Understanding the definition of endless sets In a course on theoretical computer science we have to prove if sets are endless. I have two problems with the exercise:


I don't understand exactly, what an endless set is (I find it very hard to understand, maybe because of the "endless")
I don't know, how I can prove that the given sets are endless


The definition given is this:


  A set $X$ is called endless set, if there is a endless family of sets $\{X_n\}_{n=0}^{\infty}$ with
  $... \in X_3 \in X_2 \in X_1 \in X_0 \in X$.


Can someone explain in simple words what an endless set is?
Defining endless sets, you probably assume that you don't have the Axiom of regularity.

A set is endless when it doesn't contain only simple elements, it has to contain an other set among his elements, and this other set has again to contain an other set and this third set has to contain an other set, etc..

For example, a set verifying $M = \{M\}$ is endless because it contains an other set ($M$) which contains an other set (still $M$) which contains an other set ($M$), etc.. You can take $X=X_0= X_1= ... = M$ as infinite family.

A counter example is $M = \{42\}$; $M$ doesn't contain any set. $M= \{42, \{12\}\}$ is also a counter example because $M$ contains a set ($\{12\}$) but this set doesn't contain any set.