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In their paper, "Generalizations of the Kunen inconsistency", (Annals of Pure and Applied Logic 163 (2012) 1872-1890, doi:10.1016/j.apal.2012.06.001, arXiv:1106.1951), Hamkins, Kirmayer, and Perlmutter write the following (pg. 1888—I quote Theorem 32, Corollary 34(1), and their commentary on the proofs of the theorem and the corollary preceding Theorem 32):

Theorem 32. Assume only $ZF$ [Corollary 34(1) is the $NGB$ version of this ( see pg. 1874)--my comment] There is no nontrivial elementary embedding $j$: $V$ $\rightarrow$ $V$ that is definable from parameters.

Corollary 34. Do not assume $AC$. For any transitive class $M$, there is no nontrivial elementary embedding $j$: $M$ $\rightarrow$ $V$, with a critical point, that is definable from parameters in $V$.

The essence of the proof is the classical observation that the concept of being a Reinhardt cardinal, if consistent, cannot be first order expressible, since if $\kappa$ is the least Reinhardt cardinal, witnessed by $j$: $V$ $\rightarrow$ $V$, then by elementarity $j$($\kappa$) would also be the least Reinhardt cardinal, contrary to $\kappa$ $\lt$ $j$($\kappa$). Indeed, for the same reason, there can be no consistent first-order property $\varphi$($\kappa$) implying that that $\kappa$ is Reinhardt?

also this, from the paragraph below the proof of Theorem 32:

The proof of Theorem 32 worked by observing that if $j$: $V$ $\rightarrow$ $V$ is definable in $V$, even with parameters, then the concept of being Reinhardt with respect to that definition for some parameter is first order expressible [this seems to imply that there are two tiers of functions in $NGB$--one tier for sets and one tier for proper classes (following Bernays' two separate membership relations-- $\in_{set}$ for sets and $\in_{class}$ for classes?)--my comment].

Question: What is the second-order formula that expresses for some cardinal $\kappa$, that "$\kappa$ is a Reinhardt cardinal", and is that second-order formula expressible in $NBG$?

Finally (just to note), I am using the Hamkins, Kirmayer, Prelmutter understanding of $NGB$ ( i.e., $NGB$ without Choice or Global Choice). Others, (such as Yurii Khomskii, in his slide presentation, "Alternative set theories" (pdf)) hold that Global Choice is a class axiom of $NGB$ (on the other hand, Khomskii holds that Replacement is a set axiom—does that mean that Replacement does not hold for proper classes in $NGB$?)

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  • $\begingroup$ Why the downvote? $\endgroup$
    – Thomas Benjamin
    Commented Feb 6, 2020 at 22:46
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    $\begingroup$ The answer is essentially contained in "Generalizations of the Kunen Inconsistency" starting at the last paragraph of page 5 and continuing until the end of the metamathematical preliminaries. $\endgroup$
    – Gabe Goldberg
    Commented Feb 16, 2020 at 23:27
  • $\begingroup$ @GabeGoldberg: Page 5? In the version published in Annals of Pure and Applied Logic, the quote that got rid of the downvote was on the third page of the published version. Are you referring to page 5 in some preprint version of the paper? If so, then could you supply a link to the preprint in question? I would be much obliged if you would. Thanks. I am also interested in what would happen if one added the Wholeness axiom as a class axiom of $NGB$ ($NGBC$ as well). Because "$NGB$ includes the replacement...axiom only for formulas having only first-order quantifiers, that is, $\endgroup$
    – Thomas Benjamin
    Commented Feb 18, 2020 at 22:38
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    $\begingroup$ I'm referring to the preprint on arXiv, which you linked to in the question. The only problem in formulating the axiom is to assert that a class $j$ is an elementary embedding from $V$ to $V$, since it seems to require a truth predicate. Once you know the trick for doing this, you just say "there is a class $j$ that is an elementary embedding from $V$ to $V$." The trick is explained in great detail in "Generalizations of the Kunen Inconsistency." $\endgroup$
    – Gabe Goldberg
    Commented Feb 19, 2020 at 2:58
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    $\begingroup$ @GabeGoldberg: The arXiv preprint link was part of David Robert's fine edit (thanks, Prof. Roberts). Thanks for pointing it out as well, Prof. Goldberg. The trick is, I presume, is to use Gaifman's Lemma (Lemma 2 in both papers) to show that "a class $j$ from $V$ to $V$ that is an elementary embedding is a fully elementary embedding? Can one then use Gaifman's Lemma to show the following relative consistency results: "If $Con$($NGB$) then $Con$($NGB$ + $WA_{\infty}$)", "If $Con$($NGBC$) then $Con$($NGBC$+ $WA_{\infty}$)"? $\endgroup$
    – Thomas Benjamin
    Commented Feb 19, 2020 at 23:58

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