TITLE: Absoluteness of exponential function and forcing
QUESTION [1 upvotes]: I recall reading that the exponential function $ \alpha^{\beta}$ is absolute for transitive models of ZFC.
Is it true that if we have  $ 2^{ \alpha } < \beta $ in the ground model $V$, then $( 2^{ \check{ \alpha} } < \check{\beta})^{V[G]} $ holds as well?
Does it make sense to refer to  $ 2^{\check{\alpha}}$ as an ordinal in an extension model statement as I did above?

REPLY [2 votes]: Ordinal arithmetic is not changed between transitive models. And forcing does not change the ordinals. 
So $2^\alpha$ is just the same ordinal as it was before.
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