The Bayesian theorem on convergence to the truth states that a rational inquirer believes with certainty that her degrees of belief capture the truth about a large swath of hypotheses with increasing evidence. This result has been criticized as showcasing a problematic kind of epistemic immodesty when applied to infinite hypotheses that can never be approximated by finite evidence. The central point at issue—that certain hypotheses may forever be beyond the reach of a finite investigation no matter how large one’s reservoir of evidence—cannot be captured adequately within standard probability theory. As an alternative, I propose a nonstandard probabilistic framework that, by using arbitrarily small and large numbers, makes room for the type of fine-grained conceptual distinctions appropriate for a deeper analysis of convergence to the truth. This framework allows for the right kind of modesty about attaining truth in the limit.

中文翻译：

---

重新思考趋同于真理

关于收敛于真理的贝叶斯定理指出，理性的探究者肯定地相信她的信念程度能够捕捉到关于大量假设的真相，并且证据越来越多。这一结果被批评为在应用于永远无法用有限证据近似的无限假设时展示了一种有问题的认知不谦虚。问题的中心点——无论一个人的证据库有多大，某些假设可能永远超出有限调查的范围——不能在标准概率论中充分捕捉到。作为替代方案，我提出了一个非标准的概率框架，该框架通过使用任意大小的数字，为适合于更深入地分析收敛到真相的细粒度概念区分类型腾出空间。