If consistent, is existence of a proper class of extendible cardinals provably equivalent to a $Σ^V\_5$ statement?


Recall that in ZFC, a cardinal $κ$ is extendible iff for every $λ>κ$ there is an elementary embedding $j$ of $V\_λ$ into some $V\_μ$ with $\mathrm{crit}(j)=κ$. (In ZF, one also requires $j(k)>λ$, but this does not affect the above definition in ZFC.)


Extendibles are unusual in being a natural large cardinal property with high quantifier complexity (see this [question](https://mathoverflow.net/questions/281894/examples-of-statements-with-a-high-quantifier-complexity) for other examples of high quantifier complexity). The property of being some $V\_μ$ is $Π^V\_1$ (or $Π\_1$ for short), being $λ$-extendible is $Σ\_2$, and being extendible is $Π\_3$, and so existence of an extendible cardinal is $Σ\_4$, and existence of a proper class of extendibles is $Π\_5$.


In the other direction, if $κ$ is extendible, then $V\_κ ≺\_{Σ\_3} V$, and thus (assuming consistency) existence of an extendible cardinal is not equivalent in ZFC to a $Π\_4$ statement. Analogously, I expect that existence of a proper class of extendibles is not provably (in ZFC) $Σ\_5$, but it is easy to be confused by quantifier counting.


Offered in [Q/A format](https://mathoverflow.net/help/self-answer) as I was able to solve this question while working on its exposition.



Complexity of a proper class of extendibles
No, it is not; existence of a proper class of extendible cardinals is not provable in ZFC from any consistent $Σ^V\_5$ statement.


Assume a proper class of extendibles and let $φ$ be a true $Σ\_5$ statement (or even the conjunction of all true $Σ\_5$ statements), and let a set $S$ witness $φ$, and let $κ$ be the least extendible above $S$. Then $V\_κ$ satisfies $\mathrm{ZFC} + φ \, +$ non-existence of a proper class of extendibles. ($V\_κ ≺\_{Σ\_3} V$, so $V\_κ$ satisfies all true $Π\_4$ statements with parameters in $V\_κ$, and by $S∈V\_κ$, $V\_κ ⊨ φ$.)


$V\_κ ≺\_{Σ\_3} V$ is a standard fact about extendibles (as an aside, despite its similarity, being supercompact is $Π\_2$) and follows from existence of arbitrarily high $λ$ such that $V\_κ ≺ V\_λ$ (or just $V\_κ ≺\_{Σ\_3} V\_λ$): Let $λ$ with $V\_κ ≺ V\_λ$ be large enough. By Downward Löwenheim-Skolem, $V\_λ ≺\_{Σ\_1} V$, so $V\_λ$ satisfies all true $Π\_2$ statements with parameters in $V\_λ$, so (since $λ$ is large enough relative to $κ$) $V\_λ$ (and thus $V\_κ$) satisfies all true $Σ\_3$ statements with parameters in $V\_κ$, so $V\_κ ≺\_{Σ\_3} V$.


