[FOM] Two questions on Set Theory

[Adrian-Sorin Duma]

For some Nonlinear Functional Analysis purposes, I need an axiomatic set 
theory (Tarski-Grothendieck ?) that allows the existence of a strongly
innacessible d such that d<2^|{c<d, c strongly innacessible}|.

Also, I have the following (unrelated) question: are there some "exotic"
set theories (say, with a weakened Extensionality Axiom), allowing
"many void sets" (i.e., a "hierarchy of emptiness")?

Any comment will be highly appreciated.
Best regards.

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