>>10908877hey, I see you deleted your quesiton, this is gonna be confusing
>>10908916The ordinals are in V and L has some of the same, and those all have growing sizes at all jumps.
Now there are other models of ZF inside of V: For example the state V_{w+w} is a model of ZF and a relatively small one. w+w is just the second limit ordinal.
Basically, V_w is roughly all finite sets and if you take the power set operation on that (the first one such operation giving you the continuum) a countable number of times, then you get V_{w+w}. That's basically the world of normal math except abstract algebra.
(Because in abstract algebra you want to work with e.g. the cateogry of all groups (or weirder Grothendieck kind of objects) as a set and so that's not all you may want in pure math)
Now the so called construtible universe V is very small compared to L. If you look at the continuum (say 2^|w|) in V, then there's an intersection of that with L which makes you realize that the ZF model L "doesn't see all that V sees".
If you say V=L, then you deny the existence of those other sets - although those that you throw away aren't very tangible by the nature of L being exactly the "constructible ones".
In the class L, you also have that its 2^|w|=|w_1| where w_1 is the first uncountable ordinal. So CH is proven consistent with ZF by virture of L being a model.
But there are also models in V where it can be shown to be false, so it's unprovable in the theory (and therefore in V).
Basically, the continuum 2^|w| can be pretty much what you want, the subset notion is a dangerous one.
