If some statements below are too imprecise/peculiar, please note that this is mostly due to my own lack of knowledge/understanding. Nevertheless, I will try to phrase the actual question in a more precise way.

**Background:**
Consider a statement of the form: $\forall x \in \omega_1 \, [P(x)]$. Suppose that we don't want all statements with the form in previous sentence to be necessarily either true or false. I suppose the idea here is that $\omega_1$, since it isn't countable, should be consider "too big" (so similar points would also apply to $\omega_2$, $\omega_3$ etc.).  

Now, I should perhaps mention here (at the risk of sounding very very confusing) that I have read (at few occasions) suggestions that something like $\omega_1$ should be treated as a proper class. I suppose I see some similarity of this suggestion with previous paragraph that perhaps (very loosely) we want to say that: "$\omega_1$ is something like $\mathrm{Ord}$". However, in terms of actual logic, I have no idea whether it does or does not relate to what I wrote in previous paragraph (I simply don't have enough facility with logic for this). 

I have added the previous paragraph that perhaps if someone wants to enlighten the link (in the case there is one) between the previous two paragraphs for laymen, it would be quite useful.

**Actual Question:** Consider $\mathrm{ZF}$ set theory where: **(1)** we remove the power-set axiom **(2)** we add the constructibility axiom. I want to know what is known about the proof-theoretic work along the lines of such a theory as a whole or perhaps some of its "fragments". Not sure if "fragments" is the right word but what I want to imply, loosely speaking, is weaker subsystems of the theory (that try to cover some important parts of it).

Also, a silly side question (which unfortunately also reveals my lack of understanding/knowledge). Since we are also removing power-set axiom (on top of adding constructibility), does the removal or non-removal of $\mathrm{AC}$ make any difference to the question in previous paragraph? My uneducated "guess" is probably not, but please correct if it does.