论文标题
追踪内部分类
Tracing Internal Categoricity
论文作者
论文摘要
从非正式的角度来看，公理系统的分类性意味着其非逻辑符号只有一种可能使公理为真的解释。尽管在20世纪下半叶，非列型已经变得无处不在，无论是看数字理论，几何或分析，这是Dedekind，Hilbert，Hilbert，Huntington，Peano，Peano和Veblen的这种数学理论的第一个公理化。 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.使用二阶元理论避免元理论的非列表似乎只会导致无限的回归。在本文中，我们认为内部分类会破坏这一传统图片。它适用于一阶公理和二阶公理，尽管在一阶情况下，我们只有迄今为止的示例。它并不取决于元理论，从而导致无限回归。它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，在第二阶情况下更强大。我们提出的论点暗示内部分类是分类的“正确”概念。
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.
