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In the paper

Vickers, J.; Welch, P. D., On elementary embeddings from an inner model to the universe, J. Symb. Log. 66, No. 3, 1090-1116 (2001). ZBL1025.03049.

it is stated to that if $Ord$ is Ramsey (I.e. there is a proper class $I\subseteq Ord$ of good indiscernibles), then there is a definable class $M$ and some $j: M\prec V$. But, according to Generalizations of the Kunen Inconsistency, there can be no $j: M\prec V$ for $M$ a definable class?

Does this mean that measurable cardinals are inconsistent, or is something else going on?

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    $\begingroup$ Something else is going on. In Vickers-Welch, the embedding $j : M \to V$ and the class $M$ are only definable from the class $I$ of indiscernibles witnessing that $\text{Ord}$ is Ramsey. But $I$ itself need not be definable. $\endgroup$
    – Gabe Goldberg
    Commented Oct 30, 2019 at 2:31
  • $\begingroup$ @Gabe: Are you sure that measurable cardinals are inconsistent, and just nobody noticed for 20 years? ;-) $\endgroup$
    – Asaf Karagila ♦
    Commented Oct 30, 2019 at 6:11
  • $\begingroup$ Generalizations of the Kunen inconsistency: arxiv.org/abs/1106.1951. Theorem 28 asserts that if M is definable, then there is no elementary embedding $j:M \to V$. $\endgroup$
    – Joel David Hamkins
    Commented Oct 30, 2019 at 13:37

1 Answer 1

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There is no contradiction here.

Look at Theorem $2.3$:

Suppose $I\subseteq On$ witnesses $On$ is Ramsey. Then, definably over $\langle V,\in, I\rangle$, there is a transitive class $M$, and an elementary embedding $j :\langle M,\in\rangle\rightarrow\langle V,\in\rangle$ with $j \not= id$.

Note that $I$ is involved as a parameter in the definition of $M$ and $j$. But definability in the Generalizations paper means definability in $\langle V,\in\rangle$ alone.

Contrast this with the sentence

conversely, if $On$ is Ramsey, then such a $j, M$ are definable

from the abstract, where the dependence on $I$ is unstated. "Definability" is being used as a shorthand for "definability from witnesses to the relevant hypotheses." I personally dislike this and in my opinion the abstract is a bit unclear; that said, I understand the impulse to abbreviate results in the abstract, and the corresponding theorem in the body of the paper is clearly stated. I think the takeaway is that results in the abstract or introduction should never be completely trusted (especially when taken at face value they imply something glorious which isn't itself stated).

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