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Let $T$ be a theory formalized in first order logic with equality and membership and the additional primitive constant symbol $W$, with the following axioms:

  1. Extensionality: $\forall z (z \in x \leftrightarrow z \in y) \to x=y$

  2. Class comprehension: if $\phi$ is a formula in which $x$ doesn't occur free, then all closures of: $$\exists x \forall y (y \in x \leftrightarrow \phi \land \exists z (y \in z) ) $$; are axioms.

    Define: $x=V \iff \forall y (\exists z (y \in z) \to y \in x)$

  3. Sub-world: $W \in V$

  4. Resemblance: if $\varphi^X$ is a formula in prenex normal form, whose matrix is in $L(=,\in)$, and each variable in its prefix must appear as bounded either $\in X$ or $\subseteq X$, then: $$\varphi^V \leftrightarrow \varphi^W $$, is an axiom.

  5. All axioms of "$\sf ZF + \text {CH fails everywhere}"$ relativized to $W$.

What's the consistency strength of $T$?

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    $\begingroup$ Why do you keep asking this question? $\endgroup$
    – Monroe Eskew
    Commented Mar 19, 2020 at 22:15
  • $\begingroup$ @MonroeEskew, that's the first time I asked this question. I never asked about the effect of resemblance on global failure of choice. $\endgroup$
    – Zuhair Al-Johar
    Commented Mar 19, 2020 at 23:18
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    $\begingroup$ Zuhair, I think @Monroe means that a lot of your questions are quite similar: take a theory, vary some axiom scheme, ask for consistency strength. To a professional set theorist, these questions are kind of odd. We don't see the picture in your head that motivates this sort of research, and at some point it starts to become repetitive for no good reason. We know that you're a hobbyist, and I think it's great people are interested in this on a hobbyist level. But seeing how this is not what set theorists normally do, seeing a continuous flow of these questions is just... befuddling. $\endgroup$
    – Asaf Karagila ♦
    Commented Mar 20, 2020 at 8:49
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    $\begingroup$ Based on the fact that among your last 30 questions only 8 received an answer and you get +6 summing all of their up and downvotes, the other users of the website don't appear to be interested in such questions for their own sake. As Asaf was saying those theories appear arbitrary/unmotivated, usually set theorists start by studying a problem interesting by itself which then turns out to have high consistency strength, they don't sit around randomly modifying axioms until they stumble in an interesting one. Of course you can ask any question, but if you never get answers you should wonder why $\endgroup$
    – Alessandro Codenotti
    Commented Mar 20, 2020 at 11:35
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    $\begingroup$ It seems that you are just imagining different axiom systems and telling us about them, while putting the question, "What is the consistency strength?" at the end to justify these being mathoverflow questions. I think this is not an appropriate use of mathoverflow, because it is not really about posing and solving mathematical problems. Moreover, "What is the consistency strength?" is not in general a well-defined question because the set of all theories is not linearly ordered or well-founded in the relative consistency partial order. $\endgroup$
    – Monroe Eskew
    Commented Mar 20, 2020 at 16:36

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