The first order formulas preserved under ultrafilter extensions are not recursively enumerable

Approximating Resultants of Existential Second-Order Quantifier Elimination upon Universal Relational First-Order Formulas

We investigate second-order quantifier elimination for a class of formulas characterized by a restriction on the quantifier prefix: existential predicate quantifiers followed by universal individual quantifiers and a relational matrix. For a given second-order formula of this class a possibly infinite sequence of universal first-order formulas that have increasing strength and are all entailed ...

An Axiomatization of a First-order Branching Time Temporal Logic

We introduce a first-order temporal logic for reasoning about branching time. It is well known that the set of valid formulas is not recursively enumerable and there is no finitary axiomatization. We offer a sound and strongly complete axiomatization for the considered logic.

Formal Mathematical Systems including a Structural Induction Principle

This study provides a general frame for the generation of languages in recursive systems closely related to formal grammars, for the predicate calculus and a constructive induction principle. We introduce recursive systems generating the recursively enumerable relations between lists of terms, the basic objects under consideration. A recursive system consists of axioms which are special quantif...

Monodic Fragments of Probabilistic First-Order Logic

By classical results of Abadi and Halpern, validity for probabilistic first-order logic of type 2 (ProbFO) is Π 1 -complete and thus not recursively enumerable, and even small fragments of ProbFO are undecidable. In temporal first-order logic, which has similar computational properties, these problems have been addressed by imposing monodicity, that is, by allowing temporal operators to be appl...

Most General First Order Theorems are not Recursively Enumerable