Are there axioms for $\mathrm{ZFC}$ that imply that $\aleph_1$ is very large? A bit of philosophy: under the usual definition of the aleph numbers, ZFC proves the sentence "$\aleph_1$ is an ordinal." However, in some sense $\aleph_1$ isn't really an ordinal (in my opinion), because its position in the ordinals varies greatly between different models of $\mathrm{ZFC}$. Its not "fixed." (Of course, if you believe in a true set-theoretic universe, you will assert that its $\aleph_1$ is the true $\aleph_1$ and hence that there is a fixed, true ordinal corresponding to $\aleph_1$.)
Anyway, philosophy aside, I was wondering if there are axioms for $\mathrm{ZFC}$ that either directly assert, or otherwise imply that $\aleph_1$ is very large in the ordinals. Is this even possible? (I don't think the continuum hypothesis could reasonably be construed to be such an axiom, but please comment if you think otherwise.)
 A: As Andres points out in the comments, $\sf CH$ is not the actual player here.
Consider the case that there is an inaccessible cardinal $\kappa$ which is itself a limit of inaccessible cardinals. By forcing we can arrange $\kappa$ to be $\omega_1$. This will certainly say that $\omega_1$ is large in the sense that it is the limit of inaccessible cardinals in an inner model. Moreover if $X$ is real number, then either we can "compute" the entire collapse from $X$, or that $\kappa$ is inaccessible in the model generated by $X$.
But that is not a "natural axiom". Here is one, however, $0^\#$ exists. It means that $\omega_1$ is a very large cardinal in $L$ and it is the limit of equally large large cardinals.
Similar "sharps" imply the same thing. Consequentially, any large cardinal axiom which imply the existence of sharps will say that $\omega_1$ is "quite" large. For example a measurable cardinal implies the existence of $0^\#$, and since these things are upwards absolute, then we can say that having "an inner model with a measurable" (which is a reasonable large cardinal axiom) implies that as well.
A: By the downward Löwenheim - Skolem theorem there is a countable transiive model of every theory that holds in any set-sized transitive model and large cardinal axioms generally imply that there is a transitive set that satisfies all weaker large cardinal axioms. Thus the stronger the large cardinal axioms that hold in $V$ (or in sets) the larger $\omega_1$ since every ordinal in a countable transitive model is a countable ordinal, and thus less than $\omega_1$.
