Is there a branch of mathematics that requires the existence of sets that contain themselves?

Question

I notice that Russell's paradox, Burali-Forti's paradox, and even Cantor's paradox, all depend on our tolerance of sets that contain themselves (at one level of depth or another). Thus, I was thinking if it wouldn't be a good way to stop the paradoxes, to just prohibit sets containing themselves, via a modification in the axiom scheme of comprehension, probably.

But is there some branch of mathematics, maybe something close to recursion theory, that depends on sets that contain themselves at some level of depth?

Also, is there any other paradox of naive set theory that doesn't depend on sets that contain themselves?

Thanks in advance.

Answer 1

This question from today discusses paradoxes of set theory that don't exactly depend on sets that contain themselves. But in set theory all you have is sets and containment, so if you rule those out, as a source of paradoxes or anything else, there's not much left to work with.

Peter Aczel's theory of non-well-founded sets is a mathematical theory all about sets that do contain themselves, or contain sets that contain them, and variations thereof.

In J.H. Conway's theory of combinatorial games, certain games are represented by structures that do contain themselves. A game is defined to be an ordered pair of sets of games. The left set is the set of game positions to which the Left player can move, and the right set similarly. So for example $(\emptyset, \emptyset)$ is the trivial game in which neither player has any legal moves. Some real-world game positions correspond to ${\bf On} = (\{{\bf On}\}, \emptyset)$ where the Left player can make as many moves as necessary, or:

$$\begin{eqnarray} {\bf tis} & = & (\{{\bf tisn}\}, \emptyset) \\ {\bf tisn} & = & (\emptyset, \{{\bf tis}\}) \end{eqnarray}$$

where the two players can keep moving indefinitely, but neither player can move twice in a row. These "loopy games" might be represented by non-well-founded sets.

Answer 2

In the days leading to the formalization of ZF, when set theory was still missing its F, people had the idea about set theory with elements which are not sets. These were called atoms or urelements. This theories were used to prove independence results, for example it was proved that if we begin with a model of ZF+Atoms+AC we can create a model in which the axiom of choice does not hold.

These consistency results were the drive behind Cohen's work when he developed forcing and proved that a good portion of these results hold without assuming atoms exist. After a short time mathematicians established ways of transferring the proof from a context of atoms into a context without atoms, so the old methods remain valid and useful today.

One of the people which developed and helped finalizing this method was Ernst Specker, who used sets of the form $x=\{x\}$ as the atoms. That is, we have a big class $W$ in which there is a collection $V$ of well-founded sets which make a model of ZFC, and between $V$ and $W$ there is a collection of sets of the form $x=\{x\}$.

This is a very important use of ill-founded sets, and it was used to establish some important consistency results (e.g. the existence of a vector space without a basis).

Answer 3

I'm taking a chance here ^^, this "answer" is by no means an answer to your question and isn't even related I think, it might even be a misconception of mine... Anyways, I've sometimes come across the notion of the category of categories which in my book is a structure that contains itself, albeit not a set.

Corrections are welcome ^^