Forcing MSO on Infinite Words in Weak MSO

We propose a forcing-based interpretation of monadic second-order logic (MSO) on infinite (omega) words in Weak MSO (WMSO). The interpretation is purely syntactic. We show that a formula with parameters is true in MSO if and only if its interpretation is true in WMSO. We also show that a closed formula is true in MSO if and only if its interpretation is provable under some axioms which hold for WMSO, but without axiomatizing it. We use model-theoretic arguments. Our approach is inspired from point-free topology: infinite words, seen as topological points, are approximated by filters of bounded segments. We devise forcing conditions such that the corresponding generic filters approximate Ramseyan factorizations of infinite words modulo satisfaction of formulas of a given quantifier depth. Our interpretation parallels some approaches to McNaughton's Theorem (equivalence between non-deterministic Bϋchi automata and deterministic Rabin automata) but the obtained formulas do not describe deterministic automata.