Model theory of monadic predicate logic with the infinity quantifier

Abstract

This paper establishes model-theoretic properties of texttt {M} texttt {E} ^{infty }, 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} and texttt {M} , respectively). For each logic texttt {L} in { texttt {M} , texttt {M} texttt {E} , texttt {M} texttt {E} ^{infty }} we will show the following. We provide syntactically defined fragments of texttt {L} characterising four different semantic properties of texttt {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 to a sentence varphi ^mathsf{p} belonging to the corresponding syntactic fragment, with the property that varphi is equivalent to varphi ^mathsf{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} -sentences.