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Suppose one has a model $M$$\vDash$$NGBC^{-f}$+BAFA. Does there exist a (class) forcing extension $M[G]$$\vDash$$NGBC^{-f}$+$BAFA$ that has a submodel $N$$\vDash$$NGB^{-f}$+$BAFA$+$\lnot$$AC$? Can one also construct permutation models of $NGB^{-f}$+$BAFA$+$\lnot$$AC$?

Note $NGB^{-f}$ is von Neumann-Goedel-Bernays set theory without the axiom of foundation.

$BAFA$ is Boffa's Anti-Foundation Axiom; that is, that

"...for every transitive set $t_0$ and every extensional binary relation $<$$a$,$e$$>$ that end-extends $<$$t_0$, $\in$$>$, meaning $t_0$$\subseteq$$a$ and $\in$$\upharpoonright$$t_0$=$e$$\cap$($a$$\times$$t_0$), there exists a transitive set $t$, and an isomorphism from $<$$a$,$e$$>$ to $<$$t$,$\in$$>$ that is the identity on $t_0$..."

(this from Daghighi's, Golshani's, Hamkins',and Jerabek's preprint "The Role of the Foundation Axiom in the Kunen Inconsistency", from which the notation was taken and the motivation for this question.)

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  • $\begingroup$ Suggested improvement: Give definitions of the theories. $\endgroup$
    – Monroe Eskew
    Oct 9, 2015 at 17:15
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    $\begingroup$ I think the first thing to do is to properly define "forcing" over models of BAFA! Note that even forcing over models of AFA - a much tamer antifoundation axiom - took real effort: see S. Kentaro's article onlinelibrary.wiley.com/doi/10.1002/malq.200410060/abstract. The problem is that without Foundation it's not clear how to define a name or the interpretation of a name. $\endgroup$
    – Noah Schweber
    Oct 10, 2015 at 5:19
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    $\begingroup$ Actually, this article might be more directly related: arxiv.org/pdf/0712.1968.pdf. See also users.auth.gr/tzouvara/Texfiles.htm/forcing.pdf. (By the way, is there a reason you care about the class-theoretic version instead of the set-theoretic version?) $\endgroup$
    – Noah Schweber
    Oct 10, 2015 at 5:22
  • $\begingroup$ @NoahSchweber: I care about the class-theoretic version for the same reason that Hamkins and others say that $NGBC$ can better capture the import of the Kunen inconsistency than can $ZFC$+'$j$'. If $NGBC$ can better capture the import of the Kunen inconsistency, then perhaps class forcing would be the better method for constructing models of $NGB^{-f}$+ $BAFA$+$\lnot$$AC$. Since I am not at all sure about this, I added class forcing (in parentheses) for completeness. In which case, one might want to adjust the definition of $BAFA$ to suit--replace every occurrence of 'set' in that $\endgroup$
    – Thomas Benjamin
    Oct 10, 2015 at 6:47
  • $\begingroup$ (cont.) definition with 'class'. If this idea is wrong-headed, please let me know. Thanks. $\endgroup$
    – Thomas Benjamin
    Oct 10, 2015 at 6:49

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