Internal Categoricity in Arithmetic and Set Theory
Internal Categoricity in Arithmetic and Set Theory
复制标题
算术和集合论中的内范畴性
DOI：
10.1215/00294527-2835038
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
Tong Wang
中科院分区：
文献类型：
--
作者：
J. Väänänen;Tong Wang
我们证明了二阶Peano公理的范畴性可以从综合公理得到证明。我们还表明，二阶Zermelo-Fraenkel公理的范畴，给定的顺序类型，可以证明从综合公理。因此，这些著名的范畴性结果不需要所谓的“完全”二阶逻辑，Henkin二阶逻辑就足够了。我们还解决了这些公理系统的“一致性”的问题，在二阶意义上，也就是说，这些系统的模型存在的问题。在这两种情况下，我们都给出了一个一致性证明，但自然地，我们必须假设的不仅仅是理解公理。
We show that the categoricity of second-order Peano axioms can be proved from the comprehension axioms. We also show that the categoricity of second-order Zermelo–Fraenkel axioms, given the order type of the ordinals, can be proved from the comprehension axioms. Thus these well-known categoricity results do not need the so-called “full” second-order logic, the Henkin second-order logic is enough. We also address the question of “consistency” of these axiom systems in the second-order sense, that is, the question of existence of models for these systems. In both cases we give a consistency proof, but naturally we have to assume more than the mere comprehension axioms.