追踪内部类别

摘要：

非正式地说，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。虽然非范畴性在20世纪下半叶已经变得无处不在，无论是在数论、几何还是分析中，但戴德金、希尔伯特、亨廷顿、皮亚诺和凡勃伦对这些数学理论的第一个公理化确实是范畴性的。早期的范畴公理化和更现代的非范畴公理化之间的区别的一个共同的解决方案是，后者从Skolem的Partial定理和哥德尔的不完全性定理导出它们的非范畴性，而前者是二阶的，严重依赖于元理论，其中Skolem-Gödel现象再次出现。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的倒退。在本文中，我们认为，内部范畴打破了这一传统的图片。它适用于一阶和二阶公理化，虽然在一阶情况下，我们到目前为止只有例子。它并不依赖于元理论，而这种依赖会导致无限的倒退。它涵盖了早期研究者的经典范畴性结果。在第一阶情形中，它比范畴性本身弱，而在第二阶情形中，它比范畴性本身强。我们给出的参数表明，内部范畴是“正确”的范畴概念。

英文摘要：

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 20th 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 metatheory, 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 paper we maintain that internal categoricity breaks 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. And it 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 give arguments suggesting that internal categoricity is the "right" concept of categoricity.

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