I need the proof or a reference for the proof that Sen's $\gamma$ and $\alpha$ imply rationalizability of choice (iff?).
Moreover, on Rubinstein's notes, rationalizability can be shown by Sen's $\alpha$ and the condition that the domain of the choice function contains at least all subsets of X of cardinality 3 or less. I cannot understand this last condition and how it "replaces" Sen's $\gamma$
A brief discussion on the extra assumption implied by Sen's $\beta$, which makes the choice rationalizable under preference relation would be useful.
[I have already posted the question of economics stack, but it seems it is a rather technical question, I didn't get much success]
EDIT Def If $\forall A,B \subseteq X; x \in C(A) \cap C(B) \Rightarrow x \in C(A \cup B)$, then the choice rule $C(\cdot)$ satisfies Sen’s $\gamma$ Axiom.