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.

中文翻译：

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可满足性在直觉上是错误的：来自 Dana Scott 的一个问题

可满足性或 Sat $$^{1}$$ 是元理论陈述 每个形式上直观一致的一阶句子集都有一个模型。所讨论的模型是标准一阶模型理论（Chang 和 Keisler 在模型理论，第 3 版，Dover Publications Inc.，Mineola，2012 年）中熟悉的塔斯基关系结构，但这里在直觉元数学中进行处理。我们证明 IZF（直觉主义 Zermelo-Fraenkel 集合论）和 HAS（二阶 Heyting 算术）都证明 Sat $$^{1}$$ 是完全错误的。在 Carter 的带领下（Notre Dame J Form Log 49(1):75–95, 2008），我们然后将这个结果推广到一些可证明的中间一阶逻辑，包括 Rose 逻辑（Trans Am Math Soc 61:1– 19, 1953)。这些元定理将 Sat $$^{1}$$ 的直觉主义基础意义与 Sat $$^{0}$$ 的直觉主义基础意义明显区分开来，后者是直觉主义命题逻辑的可满足性主张。同时，它们与 Test、COMP $$^{0}$$ 和 $$\mathbf{COMP }^{1}$$ 以及它们之间建立了直觉联系。其中，Test 是可测试性的方案，COMP $$^{0}$$ 和 $$\mathbf{COMP }^{1}$$ 分别是直觉命题逻辑和谓词逻辑的完备性。