Is $2^\alpha=2^\beta\Rightarrow \alpha=\beta$ a $\sf ZFC$-independence result? In a lecture recently one of my lecturers was proving something to do size of basis or something (I can't remember exactly) and somewhere near the end of the proof we had the following:
$2^\alpha=2^\beta\Rightarrow \alpha=\beta$
Now this was just in a proof about finite (or possibly countable) things so everything is cool here but this rang a bell a a results that may be independent of ZFC or something which I couldn't quite recall. I had a look but I couldn't find anything about it?
Thanks for the help

Yes, this is very much independent of the axioms of $\sf ZFC$. It is a consequence of $\sf GCH$, but not equivalent.
It is possible to have $2^{\aleph_0}=2^{\aleph_1}$.
On the other hand, we can have that for finite $n$, $2^{\aleph_n}=\aleph_{n+3}$, and otherwise $2^{\aleph_\alpha}=\aleph_{\alpha+1}$. In that case, clearly $2^\kappa=2^\lambda\implies\kappa=\lambda$, but $\sf GCH$ fails.
