We propose an axiomatic foundation of mathematics based on the finite sequence as the foundational concept, rather than based on logic and set, as in set theory, or based on type as in dependent type theories. Finite sequences lead to a concept of pure data, which is used to represent all mathematical objects. As an axiomatic system, the foundation has only one axiom which defines what constitutes a valid definition. Using the axiom, an internal true/false/undecided valued logic and an internal language are defined, making logic and language-related axioms un- necessary. Valid proof and valid computation are defined in terms of equality of pure data. An algebra of pure data leads to a rich theory of spaces and morphisms which play a role similar to the role of Category Theory in modern Mathematics. As applications, we explore Mathematical Machine Learning, the consistency of Mathematics and address paradoxes due to Godel, Berry, Curry and Yablo.

翻译：我们提出一种基于有限序列作为基础概念的数学公理化基础，而非像集合论那样以逻辑和集合为基础，或像依赖类型理论那样以类型为基础。有限序列引出了纯数据的概念，并用于表示所有数学对象。作为公理系统，该基础仅包含一条公理，用于定义何为有效的定义。利用该公理，可定义内部真/假/不可判定的三值逻辑及内部语言，从而使与逻辑和语言相关的公理变得不再必要。有效证明与有效计算均以纯数据的相等性来定义。纯数据代数衍生出丰富的空间与态射理论，其作用类似于范畴论在现代数学中的地位。作为应用，我们探索了数学机器学习、数学的一致性，并解决了由哥德尔、贝里、柯里和亚布洛引发的悖论。