We know that if $V=L$ holds, then $|\cal{P}(\omega)|=|\cal{P}(\omega)\cap \textrm{L}|=\aleph\_1$ whereas, in the presence of a measurable cardinal (in fact, even Ramsey) $|\cal{P}(\omega)\cap \textrm{L}|=\aleph\_0$. I remark that the cardinalities are of course computed in (the corresponding) $V$.


The first is just the fact that the constructible universe satisfies CH, while the second has to do with the fact that in the presence of a measurable, $\omega\_1^{L}<\omega\_1$ i.e. the existence of large cardinals makes the relative $\omega\_1^{L}$ "drop" below its "maximum possible" value (which is attained, if you want, in the "extreme case" when $V=L$).


My question is, what can we say, in general, about the beaviour of $\omega\_1^{L}$ given axioms of increasing strength above (or equal to, in strength) $V\neq L$? In particular, what happens if we just assume $V\neq L$?



Large cardinals and constructible universe
Each of the following implies that (the true) $\omega\_1$ is inaccessible in $L$, and hence that there are only countably many constructible reals: 


* The proper forcing axiom
* There is a Ramsey cardinal
* $0^\#$ exists
* All projective sets are Lebesgue measurable
* All $\Sigma^1\_3$-sets are Lebesgue measurable


(EDIT: These are just some of the well-known examples that came to my mind. This list is neither exhaustive nor canonical.)


The mere existence of a nonconstructible set, or even a nonconstructible real, does not imply that $\omega\_1^L$ is countable. There are many forcing notions in $L$ which do not collapse $\omega\_1$: adding one or many Cohen reals, destroying Souslin trees, etc. Each such forcing (over L) results in a model where $\omega\_1=\omega\_1^L$. 


In fact, "Martin's axiom plus continuum is arbitrarily large" is consistent with $\omega\_1^L=\omega\_1$. (But also with $\omega\_1^L<\omega\_1$.) 


ADDED: Preserving $\aleph\_1$ of the ground model (which may or may not be the constructible universe $L$) is a key component in many independence proofs concerned with the theory of the reals. The 
"countable chain condition", which is enjoyed by all the forcings I mentioned above, is a property of forcing notions that guarantees preservation of $\aleph\_1$; there are several other (weaker) properties which also suffice, most prominently (Baumgartner's) "Axiom A" and (Shelah's) "properness".


