# Need help articulating an idea in rigorous language

So I'm working on a personal project, and I have most of the details worked out, but I'm having difficulty expressing an idea in precise terms.

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

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Standard Zermelo-Fraenkel set theory has an axiom that precludes sets that contain themselves and/or any type of infinite regress. It's called the Axiom of Regularity.

http://en.wikipedia.org/wiki/Axiom_of_regularity

So, since you can't do what you want in standard set theory, you'd have to decide what version of set theory you're using.

On the other hand it may be that you're trying to model some situation that standard math can handle with the ideas of sequences or recursion. For example you can define one thing, then define the second thing in terms of the first; the third in terms of the second; and so forth.

If you say what you're trying to do, people can suggest a mathematical model. But an infinite regress inside a set can't be done in standard math.

Also, the Axiom of Extension says that a set is characterized purely by its elements. So if you had a set {..., 1,1,1,...}, that set is exactly the same set as {1}.

On the other hand, we do have models of structures that are discrete and extend infinitely in each direction. For example the integers ..., -3, -2, -1, 0, 1, 2, 3, ... seems to have the structure you're looking for. But of course you can't biject a singleton to an infinite set.

Sorry, I should have stated that in my project I have abandoned certain conventions. Just so there is no confusion for anyone else, I do have an extensive background in math...including a degree, so there is no need to reinvent the wheel, so to speak, in any explanation; I just need the language on strictly hypothetical basis. Thanks