@ZuhairAl-Johar the powerset of the powerset

About the question of isomorphic $\aleph_1$-dense subsets, Sierpinski showed that it is possible that there exists $2^{\aleph_0}$-isomorphism classes of $\aleph_1$-dense subsets of reals (Sierpinski assumed CH, but one should get similar result with the failure of CH by adding Cohen reals), on the other hand Baumgartner showed that PFA implies that all $\aleph_1$-dense subsets of real are isomorphic

@ZuhairAl-Johar $(\omega,+1,<)\in HPD$ and $HPD$ sees that $+1$ is injective, $0$ is not in the image and that $\omega$ is closed under $+1$ and that $<$ is well ordering, so yes (but anything after $Z_2$ pretty much falls flat). But note that we don't know whether $Th(HPD)$ is recursively axiomatible, so consistency strength doesn't mean much for now

@ZuhairAl-Johar what exactly do you mean in this context?

Considering that there is no reflective principles between $V_{\rho(x)}$ and $V$, I would imagine that not a lot, even ignoring the restriction on the parameters

I would believe that this is open

@ZuhairAl-Johar any proof in stratified-ZF will certainly transfer into ZF, as ZF is an extension of stratified-ZF, so it is also open in stratified-ZF

@ZuhairAl-Johar "it"=WPP=your version of the partition principle, that is: $AC⇒PP⇒CB^*⇒WPP$ and the opposite direction of those arrows are all open in ZF

@ZuhairAl-Johar IIRC, it is open whether it implies the dual CB theorem (which in turns it is open whether the dual CB theorem implies PP and it is open whether PP implies AC) in ZF

@ZuhairAl-Johar yes, for Reinhardt you need nee in the language of $\{∈,j\}$. If you have nee in the language of $\{∈\}$ you get a wholeness cardinal, (see also), which are weaker than $I3$ (I believe that the question of whether full wholeness axiom is strictly weaker than $I3$ or is it equivalent is open)