Satisfiability is False Intuitionistically: A Question from Dana Scott

Abstract

Satisfiability or Sat$^{1}$ is the metatheoretic statement

Every formally intuitionistically consistent set of first-order sentences has a model.

The models in question are the Tarskian relational structures familiar from standard first-order model theory (Chang and Keisler in Model theory, 3rd edn, Dover Publications Inc., Mineola, 2012), but here treated within intuitionistic metamathematics. We prove that both IZF, intuitionistic Zermelo–Fraenkel set theory, and HAS, second-order Heyting arithmetic, prove Sat$^{1}$ to be false outright. Following the lead of Carter (Notre Dame J Form Log 49(1):75–95, 2008), we then generalize this result to some provably intermediate first-order logics, including the Rose logic (Trans Am Math Soc 61:1–19, 1953). These metatheorems distinguish the intuitionistic foundational significance of Sat$^{1}$ sharply from that of Sat$^{0}$, the satisfiability claim for intuitionistic propositional logic. At the same time, they establish intuitionistic connections with and between Test, COMP$^{0}$, and $\mathbf{COMP }^{1}$. Here, Test is the scheme of Testability, and COMP$^{0}$ and $\mathbf{COMP }^{1}$ are completeness for intuitionistic propositional logic and predicate logic, respectively.