3
$\begingroup$

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.)

Improve this question
$\endgroup$
8
  • $\begingroup$ Suggested improvement: Give definitions of the theories. $\endgroup$
    – Monroe Eskew
    Commented Oct 9, 2015 at 17:15
  • 2
    $\begingroup$ 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
    Commented Oct 10, 2015 at 5:19
  • 2
    $\begingroup$ 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
    Commented 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
    Commented 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
    Commented Oct 10, 2015 at 6:49

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .