We show that an adaptation of Peano’s axioms for second-order arithmetic to the language of MSO completely axiomatizes the theory over infinite trees. This continues a line of work begun by Büchi and Siefkes with axiomatizations of MSO over various classes of linear orders.
Our proof formalizes, in the axiomatic theory, a translation of MSO formulas to alternating parity tree automata. The main ingredient is the formalized proof of positional determinacy for the corresponding parity games which, as usual, allows us to complement automata in order to deal with negation of MSO formulas. The Comprehension scheme of monadic second-order logic is used to obtain uniform winning strategies, whereas most usual proofs of positional determinacy rely on forms of the Axiom of Choice or transfinite induction.
NB: There is an error in the uniform positional strategy argument in this paper. A full version addressing this will be available soon.