I have been learning about Gödel's constructible universe $L$, and the various reasons set theorists and mathematicians in general sometimes think it isn't the "real" universe of sets, to whatever extent such a thing exists. I was interested to see a comment in this answer from Andreas Blass:
Continuing Andres Caicedo's comment about "a strong sense in which $V \neq L$: If you accept this idea (as most set theorists do), then just imagine flipping a fair coin infinitely often. The resulting binary sequence (representing heads as 0 and tails as 1) is, with probability 1, not in $L$.
I interpreted this partly as a rigorous statement following from a few common set-theoretic axioms expressed prior in the comment section (e.g. the existence of $0^\#$, the existence of certain large cardinals, and so on), and partly an informal statement to build intuition for why people may prefer to view $V \neq L$.
To try to make this rigorous, I am curious how strong the statement is that the set of reals in $L$ is of measure zero in $\Bbb R$. I guess that this is independent of ZFC, since if $V = L$ we have that the set of reals in $L$ is the entire set. If we add this as an axiom, how strong of a set theory do we get? Is this statement stronger than $V \neq L$ and weaker than the existence of $0^\#$? Or does it follow from $V \neq L$ (and thus equivalent to it)?
I am also curious if there is some meta-theoretical way to make intuitive sense of this idea, at least informally. Basically, although it is clear enough that the definable powerset isn't the true powerset, the idea of $L$ is to iterate this operation uncountably many times, and I don't have much intuition for how many reals that gets us. It seems like a reasonably common view that this, in fact, only gets us countably many reals, although $L$ "thinks" it's $\omega_1^L$-many. Trying to piece this together, this would follow from the theorem that $\omega_1^L$ is the maximum ordinal at which $L$ generates new reals, and also that there is (apparently, somehow) a real encoding $\omega_1^L$ as an order-type. The existence of the latter is clearly only possible if $\omega_1^L$ is countable, unless, perhaps, if the real has an uncountably long binary expansion which the model thinks is countable (which would be possible in a non-well-founded set theory). Thus, if there is a model of ZFC which is well-founded, which has all of the ordinals in it, and also which has a real encoding the order-type of $\omega_1^L$, one could reasonably claim that there is some meta-theoretical sense in which $L$ does indeed generate very few reals. Is this the right idea?