This is a question about ZFC (or maybe NGB), but it is motivated by Randall Holmes' article on alternative set theories, especially his elaborations on TST, bounded Zermelo set theory and pocket set theory. Pocket set theory seems to derive its strength from the limitation of size axiom:
(limitation of size) – A class is a proper class if and only if it is equinumerous with all proper classes.
I read that NBG also verifies limitation of size, and that NBG + the generalized continuum hypothesis (GCH) is equiconsistent with ZFC. I don't really like the generalized continuum hypothesis, but it seems to be an adequate way to express the analog of "limitation of size" (and some intuitions about a size based set theory) for "sub-universes" of NGB.
Question: Is bounded Zermelo set theory + limitation of size + GCH identical to NBG + GCH?
This question is slightly ill posed, because bounded Zermelo set theory doesn't have proper classes, hence it is unclear what is meant by adding limitation of size. Maybe a "better" question would be
Question: Is bounded Zermelo set theory + GCH identical to ZFC + GCH
Bonus question: What can be said about the consistency strength of pocket set theory?
The idea behind this question is that bounded Zermelo set theory is equiconsistent with the simple theory of types TST (or typed set theory), and TST essentially embodies the philosophical position of Frank P. Ramsey and Rudolf Carnap, who accepted the ban on explicit circularity, but argued against the ban on circular quantification. This gives a very good reason to expect that bounded Zermelo set theory is consistent, but how does this relate to ZFC? The "limitation of size" principle seems (to me) like an impredicative principle with the potential to lead to a set theory equiconsistent with ZFC. The formulation via GCH is "ugly", hence I felt the need for these excessive explanations and motivations.