The formal verification of the ctm approach to forcing

IF 0.6 2区数学 Q2 LOGIC Annals of Pure and Applied Logic Pub Date : 2024-01-30 DOI:10.1016/j.apal.2024.103413

Emmanuel Gunther , Miguel Pagano , Pedro Sánchez Terraf , Matías Steinberg

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Abstract

We discuss some highlights of our computer-verified proof of the construction, given a countable transitive set-model M of ZFC, of generic extensions satisfying $ZFC + \neg CH$ and $ZFC + CH$ . Moreover, let $R$ be the set of instances of the Axiom of Replacement. We isolated a 21-element subset $Ω \subseteq R$ and defined $F : R \to R$ such that for every $Φ \subseteq R$ and M-generic G, $M ⊨ ZC \cup F “ Φ \cup Ω$ implies $M [G] ⊨ ZC \cup Φ \cup {\neg CH}$ , where ZC is Zermelo set theory with Choice.

To achieve this, we worked in the proof assistant Isabelle, basing our development on the Isabelle/ZF library by L. Paulson and others.

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对 CTM 强迫方法的正式验证

我们将讨论计算机验证证明的一些要点，即在给定 ZFC 的可数传递集合模型 M 的情况下，构造满足 ZFC+¬CH 和 ZFC+CH 的泛型扩展。此外，让 R 是替换公理的实例集。我们分离出一个 21 元子集 Ω⊆R，并定义了 F:R→R，使得对于每一个 Φ⊆R 和 M 泛函 G，M⊨ZC∪F "Φ∪Ω 意味着 M[G]⊨ZC∪Φ∪{¬CH}，其中 ZC 是带选择的泽梅洛集合论。为了实现这一目标，我们使用了证明助手 Isabelle，以 L. Paulson 等人的 Isabelle/ZF 库为基础进行开发。

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来源期刊

Annals of Pure and Applied Logic 数学-数学

CiteScore

1.40

自引率

12.50%

发文量

审稿时长

200 days

期刊介绍： The journal Annals of Pure and Applied Logic publishes high quality papers in all areas of mathematical logic as well as applications of logic in mathematics, in theoretical computer science and in other related disciplines. All submissions to the journal should be mathematically correct, well written (preferably in English)and contain relevant new results that are of significant interest to a substantial number of logicians. The journal also considers submissions that are somewhat too long to be published by other journals while being too short to form a separate memoir provided that they are of particular outstanding quality and broad interest. In addition, Annals of Pure and Applied Logic occasionally publishes special issues of selected papers from well-chosen conferences in pure and applied logic.

期刊最新文献

Editorial Board Proof-theoretic methods in quantifier-free definability Generic multiplicative endomorphism of a field Π2-rule systems and inductive classes of Gödel algebras Modal logics over lattices