Reflexive Intermediate First-Order Logics

TL;DR: Red implies that the species of valid first order predicate schemata is not r.e., and hence that intuitionistic first orderpredicate logic .t 1 is incomplete, and a specific schema is constructed which is valid if RED, but unprovable.

TL;DR: It is proved that both IZF, intuitionistic Zermelo–Fraenkel set theory, and HAS, second-order Heyting arithmetic, prove Sat, the satisfiability claim for intuitionistic propositional logic, to be false outright.

Abstract: We consider a principle of constructivity RED which states that every decidable predicate over the natural numbers is (weakly) recursively enumer able (r.e.). RED is easily seen to be derived from Church's thesis CT 0 ("every construction is given by a recursive function"). Results: (3) The two results above when KREISEL-TROELSTRA [70]'s parameters. (4) LEIVANT [75].

TL;DR: It is shown that, if a suitable intuitionistic metatheory proves that consistency implies satisfiability for subfinite sets of propositional formulas relative either to standard structures or to Kripke models, then that metatheORY also proves every negative instance of every classical propositional tautology.