2 clarification in response to a remark by Carl Mummert

I think the answer to the question doesn't really have anything to do with how Cohen actually proved the independence of CH, it is more about what a model of set theory is.

There are some axioms defining what a group is. These axioms can be written in ordinary first-order propositional logic. A model of the theory is a set with a binary operation and a distinguished element such that it satisfies the group axioms. That is, a model of the theory is simply a group. Similarly, a model of set theory is simply a set with a relation (representing) such that it satisfies the axioms of set theory. Now there are theorems that follow directly from the group axioms. And these are just the things that hold for every model of the theory. They hold for every group. Other statements, such as "For all a and b, ab=ba" hold for some models but not for others. They are independent of the axioms.

Now, things are really the same in set theory. You can have models (S,e) and (S'.e') such that the continuum hypothesis holds in one (S,e), but not in (S',e). Now these are simply sets with a binary relation defined on them. Such sets exists since, loosely speaking, every consistent theory has a model.*

But this doesn't really tell us anything about the world of sets we are working with, apart from the fact that both could be true. All we know is that the theory we are working with has, provided it is consistent, a model. But there could be many models with different properties, and, since they are consistent with the axioms, we cannot use the axioms to identify a "true" model. In particular we cannot learn from them tha that something holding in a model is true, we can only use them to show something is not provable false (in our case, the CH and its negation).

*Usually, to make things simpler, one works with models which are closer to the real sets in that for $a,b\in S$ one has $a e b$ just in case $a\in b$ and includes sufficiently many elements such that one can actually treat them in some respects like usual sets. That is where the countable, transitive, standard models come from.

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I think the answer to the question doesn't really have anything to do with how Cohen actually proved the independence of CH, it is more about what a model of set theory is.

There are some axioms defining what a group is. These axioms can be written in ordinary first-order propositional logic. A model of the theory is a set with a binary operation and a distinguished element such that it satisfies the group axioms. That is, a model of the theory is simply a group. Similarly, a model of set theory is simply a set with a relation (representing) such that it satisfies the axioms of set theory. Now there are theorems that follow directly from the group axioms. And these are just the things that hold for every model of the theory. They hold for every group. Other statements, such as "For all a and b, ab=ba" hold for some models but not for others. They are independent of the axioms.

Now, things are really the same in set theory. You can have models (S,e) and (S'.e') such that the continuum hypothesis holds in one (S,e), but not in (S',e). Now these are simply sets with a binary relation defined on them. Such sets exists since, loosely speaking, every consistent theory has a model.*

But this doesn't really tell us anything about the world of sets we are working with, apart from the fact that both could be true. All we know is that the theory we are working with has, provided it is consistent, a model. But there could be many models with different properties, and, since they are consistent with the axioms, we cannot use the axioms to identify a "true" model. In particular we cannot learn from them tha something is true, we can only use them to show something is not provable false (in our case, the CH and its negation).

*Usually, to make things simpler, one works with models which are closer to the real sets in that for $a,b\in S$ one has $a e b$ just in case $a\in b$ and includes sufficiently many elements such that one can actually treat them in some respects like usual sets. That is where the countable, transitive, standard models come from.