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I've seen $\mathsf{ZFC2}$ mentioned in a few questions such as this one and I'm curious about ways to axiomatize it that have the fewest differences from plain $\mathsf{ZFC}$ as possible. My question is similar to this one, but proposes an explicit theory and is specifically about minimizing the differences from first-order $\mathsf{ZFC}$. This post to a mailing list mentions that $\mathsf{ZFC2}$ has as axioms the second-order versions of $\mathsf{ZFC}$ axioms, but doesn't go into much detail about what exactly that means.

For the purposes of this question, I'm assuming that there's rough consensus on what $\mathsf{ZFC2}$ really is as a theory and am wondering if my naive construction is equivalent to it.

For the sake of concreteness, let the language second-order logic be the same as the language of first-order logic but with second order quantifiers $\forall x : R(n)\mathop. \cdots$ and $\exists x : R(n) \mathop. \cdots$, which quantify over relations of arity $n$. A bare quantifier $\forall x \mathop. \cdots$ is a first-order quantifier.

Does naively changing the axiom schema of $\mathsf{ZFC}$ so that $\varphi$ ranges over all second-order formulas give an equivalent theory to $\mathsf{ZFC2}$?

$\mathsf{ZFC2}$ has two axiom schemas: the axiom schema of specification and the axiom schema of replacement.

If we treat $\forall x \mathop. \cdots$ and $\exists x \mathop. \cdots$ as implicitly first-order quantification, we can trivially transfer all the non-schematic axioms of $\mathsf{ZFC}$ to a second order setting.

If we then take another look at the axiom schema of specification and the axiom schema of replacement and allow the synctactic variable $\varphi$ to range over all the second-order sentences, do we get an axiomatization of $\mathsf{ZFC2}$ or a theory equivalent to $\mathsf{ZFC2}$?

Greg Nisbet
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    (This is a duplicate, but let me very briefly summarize the situation:) "Schematic" theories such as $\mathsf{PA}$ or $\mathsf{ZFC}$ can be thought of as general instances, corresponding to first-order logic, of broader "logics-to-theories" maps. In the case of the "ZFC-generalizer" $\mathcal{ZFC}$ (so $\mathsf{ZFC}=\mathcal{ZFC}(FOL)$), it is consistent that $\mathcal{ZFC}(SOL)$ has a countable model and so is very different from the theory commonly called "second-order ZFC." On the other hand, $\mathsf{ZFC}$ + "$\mathcal{ZFC}(SOL)$ has a model" is equiconsistent with an inaccessible. – Noah Schweber Jun 10 '21 at 02:39
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    In order, see: this answer of mine re: "schematic theories," this answer of Lietz re: countable models of $\mathcal{ZFC}(SOL)$, and this answer of Lietz (to the question of which this is a duplicate) re: the consistency strength bit. Incidentally, complementing the second link I showed that it is consistent that $\mathcal{ZFC}(SOL)$ has no countable models. – Noah Schweber Jun 10 '21 at 02:42
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    All in all, the situation is surprisingly subtle! – Noah Schweber Jun 10 '21 at 02:43
  • I'm really confused as to why this question is a duplicate. I'm trying to understand what the standard definition of $\mathsf{ZFC2}$ is and whether you can get it by extending specification and replacement with quantification over $\varphi$s with second order parameters, and quantifiers as well. I think the linked question is about an alternative $\mathsf{ZFC2}$? – Greg Nisbet Jun 10 '21 at 02:46
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    The posts I've linked show that your proposed change results in something rather different from what is commonly called "second-order ZFC" (unless I'm missing something, $\mathcal{ZFC}(SOL)$ is exactly the alternative theory your thinking about). Doesn't that answer your question? – Noah Schweber Jun 10 '21 at 02:47
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    The "alternative ZFC2" in the linked question is exactly the same as yours. – Eric Wofsey Jun 10 '21 at 02:48
  • The alternate $\mathsf{ZFC2}$ that my thing is equivalent to is $\mathcal{ZFC}(\text{Second Order Logic})$ and not $\mathsf{ZFC}^\text{scheme}_2$, right? I think I see it now. Thanks. – Greg Nisbet Jun 10 '21 at 03:09
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    @GregoryNisbet $\mathcal{ZFC}(SOL)$, $\mathsf{ZFC}_2^{def}$, and $\mathsf{ZFC}_2^{scheme}$ are three names for the same thing; this thing is different in a rather complicated way from second-order ZFC (which is usual ZFC with powerset replaced with essentially "$\forall x\exists y\forall Z(Z\subseteq x\leftrightarrow \exists z(z\in y\wedge z=Z))$" and replacement replaced with essentially "$\forall x,F\exists y(y={F(z):z\in x})$," where I'm using capital vs. lowercase for second-order vs. first-order variables). – Noah Schweber Jun 10 '21 at 03:14
  • @NoahSchweber Thank you for the explanation and for your help. In the powerset axiom, it's interesting that equality can cross sort boundaries ... I've never seen that before. – Greg Nisbet Jun 10 '21 at 03:22
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    @GregoryNisbet Well, it's shorthand. You have the built-in-logic elementhood relation connecting second-order and first-order objects, call it "$\varepsilon$," and the usual non-logical binary relation symbol of set theory, call it "$\in$." Then "$A=a$" is an abbreviation for "$\forall b(b\varepsilon A\leftrightarrow b\in a)$." (FWIW there may be interesting subtleties around second-order set theories without extensionality, I don't know much about non-extensional theories at all.) – Noah Schweber Jun 10 '21 at 03:32

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