What are examples of models for which the Continuum Hypothesis is true/false?

Question

I'm not a set theorist so pardon my improper language. I'm trying to make sense of the unprovability of the Continuum Hypothesis. What I've come to understand is this: since set theory is broad and capable of modeling many different mathematical structures, CH can taken as either true or false, depending on what we're working with. This is leading me to think -- shouldn't there be specific examples of structures modeled by set theory where CH is false, and therefore an actual example of an uncountable set smaller than $\mathbb{R}$ ? I'm not sure if I'm just understanding this wrong; if that's the case then please let me know! But otherwise, what I'm asking is: what is an example of a set of this cardinality?

Answer 1

Given a model V, you can always look at all the constructible sets within V (this is called L) and L satisfies CH. Using forcing techniques, one can create a model V[G] that satisfies "2^omega = w_2" (i.e. there is precisely one uncountable level (w_1) which is smaller than w_2 (which is the size of the real numbers).

I need to say that to get to the point where one can understand this "generic model V[G]" that satisfies "CH is false" is years of study