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    $\begingroup$ (Turning my misinterpretation into a comment): Note that if $W$ were a relation symbol, this theory would be quite weak - in particular, consistent relative to PA. Let $M$ be a nonstandard model of PA, $\alpha$ a nonstandard $Ackermann(M)$-ordinal, consider the the expansion of $(V_{\alpha+1})^{Ackermann(M)}$ by interpreting $W$ as the well-founded part of $Ackermann(M)$, and note that set comprehension follows from overspill. Having $W$ an actual object changes things: e.g. $W$ satisfies Infinity since it's the smallest adjunction-closed set. $\endgroup$
    – Noah Schweber
    Commented Jun 17, 2019 at 20:19
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    $\begingroup$ Have you proved that $W$ is a model of ZFC in your theory? Or do you believe you can prove this? $\endgroup$
    – Will Sawin
    Commented Jun 17, 2019 at 20:53
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    $\begingroup$ If I'm not mistaken, you can construct a model of your theory from ZFC with an inaccessible. If $\kappa$ is inaccessible, you let the universe be $V_{\kappa+1}$ and let $W$ be $V_\alpha$ for $\alpha<\kappa$ such that any ordinal less than $\kappa$ definable in $V_{\kappa+1}$ with parameters from $V_\alpha$ is less than $\alpha$. (You can prove such an $\alpha$ exists using inaccessibility of $\kappa$.) $\endgroup$
    – Eric Wofsey
    Commented Jun 17, 2019 at 22:24
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    $\begingroup$ These axioms are weird in that they imply the existence of many elements that are not sets (since $W$ needs to not be definable with parameters from $W$), but say very little about such elements. For instance, if you have any model and a class $C$ in the model which is not an element, it seems you can get a new model by just making $C$ an element: i.e., for every class $D$ that you had, add a new class $D\cup\{C\}$. (Although I suppose if $C$ is not definable with parameters from $W$, this might somehow mess up set comprehension.) $\endgroup$
    – Eric Wofsey
    Commented Jun 18, 2019 at 2:54
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    $\begingroup$ @NajibIdrissi, to my trivial knowledge I've first presented here as a question. I don't know if it was on table before this attempt. I've presented before a similar axiomatization to FOM but with implication instead of the biconditional in set comprehension, and it was only the two schemas. $\endgroup$
    – Zuhair Al-Johar
    Commented Jun 18, 2019 at 14:15