Infinitely many nonequivalent unprovable statements in ZFC because of Gödel's incompleteness theorem?

Question

Am I right in thinking that there are countably infinitely many nonequivalent unprovable statements in ZFC due to the Gödel's first incompleteness theorem? If not, why?

Answer 1

First of all, "countably" is redundant, because there are only countably many sentences in the first place. With that out of the way, you are correct. Assume there are only finitely many unprovable statements (up to equivalence modulo ZFC) $\sigma_1 \ldots \sigma_n$, and that ZFC is consistent. Then take $\text{ZFC}^\star$ some complete, consistent extension of ZFC. Since there are only finitely many $\sigma_i$, we can effectively axiomatize $\text{ZFC}^\star$ by taking the axioms of ZFC together with one of $\sigma_i$ or $\neg \sigma_i$ for each $i$ (depending on whether $\text{ZFC}^\star \models \sigma_i$), contradicting the first incompleteness theorem.