
There is not even a DEFINITION of a notation with an infinite number
of braces. All terms of the language of set theory are finite. The
LANGUAGE itself does not allow a term that has infinitely many braces.

I think I see Enkai's problem here.

Let's compare the Zermelo ordinals with the von Neumann ordinals:

Zermelo:
0 = {}
1 = {{}}
2 = {{{}}}
3 = {{{{}}}}
....

von Neumann:
0 = {}
1 = {{}}
2 = {{},{{}}}
3 = {{},{{}},{{},{{}}}}
....
omega

It's not obvious to Enkai why infinite von Neumann
ordinals exist, but not infinite Zermelo ordinals. To
him, if one is forbidden then so should the other. (It's
obvious to me that Enkai's lines:

[[...[[[x]]]...]] = [...[[[x]]]...]

and

z = [y] = [[x]] = [[[w]]] = ...

suggest infinite Zermelo ordinals.)

BTW, I once tried to form a rigorous theory of the form
ZF-Foundation/Regularity+"infinite Zermelo ordinals exist."
.

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