Model theory of monadic predicate logic with the infinity quantifier

This paper establishes model-theoretic properties of ^∞ , a variation of monadic first-order logic that features the generalised quantifier ∃ ^∞ (‘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 ( and , respectively). For each logic ∈{ , , ^∞} we will show the following. We provide syntactically defined fragments of characterising four different semantic properties of -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 φ to a sentence φ ^𝗉 belonging to the corresponding syntactic fragment, with the property that φ is equivalent to 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 -sentences.