[redvenom]

This response is not quite right. "It's only completeness [...]" sentence does not make sense.

CH is independent of ZFC, period, as proved by Cohen. Talking about 'semantic level' does not make sense.

CH is an example of the incompleteness of ZFC. There are models of ZFC in which CH is true and models in which CH is false.

[legolas2412]

Trying to understand logic. What is a model, and how does it differ from theory?

For specific example, what is a model of ZFC? Is it just another theory, one which includes ZFC and few more axioms? Why not call it a derived theory or a subset theory?

[redvenom]

This explanation is somewhat informal, but I gets the point across: a theory is a set function symbols, relation symbols, and axioms governing them. A model is a specific set with an interpretation of those function and relation symbols that satisfies every axiom of the theory.

It's much easier to understand if we take an example. An example of a theory is the single sentence:

"There exists an X and there exists a Y such that X is not equal to Y."

(Of course typically in logic you would use logic symbols, but here I am writing out in an English sentence.)

Now, a model of this theory is the set {1,2}. Another model is the set {1,2,3}. More generally: any set with at least two elements is a model of that theory. The "function symbols" and "relation symbols" can be introduced in the language to talk about operations like addition and multiplication.

For example, the theory of groups uses the language of groups with a binary function symbol representing group multiplication. Any group (such as the integers with addition or invertible matrices with matrix multiplication) is a model of that theory.

So: theories are sets of axioms in some language, and models are sets together with actual functions/relations that satisfy those axioms.

Models of ZFC are a little bit counterintuitive. But they are single sets that interpret all the axioms of ZFC, rather than actual sets that we use in informal mathematics. Models of ZFC can be quite unusual because of the incompleteness theorem, and there are infinitely many models because of this (such as some in which CH is true, etc.).