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Based on our intuition about von-Neuman's rank, there is a standard view to describe a model of ZFC as a large V-shape world.

When we remove the Axiom of Foundation (AF) from ZFC and replace it with an anti-foundation axiom like (AFA, BAFA, SAFA, etc.) we expand the V-shape world of ZFC by adding some new non-well founded sets. I wonder what is the true intuition about these expanded worlds? How should we imagine them? Which one of these descriptions are more useful?

Question: Please introduce some references for intuitive diagrams which describe the shape of the world of theories ZFC-AF+AFA, ZFC-AF+BAFA & ZFC-AF+SAFA. Any reference for diagrams which describe the shape of other non-well founded axiomatic systems which are essentially different from ZFC like NF and NFU are also welcome.

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Hope this reference helps: "Broadening the Iterative Conception of Set" by Mark F. Sharlow, Notre Dame Journal of Formal Logic, Volume 42, Number 3, 2001, pp.149-170. According to the abstract, "the modified conception maintains most of the features of the iterative conception of set, but allows for some non-wellfounded sets. It is suggested that this modified iterated conception of set supports the axioms of Quine's set theory NF". Pay particular attention to Section 4--"Loosening Up the hierarchy" and figures 1-4.

As regards the relation of the iterative concept of set with, say, AFA, I refer you to a quote from Stephen G. Simpson regarding this very topic (this from one of his postings):

"AFA in no way invalidates the iterative concept of set. The intended model V of ZFC (including the axiom of foundation) is a canonical inner submodel of the intended model $V^{*}$ of

$ZFC^{*}$=ZFC-Foundation +Anti-foundation [AFA--my comment].

Namely, V is the well-founded part of $V^{*}$. And all of the f.o.m. action takes place in V. And each of V and $V^{*}$ is canonically recoverable from the other. (The elements of $V^{*}$ are just the isomorphism types of directed graphs in V [this suggests a way (at least to me) to generalize the iterative concept of set--let each stage of the hierarchy form the isomorphism types of new subdigraphs, including the ones that violate the Axiom of Foundation--my comment].) If we refer to the elements of V as 'sets' and the elements of $V^{*}$ as 'hypersets' (Sazonov) or 'schmets' (Anderson) or whatever, there is no conflict." (www.cs.nyu.edu/pipermail/fom/2000-May/004007.html)

At least it is something to think about.

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  • $\begingroup$ Thank you very much for your very useful references, Thomas! $\endgroup$
    – user180918
    Commented Nov 26, 2014 at 10:15
  • $\begingroup$ @AliSadeghDaghighi: No problem. Question: assuming ZF-Foundation+BAFA, can one prove that there exist Reinhardt Cardinals? BAFA is supposed to produce elementary embeddings of V to itself. $\endgroup$
    – Thomas Benjamin
    Commented Nov 26, 2014 at 12:21
  • $\begingroup$ Within $ZFGC^{-f}+BAFA$ the following hold: (a) Every isomorphism of transitive sets can be extended to an automorphism of the universe. In particular, there exist nontrivial automorphisms. (b) There is a definable class non-trivial elementary embedding $j : V \rightarrow V$, which is not an automorphism. But note that the use of Global choice in (a), (b) is essential because within $ZFC^{-f}+BAFA$ we have: (c) There is no nontrivial $\Delta_0$-elementary embedding $j : V \rightarrow V$ of the universe with itself that is definable in the first-order language of set theory. $\endgroup$
    – user180918
    Commented Nov 26, 2014 at 14:20

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