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  • $\begingroup$ Your “not quite the same thing as second-order ZFC” links to another question which itself links to another question in which I think the relevant part is a muddled discussion in the comments to the question. This is a bit hard to follow and nothing is said clearly. So, even if it's not directly germane to the present question, could you add a footnote (here, or to one of the linked questions) briefly explaining the difference between “ZFC with Separation and Replacement schemes modified to allow formulas of second-order logic” and “second-order ZFC” because I thought they were the same? 🙏 $\endgroup$ Commented Sep 24, 2022 at 20:58
  • $\begingroup$ @Gro-Tsen Second-order $\mathsf{ZFC}$ has the second-order powerset axiom and the second-order (single-sentence) replacement axiom, and is stronger than the first logic asked about in this question. In particular, set models of second-order $\mathsf{ZFC}$ are (up to isomorphism) just the $V_\kappa$s for $\kappa$ strongly inaccessible, whereas the class of models of $\mathfrak{ZFC}(\mathsf{SOL})$ is significantly harder to describe. $\endgroup$ Commented Sep 24, 2022 at 21:00
  • $\begingroup$ @Gro-Tsen (Sorry, "$\mathfrak{ZFC}$" should be "$\mathscr{ZFC}$" in my previous comment.) See also the first part of this old MSE answer of mine. $\endgroup$ Commented Sep 24, 2022 at 21:09
  • $\begingroup$ Aaaaah, you're saying “second-order ZFC” has versions of Separation and Replacement where instead of having a scheme over all (first-order) formulas we have a single axiom with the scheme replaced by a second-order quantification; whereas here you consider a scheme over all second-order formulas. Right? But I still fail to see why they're not equivalent: doesn't second-order comprehension ensure that anything we can write with a second-order formula define a second-order object, and conversely, any second-order object can be considered as a single-variable formula? $\endgroup$ Commented Sep 24, 2022 at 21:24
  • $\begingroup$ Ah wait, when you write a scheme ranging over second-order formulas, do you allow second-order parameters in the formulas in the scheme, which are then outwardly universally quantified just like the first-order parameters are? $\endgroup$ Commented Sep 24, 2022 at 21:25