Barton Neil, Friedman Sy-David
Kurt Gödel Research Center for Mathematical Logic (KGRC), University of Vienna, Währinger Straße, 25, 1090 Vienna, Austria.
Synthese. 2020;197(2):623-649. doi: 10.1007/s11229-017-1336-9. Epub 2017 Mar 4.
Discussion of new axioms for set theory has often focused on conceptions of , and how these might relate to the iterative conception of set. This paper provides critical appraisal of how certain maximality axioms behave on different conceptions of ontology concerning the iterative conception. In particular, we argue that forms of multiversism (the view that any universe of a certain kind can be extended) and actualism (the view that there are universes that cannot be extended in particular ways) face complementary problems. The latter view is unable to use maximality axioms that make use of , where the former has to contend with the existence of extensions violating maximality axioms. An analysis of two kinds of multiversism, a Zermelian form and Skolemite form, leads to the conclusion that the of maximality captured by an axiom differs substantially according to background ontology.
关于集合论新公理的讨论常常聚焦于 的概念，以及这些概念与集合的迭代概念可能存在怎样的关联。本文批判性地评估了某些极大性公理在关于迭代概念的不同本体论概念下的表现。特别是，我们认为多元主义的各种形式（即认为任何特定种类的全域都可以被扩展的观点）和现实主义（即认为存在一些全域不能以特定方式被扩展的观点）面临着互补的问题。后一种观点无法使用利用 的极大性公理，而前一种观点则必须应对违反极大性公理的扩展的存在。对两种多元主义形式，即策梅洛形式和斯科伦形式的分析得出结论：根据背景本体论，公理所捕捉到的极大性的 有很大差异。
