A simple maximality principle
A simple maximality principle
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简单的极大原则
DOI：
10.2178/jsl/1052669062
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发表时间：
2000
影响因子：
0.6
通讯作者：
J. Hamkins
中科院分区：
文献类型：
--
作者：
J. Hamkins
摘要在本文中，遵循Christophe Chalons的思想，我提出了一种新的强制公理，即最大性原则，它断言任何在某个强制扩展V中成立的句子φ和所有后续扩展V * 已经在V中成立。事实上，这样的句子也必须在V的所有强制扩展中成立。因此，从模态角度来看，最大值原理用（Dad □ φ）<$□ φ表示，与模态理论S5等价。本文证明了极大值原理与ZFC是相对一致的。通过允许真实的参数出现在φ中而得到的最大值原理的黑体版本，与断言Vδ <$V对于不可达基数δ的方案是等相容的，这反过来又与断言ORD是Mahlo的方案是等相容的。沿着这些路线的最强原理是□，它断言在V和所有强制扩张中成立。由此，可以得出0 #存在，x#对每个集合x都存在，射影真理是强迫不变的，Woodin基数是一致的等等。许多悬而未决的问题仍然存在。
Abstract In this paper, following an idea of Christophe Chalons, I propose a new kind of forcing axiom, the Maximality Principle, which asserts that any sentence φ holding in some forcing extension Vℙ and all subsequent extensions Vℙ*ℚ holds already in V. It follows, in fact, that such sentences must also hold in all forcing extensions of V. In modal terms, therefore, the Maximality Principle is expressed by the scheme (◊ □ φ) ⇒ □ φ, and is equivalent to the modal theory S5. In this article, I prove that the Maximality Principle is relatively consistent with ZFC. A boldface version of the Maximality Principle, obtained by allowing real parameters to appear in φ, is equiconsistent with the scheme asserting that Vδ ≺ V for an inaccessible cardinal δ, which in turn is equiconsistent with the scheme asserting that ORD is Mahlo. The strongest principle along these lines is □ , which asserts that holds in V and all forcing extensions. From this, it follows that 0# exists, that x# exists for every set x, that projective truth is invariant by forcing, that Woodin cardinals are consistent and much more. Many open questions remain.