Model theory of monadic predicate logic with the infinity quantifier

Abstract

AbstractThis paper establishes model-theoretic properties of $$\texttt {M} \texttt {E} ^{\infty }$$ M E ∞ , a variation of monadic first-order logic that features the generalised quantifier $$\exists ^\infty $$ ∃ ∞ (‘there are infinitely many’). We will also prove analogous versions of these results in the simpler setting of monadic first-order logic with and without equality ($$\texttt {M} \texttt {E} $$ M E and $$\texttt {M} $$ M , respectively). For each logic $$\texttt {L} \in \{ \texttt {M} , \texttt {M} \texttt {E} , \texttt {M} \texttt {E} ^{\infty }\}$$ L ∈ { M , M E , M E ∞ } we will show the following. We provide syntactically defined fragments of $$\texttt {L} $$ L characterising four different semantic properties of $$\texttt {L} $$ L -sentences: (1) being monotone and (2) (Scott) continuous in a given set of monadic predicates; (3) having truth preserved under taking submodels or (4) being truth invariant under taking quotients. In each case, we produce an effectively defined map that translates an arbitrary sentence $$\varphi $$ φ to a sentence $$\varphi ^\mathsf{p}$$ φ p belonging to the corresponding syntactic fragment, with the property that $$\varphi $$ φ is equivalent to $$\varphi ^\mathsf{p}$$ φ p precisely when it has the associated semantic property. As a corollary of our developments, we obtain that the four semantic properties above are decidable for $$\texttt {L} $$ L -sentences.