关于概率与Hilbert VI问题的注记
A note on probability and Hilbert's VI problem
摘要：
这项工作是由于概率论中一些基本实体的常见用法缺乏数学连贯性而引起的，这是一种内在的矛盾风险。在理清这些复杂性的同时，我们意识到，同样的问题已经被提出过很多次，但只有部分解决方案，特别是布尔、希尔伯特、德·菲内蒂和雷尼等人。特别是，在使用概率论的基础一致性的恢复似乎是一个在希尔伯特VI问题的解决方案中缺失的一块。在这里，我们解决了这个问题的一个新的形式化的概率论的基础上的最小集合的公理与额外的上下文相关的条件，其整体的一致性，然后语义验证。在初等概率中，即涉及多个事件的布尔组合的概率，我们的理论导致代数化，并使用Tarski Seidenberg还原，证明所有问题的可判定性。另一方面，初等概率中的不一致性等价于（经过适当重新定义）套利或荷兰书。在连续情况下，这导致非标准分析。
英文摘要：
This work has been prompted by the surprising lack of mathematical coherence in the common usage of some of the fundamental entities in the theory of probability, with an inherent risk of contradiction. While disentangling the intricacies, we realized that the same issue has been raised many times, with only partial solutions, notably by Boole, Hilbert, De Finetti and Renyi, among others. In particular, a restoration of foundational coherence in the usage of probability theory appears to be a missing piece in the solution of Hilbert VI problem. Here we solve the problem by a new formalization of probability theory based on a minimal collection of axioms with additional context dependent conditions, whose overall consistency is then semantically verified. In Elementary Probability, i.e. probabilities involving boolean combinations of finitely many events, our theory leads to algebraization and, using Tarski Seidenberg reduction, to a proof of decidability of all problems. Inconsistency in Elementary Probability, on the other hand, is equivalent to, suitably redefined, arbitrage or Dutch Book. In the continuous case this leads to nonstandard analysis.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
