Example of set which contains itself
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.
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:
- Is the statement $A \in A$ true or false?
- Where is axiom of regularity actually used?
$x = \{ x \}$ ...
... but actually one of the axioms of ZFC (the "usual" axioms of set theory) has the immediate consequence that no set has itself as a member.
Aczel's anti-foundational axiom is a classic example: http://en.wikipedia.org/wiki/Aczel%27s_anti-foundation_axiom
"It states that every accessible pointed directed graph corresponds to a unique set. In particular, the graph consisting of a single vertex with a loop corresponds to a set which contains only itself as element, i.e. a Quine atom."
Example, $R=\{{R,2,4,6,8,10,...\}}$ it can just be written explicitly like this.