Elementary Set Theory – Example of a Set That Contains Itself

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I am trying to understand Russells's paradox

How can a set contain itself? Can you show example of set which is not a set of all sets and it contains itself.

Best Answer

In modern set theory (read: ZFC) there is no such set. The axiom of foundation ensures that such sets do not exist, which means that the class defined by Russell in the paradox is in fact the collection of all sets.

It is possible, however, to construct a model of all the axioms except the axiom of foundation, and generate sets of the form $x=\{x\}$. Alternatively there are stronger axioms such as the Antifoundation axiom which also imply that there are sets like $x=\{x\}$. Namely, sets for which $x\in x$.

For the common mathematics one can assume the foundation is based on ZFC or not (because there is a model of ZFC within a model of ZFC-Foundation), so there is no way to point out at a particular set for which it is true.

Also interesting:

  1. Is the statement $A \in A$ true or false?
  2. Where is axiom of regularity actually used?
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