Sorry if this is a silly question. I was wondering, under what axioms of set theory is it true that if $\alpha$,$\beta$ are cardinals, and $2^\alpha=2^\beta$, then $\alpha=\beta$? Do people use these conditions to prove interesting results?


This question is prompted from a recent perusing of Johnson's "Topics in the Theory of Group Presentations", where in the first few pages he "proves" free groups of different rank are non-isomorphic:
the number of mappings from a free group of rank $\omega$ to the group $\mathbb{Z}/2\mathbb{Z}$ is $2^\omega$, which would be invariant under isomorphism; and then he assumes the topic of my question: $2^\alpha=2^\beta$ implies $\alpha=\beta$.


But I remember reading something a few years ago about a student of R.L. Moore (I only remember his last names was Jones) "proving" the Moore Space conjecture, and using that $\alpha > \beta$ implied $2^\alpha > 2^\beta$, but that this was incorrect.


Anyway, I realized I don't know anything about when this is true or false, so I thought I'd ask.



When $2^\alpha = 2^\beta$ implies $\alpha=\beta$ ($\alpha,\beta$ cardinals)
François gives the correct affirmative answer. For the negative side, the usual method of proving that the negation of the Continuum Hypothesis is consistent with ZFC is to use the method of forcing to add Aleph2 many Cohen reals, so that 2ω = ω2 in the forcing extension V[G]. In this model V[G], it is also true that 2ω1 = ω2. Thus, this model shows that it is not necessarily true that different-sized sets have different sized power sets. The case of symmetric groups is likely more interesting than free groups, because in this model, the symmetric groups Sω and Sω1 have the same cardinality ω2. Nevertheless, these two groups are not isomorphic, as explained in [this MO question](https://mathoverflow.net/questions/12943). 


The general answer about what can be true for the continuum function κ --> 2κ is exactly provided by [Easton's Theorem](http://en.wikipedia.org/wiki/Easton's_theorem). This remarkable theorem states that if you have any class function E, defined on the regular cardinals κ, with the properties that 


* κ ≤ λ implies E(κ) ≤ E(λ)
* κ < E(κ)
* κ < Cof(E(κ))


then there is a forcing extension V[G] in which 2κ = E(κ) for all regular cardinal κ. In particular, this shows that the sizes of the power sets (on regular cardinals) are restricted to obey only and exactly the hypotheses listed explicitly above. Each of these properties corresponds to a well-known fact about cardinal exponententiation.


Using Easton's theorem, we can build models of set theory where 2κ = κ++ for all regular κ. The added power of the Woodin/Foreman result mentioned by François is that they also get this for singular cardinals. 


The point now is that there are innumerable examples provided by Easton's theorem that satisfy your hypothesis that the continuum function is one-to-one. If one begins with a model of GCH and selects *any* injective Easton function E, then the resulting model of set theory V[G] will have E as it's continuum function κ --> 2κ = E(κ) for regular κ, and the GCH will continue to hold at singular κ, preserving injectivity. So one is quite free to satisfy your hypothesis while having any kind of crazy failures of GCH. 


