When is $x=\{ x\}$?

Inspired by this question: When/for which $x$ do we have $x=\{ x\}$ ?

Solution 1:

Most mathematicians assume the background set theory to be ZFC. In this theory every non-empty set has an $\in$-minimal element.

This is known as the axiom of regularity, or axiom of foundation. It asserts that if $A$ is a non-empty set, then there is some $z\in A$ such that $z\cap A=\varnothing$.

In turn this implies that $x\notin x$ otherwise if $x=\{x\}$ then for all $y\in x$ we have $y=x$ and therefore $y\cap x\neq\varnothing$.

However in other set theories, for example Quine's New Foundations, and we can even create a model of ZF-Reg, that is all the axioms of ZF, except regularity. In such model there is a definable class which is a model of full ZF.

The constructions are quite technical, but we can end up in a very wonderful situation where $a\in a$ or even $P(a)\in a$, where $P$ is used to denote the power set.

These models were quite useful in independence proofs back in the late 1950's and early 1960's, mostly due to Specker (and his students) which used the sets $x=\{x\}$ as urelements (non-sets elements) relatively to the inner model of ZF. One noted example of a proof used this method was a vector space without a basis.