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)?

i.e.") – Trevor Wilson Nov 30 '13 at 17:20single statement$\text{GCH}$ as atheorywith class many sentences. Obviously when we want to write it formally we should use legitimated notation like what I used in (a) and (b). – Saint Georg Dec 1 '13 at 4:04