Informally speaking, the categoricity of an axiom system means that its non-logical symbols have only one possible interpretation that renders the axioms true. Although non-categoricity has become ubiquitous in the second half of the twentieth century whether one looks at number theory, geometry or analysis, the first axiomatizations of such mathematical theories by Dedekind, Hilbert, Huntington, Peano and Veblen were indeed categorical. A common resolution of the difference between the earlier categorical axiomatizations and the more modern non-categorical axiomatizations is that the latter derive their non-categoricity from Skolem's Paradox and Gödel's Incompleteness Theorems, while the former, being second order, suffer from a heavy reliance on meta-theory, where the Skolem–Gödel phenomenon re-emerges. Using second-order meta-theory to avoid non-categoricity of the meta-theory would only seem to lead to an infinite regress. In this article we maintain that internal categoricity breaks with this traditional picture. It applies to both first- and second-order axiomatizations, although in the first-order case we have so far only examples. It does not depend on the meta-theory in a way that would lead to an infinite regress. It also covers the classical categoricity results of early researchers. In the first-order case it is weaker than categoricity itself, and in the second-order case stronger. We provide arguments to suggest that internal categoricity is the “right” concept of categoricity.

中文翻译：

---

追踪内部分类

通俗地说，公理系统的范畴性意味着它的非逻辑符号只有一种可能的解释，使公理为真。尽管非范畴性在 20 世纪下半叶变得无处不在，无论是数论、几何还是分析，戴德金、希尔伯特、亨廷顿、皮亚诺和凡勃伦对这些数学理论的第一次公理化确实是范畴化的。较早的分类公理化和更现代的非分类公理化之间差异的一个共同解决方案是，后者从斯柯勒姆悖论和哥德尔不完备性定理推导出它们的非分类性，而前者是二阶的，严重依赖于元理论，其中 Skolem-Gödel 现象重新出现。使用二阶元理论来避免元理论的非范畴性似乎只会导致无限回归。在本文中，我们认为内部分类打破了这种传统的图景。它适用于一阶和二阶公理化，尽管在一阶情况下我们到目前为止只有例子。它不依赖于会导致无限回归的元理论。它还涵盖了早期研究人员的经典分类结果。在一阶情况下它比分类本身弱，而在二阶情况下强。我们提供的论据表明内部范畴性是范畴性的“正确”概念。在本文中，我们认为内部分类打破了这种传统的图景。它适用于一阶和二阶公理化，尽管在一阶情况下我们到目前为止只有例子。它不依赖于会导致无限回归的元理论。它还涵盖了早期研究人员的经典分类结果。在一阶情况下它比分类本身弱，而在二阶情况下强。我们提供的论据表明内部范畴性是范畴性的“正确”概念。在本文中，我们认为内部分类打破了这种传统的图景。它适用于一阶和二阶公理化，尽管在一阶情况下我们到目前为止只有例子。它不依赖于会导致无限回归的元理论。它还涵盖了早期研究人员的经典分类结果。在一阶情况下它比分类本身弱，而在二阶情况下强。我们提供的论据表明内部范畴性是范畴性的“正确”概念。它还涵盖了早期研究人员的经典分类结果。在一阶情况下它比分类本身弱，而在二阶情况下强。我们提供的论据表明内部范畴性是范畴性的“正确”概念。它还涵盖了早期研究人员的经典分类结果。在一阶情况下它比分类本身弱，而在二阶情况下强。我们提供的论据表明内部范畴性是范畴性的“正确”概念。