无句法的证明
逻辑
2007-05-23 v3 组合数学
摘要
“[数学家对逻辑的关心不比对数学的关心更多。”奥古斯都·德·摩根，1868年。证明传统上是句法的、归纳生成的对象。本文提出了命题演算（命题逻辑）的一种抽象数学表述，其中证明是组合的（图论的），而非句法的。它将命题P的组合证明定义为图同态h : C -> G(P)，其中G(P)是与P相关的图，C是着色图。主要定理是可靠性和完备性：P为真当且仅当存在组合证明h : C -> G(P)。
引用
@article{arxiv.math/0408282,
title = {Proofs Without Syntax},
author = {Dominic Hughes},
journal= {arXiv preprint arXiv:math/0408282},
year = {2007}
}
备注
Appears in Annals of Mathematics, 2006. 5 pages + references. Version 1 is submitted version; v3 is final published version (in two-column format rather than Annals style). Changes for v2: dualised definition of combinatorial truth, thereby shortening some subsequent proofs; added references; corrected typos; minor reworking of some sentences/paragraphs; added comments on polynomial-time correctness (referee request). Changes for v3: corrected two typos, reworded one sentence, repeated a citation in Notes section