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策梅洛与无限逻辑

已有 632 次阅读 2022-7-4 14:45 |个人分类:解读哥德尔不完全性定理|系统分类:科研笔记

一下译文来自Jerzy Pogonowski的文章“Mathematics is the Logic of the Infinite : Zermelo’s Project of Infinitary Logic”的第七节【1】。


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在我看来,以下因素对泽梅洛的无限逻辑项目的动机至关重要。

1. 策梅洛对无限的性质的看法:他声称,数学是无限的逻辑。

2. 策梅洛认为,他的集合理论是整个数学的根本。特别是,他把公式和证明设想为来自集合的累积层次的有根基的构造。

3. 策梅洛对斯科莱姆的拒绝。策梅洛认为,对集合理论的研究不应局限于其单一模式,在数学证明程序中不应接受有限主义的立场。

4. 策梅洛认为,不应该对分离公理中出现的公式进行限制(特别是对一阶语言的限制是不合理的)。


策梅洛是一个真正的数学家,他很少表达哲学观点,总是强调他主要对所研究问题的数学方面感兴趣。他在1921年为1929年春季在华沙的演讲准备的短文 "Thesen ̈̈ das Unendliche in der Mathematik",在Taylor 2002van Dalen, Ebbinghaus 2000中得到了分析。该文内容如下: 


关于数学中的无限性的论述:

I) 每一个真正的数学命题都具有无限的性质,也就是说,它涉及到一个无限的领域,并被认为是无限多基本命题的集合。

II) 无限性不是在现实中从物理或心理上给予我们的,它必须作为柏拉图意义上的理念假设来把握。

III) 由于无限命题永远不能从有限命题中推导出来,任何数学理论的公理也必须是无限的,而且这种理论的一致性只能通过提出相应的无限多基本命题的一致性系统来证明,别无他法。

IV) 传统的亚里士多德逻辑,根据其性质,是有限的,因此不适合作为数学科学的基础。于是,有必要建立一个扩展的无限柏拉图逻辑,建立在某种无限的直觉之上例如,与选择公理的问题有关但自相矛盾的是,直觉派因习惯的力量而拒绝接受这种逻辑。

V) 每个数学命题都必须被视为(无限多的)基本命题的集合,基本关系,通过合取、析取和否定的方式,而一个命题对其他命题的每一次推导,特别是每一次证明,只不过是基本命题的重新组合 


如何理解这些主张呢?我们应该牢记,策梅洛从未使用过现代意义上的形式化语言。下面这段话经常被人引用:

- 当时,我可以依赖的普遍承认的数理逻辑并不存在今天也不存在,每个基础研究者都有自己的逻辑。


策梅洛用比喻的方式将数学描述为无限的逻辑比较一下Nachlaß的以下说明,这与他1929527日至68日在华沙所做的九个讲座中的第三个(有限域和无限域)有关:

 - 一个纯粹的有限主义的数学,其中没有任何东西真正需要证明,因为一切都已经可以通过使用有限模型来验证,这将不再是真正意义上的数学。相反,真正的数学根据其性质是无限的,并建立在无限领域的假设之上;它甚至可以被称为无限的逻辑 


在我看来,“Thesen ̈̈ Das Un Endliche in der Mathematik”最重要的方面是本体论的主张,即数学主要处理无限的结构。策梅洛认为,任何借助于有限语言工具来表达数学依赖性的尝试都只是一种近似。


参考文献:

https://www.researchgate.net/publication/357582750_Mathematics_is_the_Logic_of_the_Infinite_Zermelo's_Project_of_Infinitary_Logic


原文:


7. Zermelo’s infinitary logic


In my opinion the following factors are essential to Zermelo’s motivation for his project of infinitary logic:

1. Zermelo’s views on the nature of infinity: he claimed that Mathematics is the logic of the Infinite.

2. Zermelo’s belief that his set theory is fundamental to the entirety of mathematics. In particular, he conceived formulas and proofs as well-founded constructs from the cumulative hierarchy of sets.

3. Zermelo’s rejection of Skolemism. Zermelo believed that investigations into set theory should not be restricted to its single model and that one should not accept the finitistic standpoint in mathematical proof procedures.

4. Zermelo’s belief that no restriction should be imposed on formulas occurring in the axiom of separation (in particular the restriction to a first-order language is not justified) 


Zermelo was primarily a genuine mathematician and only rarely did he express philosophical opinions, always stressing the fact that he was mainly interested in mathematical aspects of the investigated problems. His “Thesen  ̈uber das Unendliche in der Mathematik”, a short note (includedin his Nachlaß) prepared in 1921 for his lectures in Warsaw in Spring 1929, was analyzed in Taylor 2002 and van Dalen, Ebbinghaus 2000. The text runs as follows: 


Theses concerning the infinite in mathematics

  1. Every genuinely mathematical proposition is “infinitary” in character, that is, is concerned with an infinite domain and is to be considered a collection of infinitely many “elementary propositions”.

  2. The infinite is not given to us physically or psychologically in reality, it must be grasped as “idea” in Plato’s sense and “posited”.

  3. Since infinitary propositions can never be derived from finitary ones, the“axioms” of any mathematical theory, too, must be infinitary, and the“consistency” of such a theory can be “proved” by no other means than the presentation of a corresponding consistent system of infinitely many elementary propositions.

  4. Traditional “Aristotelian” logic is, according to its nature, finitary, and hence not suited for the foundation of mathematical science. Whence the necessity of an extended “infinitary” or “Platonic” logic that rests on some kind of infinitary “intuition” – as, e.g., in connection with the question of the “axiom of choice” – but which, paradoxically, is rejected by the “intuitionists” by force of habit.

  5. Every mathematical proposition must be considered a collection of (infinitely many) elementary propositions, the “fundamental relations”, by means of conjunction, disjunction and negation, and every deduction of a proposition from other propositions, in particular every “proof”, is nothing but a “regrouping” of the elementary propositions 

(Zermelo 1921; citing the translation in Ebbinghaus, Fraser and Kanamori 2010, 307)


How are these claims to be understood? One should bear in mind that Zermelo never used formalized languages in the modern sense of the term. The following passage is very often cited: 


At the time, a universally acknowledged “mathematical logic” on which I could have relied did not exist – nor does it exist today when every foundational researcher has his own logistic. (Zermelo 1929, 340; citing the translation inEbbinghaus, Fraser and Kanamori 2010, 359)


Zermelo used a metaphorical description of mathematics as the logic ofthe Infinite – compare the following note from Nachlaß, related to the third(Finite and infinite domains) out of nine his lectures presented in Warsaw between 27 May and 8 June 1929


A purely “finitistic” mathematics, in which nothing really requires proof since everything is already verifiable by use of the finite model, would no longer be mathematics in the true sense of the word. Rather, true mathematics is infinitistic according to its nature and rests on the assumption of infinite do-mains; it may even be called the “logic of the infinite”.  


In my opinion the most important aspect of the “Thesen  ̈uber das Un-endliche in der Mathematik” is the ontological claim that mathematics deals primarily with infinite structures. Any attempts at expressing mathematical dependencies with the help of finitary linguistic tools were considered by Zermelo mere approximations. 




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