I am getting somehow confused of compactness theorem.

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> The compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory, as it provides a useful method for constructing models of any set of sentences that is finitely consistent.
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I am not sure if I am mistaken, but what it basically seems to say is that if every finite subset of a set of first-order sentences has a model that assigns truth of the sentences, then the set would definitely have a model.

What's the point of having compactness theorem, when you basically have to find that **every** finite subset has a model? There are at least $\aleph\_0$ subsets.

The next question: Aren't there only $\aleph\_0$ first-order sentences, as each sentence is finite-sized?

And how can we not construct a model of any set of some sentences, when what we can just do is arbitrarily assign each sentence as true or false?
(Edit: so, I am asking, what's the point of referring to every finite subset?)

And finitely consistent? What does this mean?

Edit: I know that for any set of some sentences, models for the set can differ in their domain - that is components.