Assuming the axiom of choice it is very easy to see that $\aleph\_1$ is a regular Joe of a successor cardinal. It is not very large in any way except the fact that it is the first uncountable cardinal.


If however we begin with a model of ZFC+Inaccessible, we can construct models of ZF in which $\aleph\_1$ is somewhat inaccessible in the sense that $\aleph\_1\nleq 2^{\aleph\_0}$ If, on the other hand, we start with a model of ZF whic has this property then there exists an inner model with an inaccessible cardinal.


It can be that $\aleph\_1$ is a measurable cardinal, you can even have that every subset of $\omega\_1$ contains a club, or is non-stationary; and it is possible for $\aleph\_1$ to have the tree property (I only know of models by Apter in which all successor cardinals have the tree property; but that would require a proper class of very large cardinals).


In general we say that $\aleph\_1$ is P-large for a large cardinal property P, if it is consistent with ZF that $\aleph\_1$ has property P, and from such model we can produce a model of ZFC+$\kappa>\aleph\_0$ has property P. 


**Question:** Is there a limit on how P-large can $\aleph\_1$ be? (e.g. P can be tree property/$\kappa$-complete ultrafilter/supercompact measures/etc.) and are there properties P such that for $\aleph\_1$ to have them we require *more* than ZFC+P?



What sort of large cardinal can $\aleph_1$ be without the axiom of choice?
*-----edited to include corrections, thanks Joel and Tanmay-----*


From a model of "ZFC + $large(\kappa)$" you can get a model of "ZF + $large(\kappa)$ + $\kappa=\omega\_1$" if $large(\ )$ is a large cardinal property that is preserved under small forcing and can be written in the form 


"for every set of ordinals $X$, there is a set $Y$ such that $\phi(X,Y)$ holds",
for some upwards absolute formula $\phi$,


so for example weakly compact, Ramsey, measurable. The model you'd use is basically Jech's model for making $\omega\_1$ measurable in ["$\omega\_1$ can be measurable"](http://link.springer.com/article/10.1007/BF02771215).


For details see the comments below, Tanmay's answer and my thesis ["Symmetric Models, Singular Cardinal Patterns, and Indiscernibles"](http://hss.ulb.uni-bonn.de/2012/2777/2777.pdf) Chapter 1, section 3.3. These large cardinals are there called "preserved under symmetric forcing".


For the other way around (a limit on how large can $large(\ )$ be), as Trevor Wilson said, measurable in ZF gives measurable in ZFC, and I guess similar arguments would work for large cardinal properties that are "preserved under symmetric forcing". 
I guess this is not a very good limit though and I can't come up with/remember something better right now. If I do I'll come back to answer. 


