[QUOTE=Mr. P-1;250038]You're almost there.

If A is a 1-element set, then the only subsets of A are TEX]\empty[/TEX] and A itself.

A cannot be an element of A. What could be an element of A?[/QUOTE]

anything but A or [TEX]\empty[/TEX] ? as long as A can be a collection or a set of sets

[QUOTE=CRGreathouse;249982]The set S itself isn't a member of S -- ZF doesn't allow that, so that won't work.[/QUOTE]

I would argue that Z, rather than ZF is a closer formalisation of the informal set theory we've been doing here, because 1. nothing has been said which relies upon the axioms of replacement or regularity, and 2, we have admitted individual elements, which most treatments of ZF do not admit.

S[TEX]\in[/TEX]S still doesn't work, even in Z, because while not forbidden, it is impossible to prove such a set exists.

[QUOTE=science_man_88;250040]anything but A or [TEX]\empty[/TEX] ? as long as A can be a collection or a set of sets[/QUOTE]

Stick with your idea of a 1-element set. A has two subsets: [TEX]\empty[/TEX] and A. A's sole element is a subset. The element is not A. What is the element?

[QUOTE=Mr. P-1;250043]Stick with your idea of a 1-element set. A has two subsets: [TEX]\empty[/TEX] and A. A's sole element is a subset. The element is not A. What is the element?[/QUOTE]

A[1] in PARI terms, last time I checked.

[QUOTE=science_man_88;250044]A[1] in PARI terms, last time I checked.[/QUOTE]

A is a one-element set with two subsets: [TEX]\empty[/TEX] and A. One of the two subsets I have just named is an element of A. The element is not A. What is the element?

[QUOTE=Mr. P-1;250048]A is a one-element set with two subsets: [TEX]\empty[/TEX] and A. One of the two subsets I have just named is an element of A. The element is not A. What is the element?[/QUOTE]

[TEX]\empty[/TEX] is what you seem to imply.

[QUOTE=science_man_88;250049][TEX]\empty[/TEX] is what you seem to imply.[/QUOTE]

If [TEX]\empty[/TEX] is an element of a 1-element set. What is the set?

[QUOTE=Mr. P-1;250050]If [TEX]\empty[/TEX] is an element of a 1-element set. What is the set?[/QUOTE]

[TEX]\empty[/TEX] by the sounds of it.

[QUOTE=Mr. P-1;250042]I would argue that Z, rather than ZF is a closer formalisation of the informal set theory we've been doing here[/QUOTE]

Yep, agreed. But I didn't want to bring up issues of the foundation axiom and I wanted to help sm along the path so I wrote what was needed.

[QUOTE=science_man_88;250052][TEX]\empty[/TEX] by the sounds of it.[/QUOTE]

[TEX]\empty[/TEX]'s not a member of that set, though; that has no elements.

[QUOTE=science_man_88;250044]A[1] in PARI terms, last time I checked.[/QUOTE]

this is the final thing I come up with if it's not that I have no idea because I obviously don't know the question.