哥德尔不完全性定理的推广形式及其哲学影响

Generalisations and Their Philosophical Impacts of G?del’s Incompleteness Theorems

本文主要有五方面内容:一是将哥德尔不完全性定理涉及的一致性、语法完全性、ω-一致性、相对于N的可靠性、相对于N的完全性、可定义性等元理论性质推广成更一般的形式,并对其性质进行深入研究;二是简要回顾Salehi和Seraji所证推广的哥德尔第一不完全性定理,并就其关键定理给出更简洁易读的新证明,同时额外证明2组推广的哥德尔第一不完全性定理:任给n> 0,如果T是包含罗宾森算术的、Σn+1-可定义的(Πn-可定义的)、Πn+1-可靠的算术理论,那么T不是Πn+1-决定的;三是简要回顾Seraji和本文作者所证推广的哥德尔第二不完全性定理,并给出新证明,同时额外证明2组推广的哥德尔第二不完全性定理:任给n> 0,如果T是包含皮亚诺算术的、Σn+1-可定义的(Πn-可定义的)、Πn+1-可靠的算术理论,那么T不能证明自身Πn+1-可靠性;四是用两种方法再证明4组与一致性相关的推广的哥德尔第二不完全性定理:任给n> 0,如果T是包含皮亚诺算术的、一致的、Σn+1-可定义的(Πn-可定义的)、Σn+1-完全的(Πn-完全的)算术理论,那么T不能证明自身一致性,同时给出2组可证自身一致性的算术理论;五是基于推广的哥德尔不完全性定理,从对形式化方法局限的反驳、对反机械主义的支持、对数学家地位的维护等三个方面重新审视哥德尔不完全性定理所产生的哲学影响。

There are five parts in this paper:(1) generalize meta-theoretic properties such as consistency, syntactical completeness, ω-consistency, soundness with respect to N, completeness with respect to N, definability involved in Godel’s incompleteness theorems to more generalized ones which would be researched in details;(2) recall the generalized Godel’s first incompleteness theorems proved by Salehi and Seraji the crtical one of which will be provided a new and readable proof, and show two more results: for all n > 0 if T is a Σn+1-definable(Πn-definable) and Πn+1-sound extension of Robinson arithmetic then T is not Πn+1-deciding;(3) recall the generalized Godel’s second incompleteness theorems proved by Seraji and me which would be also provided a new and readable proof, and show two more results: for all n > 0 if T is a Σn+1-definable(Πn-definable) and Πn+1-sound extension of Peano arithmetic then T cannot show self Πn+1-soundness;(4) with two different approach prove four generalized Godel’s second incompleteness theorems involved consistency: for all n > 0 if T is a Σn+1-definable(Πn-definable) and Σn+1-complete(Πn-complete) extension of Peano arithmetic then T cannot show self consistency, and further two collection of theories which can prove self consistency will be defined;(5) review the philosophical impacts of Godel’s incompleteness theorems, we will based on generalized Godel’s incompleteness theorems refute the limitations of formal methods, back anti-mechanists up and defend the status of mathematicians.