# P. Halmos Naive Set Theory Exercise 1 section 3

There was this question asked a little while ago, and it was not closed:

Consider the sets $∅,\lbrace ∅ \rbrace,\lbrace\lbrace ∅ \rbrace\rbrace$ etc.; consider the pairs such as $\lbrace ∅ ,\lbrace ∅ \rbrace \rbrace$ formed by any two of them; Consider the pairs formed any two such pairs, or else the mixed pairs formed by any singleton and any pair; and proceed so on ad infinitum. Are all sets obtained in this way unique from each other?

The question was never answered, so I am asking it again, but in a slightly different way.

I am pretty stuck on this question because I don't quite get what it is asking.

I was wondering, what does it mean by all sets; does this include the singletons $∅,\lbrace ∅ \rbrace,\lbrace\lbrace ∅ \rbrace\rbrace$ ? And also, do sets being "distinct" mean that they are not equal? And finally, if someone could answer the question, but give hints, with proof, I would appreciate it.

We have only looked at the Axiom of Specification, Extension and Pairing.

EDIT: Here is the best proof that I could think of, please let me know where I went wrong, or could improve:

It is obvious that by the remarks of Section 3, there exists sets $A = \lbrace a_1, a_2 \rbrace$ and $B = \lbrace b_1, b_2 \rbrace$ created in the specified way. And, by the equivalent formulation of the Axiom of Pairing and remarks of Section 3, $A$ and $B$ are the only such sets which contain $a_1$, $a_2$ and $b_1$, $b_2$, respectively. Now suppose, to the contrary, that $A$ and $B$ are not distinct; i.e. $A = B$. However, by the Axiom of Extension, $A = B$ implies $a_1, a_2 \in B$, and $b_1, b_2 \in A$; this is a contradiction, since by the previous remarks, $A$ and $B$ were the only sets which contained $a_1$, $a_2$ and $b_1$, $b_2$, respectively.

## 1 Answer

It does not matter whether singletons are included or not, because any formed pairs will have two elements and a singleton will be either the empty set or a set with one element.

The sets formed are distinct: any set formed has exactly two elements, so for two sets to be the same the elements themselves have to be the same, which means the sets have to be formed by the same elements.

I have not read the book but I think the point of this question is trying to clarify the difference between "set as an element" and "subset".