Is the following statement consistent in $\mathsf{ZFC}$?
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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Sign up to join this communityIs the following statement consistent in $\mathsf{ZFC}$?
For every ordinal $\beta$ there is an ordinal $\lambda_0$ such that for all ordinals $\lambda\geq\lambda_0$ we have $2^{\aleph_{\lambda}}\geq \aleph_{\lambda+\beta}.$
Yes, in the Foreman-Woodin model for the global failure of $GCH$ your statement is true.
See The generalized continuum hypothesis can fail everywhere.