Truth assignments of a subset of sentence symbols

Question

I am trying to prove that truth assignments of an infinite subset of sentence symbols are uncountable. I am new to mathematical logic and I am kind of confused. I learned the compactness theorem but I'm not sure how to apply it to this proof.

Answer 1

Suppose the $x_n$, $n \in \mathbb{N}$ are the symbols you're assingning truth values to. So any such assigmenment is really a function $a: \mathbb{N} \to \{0,1\}$, where $a(n) =1$ if we assign $x_n $ the "true" value and $0$ if we assign it false.

Now in set theory it is shown that this set of functions has size the powerset of $\mathbb{N}$, which is uncountable by Cantor's theorem.