Is the following statement consistent in $\mathsf{ZFC}$?



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> > For every ordinal $\beta$ there is an ordinal $\lambda\_0$ such that for all ordinals $\lambda\geq\lambda\_0$ we have $2^{\aleph\_{\lambda}}\geq \aleph\_{\lambda+\beta}.$
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Is it consistent that the gaps between cardinals $\kappa$ and $2^\kappa$ "get larger and larger"?
Yes, in the Foreman-Woodin model for the global failure of $GCH$ your statement is true.


See [The generalized continuum hypothesis can fail everywhere.](http://www.jstor.org/stable/2944324?origin=crossref&seq=1#page_scan_tab_contents) 


