Given a set $A \subseteq \mathbb N$ the only way I know if it's infinite is by checking $n \in A$ for infinitely many different $n$ which is impossible. Similarly I cannot prove there exist any finite sets either since even if I've checked for $1,2,\ldots,n$ we could still have $(n+1) \in A$. Furthermore even if there were both finite sets and infinite sets, it would be impossible to figure out when two sets are equal without again checking infinitely many $n$ which is impossible.