It is well known that working in the frame of $\text{ZF}$, the Generalized Continuum Hypothesis ($\text{GCH}$) implies the Axiom of Choice ($\text{AC}$), i.e. $\text{ZF}+\text{GCH}\vdash \text{AC}$. 


But if we consider $\text{GCH}$ as a theory with ordinal many statements like $\text{GCH}=\{\text{CH}\_{\alpha}~|~\alpha\in \text{Ord}\}$ such that $\text{CH}\_{\alpha}$ is the statement $2^{\aleph\_{\alpha}}=\aleph\_{\alpha +1}$, then there is a natural question as follows:


**Is assuming *all* of these *strong* statements really necessary to prove a *weak* proposition like Axiom of Choice?** 


Precisely: 


**Question (1):** Is there a class $\text{C}\subsetneq \text{Ord}$ such that: 


**(a)** The assumption $\{\text{CH}\_{\alpha}~|~\alpha\in \text{C}\}$ is *strictly weaker* than the assumption $\{\text{CH}\_{\alpha}~|~\alpha\in \text{Ord}\}$, i.e. 


$\text{ZF}+\forall \alpha\in \text{C}~~~\text{CH}\_{\alpha}\nvdash \forall \alpha\in \text{Ord}~~~\text{CH}\_{\alpha}$ 


**(b)** The assumption $\{\text{CH}\_{\alpha}~|~\alpha\in \text{C}\}$ is *sufficient* to prove $\text{AC}$, i.e.


$\text{ZF}+\forall \alpha\in \text{C}~~~\text{CH}\_{\alpha}\vdash \text{AC}$


**Question (2):** If the answer of the question (1) is positive, can we choose $\text{C}$ to be a *set* not a *proper class*?


**Question (3):** What are the *minimal* classes (by inclusion order) like $\text{C}$ in the question (1)?



Minimal Generalized Continuum Hypothesis & Axiom of Choice
All you need for AC in the standard argument from GCH is that the GCH holds for an unbounded class of cardinals. The reason is that this is sufficient to conclude that any set of sets of ordinals is well-orderable, and this is sufficient to imply AC. 


So the answer to question 1 is yes; any unbounded class $C$ suffices. 


Meanwhile, merely knowing the GCH holds for cardinals below some cardinal is insufficient, since one can build the analogue of the symmetric models for $\neg\text{AC}$ above any cardinal, while preserving GCH below. Thus, there also can be no minimal $C$, since every sufficient $C$ is unbounded, and we may omit any proper initial segment of it and still have a sufficient $C$. 


