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I used to think naively the construction is straightforward, which is, if we add one layer innermost each time, then we could have one that corresponds to $\omega$ in Neumann's representation, which is. of course, constructible in ZF excluding axiom of foundation .

But I was told I can't define the successor by simply adding $\{\}$ in the center.

So my question is what is the correct construction of $\{\{\{...\}\}\}$ in ZF in the absence of axiom of foundation?

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Is ZF-AF meant to be ZF minus foundation, or ZF minus foundation plus antifoundation? – Clive Newstead Dec 7 '12 at 9:53
@CliveNewstead: I initially mean ZF-AF is ZF minus foundation. – Metta World Peace Dec 7 '12 at 9:55
Metta, often AF denotes anti foundation whereas FND or REG denote the axiom of foundation (or regularity). – Asaf Karagila Dec 7 '12 at 10:24
@AsafKaragila : Thank you for pointing that out. I was wondering why people ask about this. I will fix it. – Metta World Peace Dec 7 '12 at 10:28
up vote 4 down vote accepted

(I read this question as asking about constructing such sets in ZF-Foundation alone)

Essentially you are asking for a set $x=\{x\}$ in ZF without foundation.

The problem is that just without foundation you don't get enough. Every model of ZF is a model of ZF without foundation.

There are methods of constructing models in which the axiom of foundation fails, and you can control this failure quite nicely. These are methods which are similar to methods in which the axiom of choice is negated.

The most celebrated method is due to Specker and uses atoms and results in sets for which $x=\{x\}$.

Edit: I spoke with my teacher who taught me the method to generate models without the axiom of foundation from models with atoms, and he said that he came up with the method by himself. He figured that someone else had written about it before, so he didn't bother to publish it. While I am certain that Specker worked on similar ideas I could not find a reference. I will take it upon me to write a note on the method my teacher used.

I will update this when the note is prepared, for whoever is interested.

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Ah, many known unknowns, even more unknown unknowns. – Metta World Peace Dec 7 '12 at 10:14
Indeed. Note that you are essentially asking for a consistency result, and those are often complicated. Using atoms (urelements) to generate non-well founded models is relatively easy. I will try to look up a reference for that later this weekend and add it to my answer. – Asaf Karagila Dec 7 '12 at 10:23
Thank you very much for your effort. – Metta World Peace Dec 7 '12 at 10:31

The correct construction is to apply AF to the pointed
graph with exactly one vertex and exactly one self-loop.

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Apologize for my failure to propose the problem in an unambiguous way. – Metta World Peace Dec 7 '12 at 10:37

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