Hello I am having difficulty with this question, I am not even sure what strateg

Question

Hello I am having difficulty with this question, I am not even sure what strategy one would go about proving something like this:

Suppose L is a language which includes an infinite list c1,c2,? of constant symbols. Let ? be a set of sentences ?={ci?cj?i,j?N,i<j}. Let A be a sentence such that ??A. Prove that A has a finite model.

I am not sure whether I would prove this via a contradiction (i.e., assume A has an infinite model, or if I show a finite model that works or some how assume that we can have an infinite model and then use some sort of compactness to show it can be finite. I am a little all over the place with this question, please any help would be great!

Explanation / Answer

Note that you're looking for a model of {A}, not of ??{A} (which obviously has no finite model since all the constants need to be pairwise distinct).

Proof by contradiction would be

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