This is not my area but a question occurred to me that I can not find the answer to. There is a very strong axiom of constructibility which ironically gives us highly non-constructive math (GCH is one of its implications). What would be an equally strong axiom in the opposite direction? And I mean direction in a philosophical sense, so what would be the strongest axiom that constructivists/intuitionists would approve of?

My first idea was to find the largest $\kappa$ such that $2^{\aleph_0} = \aleph_{\kappa}$ is consistent with ZF but this set is unbounded ($\kappa$ can be any finite number) and $2^{\aleph_0} < \aleph_{\omega}$. Which brings up the question, how much fundamental difference are there between CH and $2^{\aleph_0} = \aleph_{118}$ for example?