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Premise:

We start with a set which has only itself as an element, which results in the following infinite regress:

{{{...}}}

Now what I would like to do is create a bijection between this infinite regress and some general singleton set.

For example: Say we have a set, call it S, such that S={1}. Now I want to create a bijection between S and the set above such that the resulting set is {...1,1,1,1...}.

How can I express this relationship in rigorous terms -with notation common to set theory.

Any thoughts from the community would be very appreciated.

Shalom