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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 $V\neq L$?

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up vote 11 down vote accepted

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".

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Also, my favorite, if every uncountable $\Pi^1_1$-set has a perfect subset then $\aleph_1$ is inaccessible in $L$. –  Dave Marker Jul 12 '11 at 14:06
    
Thanks! This result (of Solovay? or is it older?) predates Shelah's theorem on $\Sigma^1_3$-measurability.Was this perhaps the first large cardinal whose existence follows from a statement in descriptive set theory? –  Goldstern Jul 12 '11 at 21:29
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According to Kanamori's book (p.135) Specker gets at least part of the credit for the result mentioned by Dave. –  Ali Enayat Jul 12 '11 at 23:55
    
Thanks a lot everybody! –  kvagk Jul 13 '11 at 10:40
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