OSA: Mathematical Logic

Question 1

Which of the following claims are correct?

question	answer	explanation
For every finite signature \(S\), the set of valid first-order sentences over \(S\) is recursively enumerable.	yesno	This set can be enumerated using a proof calculus such as sequent calculus, which is correct and complete by Gödel's Completeness Theorem.
The first-order theory of the natural numbers (with addition and multiplication) is finitely axiomatisable.	yesno	If it were finitely axiomatisable then, being a complete theory, it would also be decidable.
The axiom system ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice) axiomatises a complete first-order theory.	yesno	Since the axioms of ZFC are recursively enumerable, if their deductive closure were complete, it would also be decidable.
There exist two non-isomorphic countable models of first-order Peano Arithmetic.	yesno	If any two countable models of PA were isomorphic (i.e. if PA were omega-categorical), then PA would be a complete theory.
There exists a sound and complete proof calculus for second-order logic.	yesno	Second-order logic does not have the compactness property, which is a consequence of the existence of a finitistic sound and complete proof calculus.