Model theory of monadic predicate logic with the infinity quantifier

This paper establishes model-theoretic properties of ME infinity, a variation of monadic first-order logic that features the generalised quantifier there exists(infinity) Mere 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 (ME and M, respectively). For each logic L is an element of {M, ME, ME infinity) we will show the following. We provide syntactically defined fragments of L characterising four different semantic properties of 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 phi to a sentence phi(p) belonging to the corresponding syntactic fragment, with the property that phi is equivalent to phi(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 L-sentences.