I recently started studying set theory and I have seen this exercise in the textbook. The only example I could think of was the empty set. This is how I think:

A number(for example 1), cannot be a subset of X since a number is not a set. So I need to have sets as the elements of X.

Then, I choose a element like $\{1\}$ for example. But then X must have the subset $\{1\}$, which means it has 1 as an element. This also cannot happen because of what I said before.

And even if choose an element of X as something like $\{\{\{\{1\}\}\}\}$, it will require $\{\{\{1\}\}\}$ to be an element of the set. Which then will require $\{\{1\}\}$ and eventually I come back to what I started with.

Can you show me what is wrong in the way I think of this problem, and can you show some examples of these kinds of sets?

  • $\begingroup$ In the usual set theory formalism, a number is a set. $\endgroup$ – WoolierThanThou Sep 26 '19 at 16:41
  • $\begingroup$ I think in the book I was studying it was not the case that a number is a set. Or at least nothing like that was mentioned so far. It was Cantor's set theory. $\endgroup$ – zzlawlzz Sep 26 '19 at 16:46
  • $\begingroup$ Well, if a number isn't a set, then what is it? The reason for the formalism is that you can encode the mathematical structures you'd like to consider using only sets. $\endgroup$ – WoolierThanThou Sep 26 '19 at 16:47
  • $\begingroup$ Your analysis is working its way toward what's known as the Axiom of Foundation: $\forall y \exists x \in y ~(x \cap y = \emptyset).$ $\endgroup$ – Robert Shore Sep 26 '19 at 17:01

Your argument is right that there must not be any "urelements" hidden anywhere inside the set. But here are some examples: $$\emptyset,\qquad\{\emptyset\},\qquad \{\emptyset,\{\emptyset\}\},\qquad \{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\} $$ and many more.

Note that if $S$ is a set with the desired property, then so is $S\cup\{S\}$.

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  • $\begingroup$ It is worth pointing out the relation of this answer to the set-theoretic definition of natural numbers. $\endgroup$ – JMoravitz Sep 26 '19 at 16:42
  • $\begingroup$ Is this only possible using empty sets? Are there any other sets that I can use with the property you have given? $\endgroup$ – zzlawlzz Sep 26 '19 at 16:43
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    $\begingroup$ @RüzgarAyan you might be interested in reading this question:Is mathematics just a bunch of nested empty sets? $\endgroup$ – JMoravitz Sep 26 '19 at 16:47
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    $\begingroup$ @RüzgarAyan A transitive set not reachable by the method I described is $\{ \{\{\{\{\emptyset\}\}\}\}, \{\{\{\emptyset\}\}\}, \{\{\emptyset\}\},\{\emptyset\},\emptyset\}$, but that too has nothing but emptyset at its "bones" - necessarily by the OP argument $\endgroup$ – Hagen von Eitzen Sep 26 '19 at 16:49

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