Model theory of monadic predicate logic with the infinity quantifier

Carreiro, Facundo and Facchini, Alessandro and Venema, Yde and Zanasi, Fabio (2022) Model theory of monadic predicate logic with the infinity quantifier. Archive for Mathematical Logic, 61 (3-4). pp. 465-502.

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Abstract

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. For each logic we will show the following. We provide syntactically defined fragments 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 S to a sentence S' belonging to the corresponding syntactic fragment, with the property that S is equivalent to S' 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.

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