# 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?

• This notion of endlessness essentially means that (if endless sets exist) the element-of relation is not well-founded, i.e. we must be working in a version of set theory which does not satisfy the axiom of regularity. Which axioms are you using? – Klaus Draeger Jun 29 '15 at 12:26
• I removed the second question, since we only want one question per post. – Raphael Jun 29 '15 at 12:40
• @Raphael: These were closely related questions about the same concept - are we really that strict about this rule? Is this a recent development? – Klaus Draeger Jun 29 '15 at 12:58
• @KlausDraeger Depends. The way I see it, these were not strictly related; one is an understanding question (which we like) and the other a problem dump (which we don't like). Therefore, I chose to remove the part we don't like; it is also probably obsolete once the OP understands the definition (since the exercise is very easy). Note that I adapted the answer, leaving the examples there. The only problem wrong with this question now is that it's a pure mathematics question, and hence offtopic here. – Raphael Jun 29 '15 at 14:27
• @KlausDraeger Regarding your first comment, I think it's unlikely that the OP is familiar with well-foundedness or the axiom of regularity. If they had this kind of penetration of the material, they'd probably not have to ask this question. – Raphael Jun 29 '15 at 14:29

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.