If $\kappa$ is a strongly compact cardinal, then the singular cardinal hypothesis holds above $\kappa$. Hence the existence of large cardinals at the level of "strongly compact" or above is incompatible with even (apparently) mild [large powerset axioms](https://mathoverflow.net/questions/164673/why-isnt-there-more-interest-in-large-powerset-axioms) like "$2^\kappa$ always strictly exceeds $\kappa^+.$"


This raises the question:



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> Is there a "large powerset axiom" $\varphi$ so extreme that $\mathrm{ZFC}+\varphi$ disproves the existence of strongly inaccessible cardinals? Let us also require that $\mathrm{ZFC}+\varphi$ does not prove $\neg \mathrm{Con}(ZFC + \varphi).$
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Is there a "large powerset axiom" so extreme that it disproves the existence of strongly inaccessible cardinals?
Foreman's maximality principle is as you have requested, though it is not yet known if it is consistent or not.


**Foreman's maximality principle:** Any non-trivial forcing notion either it adds a real or colapses some cardinals.


It follows from it that:


1) $GCH$ fails everywhere,


2) there are no inaccessible cardinals.


This principle is stated in the following paper:


Foreman, Magidor, Shelah, **"$0^\sharp$ and some forcing principles"**, J. Symbolic Logic, 51 (1986) 39-46.


See also "[Questions about $\aleph\_1-$closed forcing notions](https://mathoverflow.net/questions/64280/questions-about-aleph-1-closed-forcing-notions)".


