Incompatible Ω-Complete Theories
Incompatible Ω-Complete Theories
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不相容的 Ω 完全理论
DOI：
10.2178/jsl/1254748685
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
W. Woodin
中科院分区：
文献类型：
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作者：
P. Koellner;W. Woodin
1985年，第二作者证明了如果存在一类可测的Woodin基数，并且是V满足CH的一般扩展，那么它们都是一致的Σ12-statements。就强逻辑Ω-logic而言，这可以重新表述为：在上述大的基本假设下，对于Σ12， ZFC + CH为Ω-complete。此外，CH是唯一具有此特性的Σ12-statement，因为任何具有此特性的其他Σ12-statement都是Ω-equivalent到CH over ZFC。寻找ZFC的其他优势是很自然的，它们具有更大程度的Ω-completeness。例如，可以要求递归可枚举公理A，使得相对于大基数公理ZFC + A对于所有三阶算术都是Ω-complete。更进一步说，对于集合集合的每个可指定部分Vλ（例如，可以将Vλ作为满足存在一个适当的大基数类的最小层次），可以要求递归可枚举公理a，使得相对于大基数公理ZFC + a对于Vλ的理论是Ω-complete。如果这样的理论存在，相互扩展，并且在某种意义上是唯一的，任何其他这样的理论B具有与A相同的Ω-completeness水平，实际上是Ω-equivalent到A / ZFC，那么这将表明存在一个唯一的Ω-complete图像的连续片段集合的宇宙，它将为公理补充大基本公理提供一个非常有力的例子。在本文中，我们证明唯一性必须失效。特别是，我们表明，如果有一个这样的理论Ω-implies CH，那么就有另一个Ω-implies¬-CH。
Abstract In 1985 the second author showed that if there is a proper class of measurable Woodin cardinals and and are generic extensions of V satisfying CH then and agree on all Σ12-statements. In terms of the strong logic Ω-logic this can be reformulated by saying that under the above large cardinal assumption ZFC + CH is Ω-complete for Σ12. Moreover, CH is the unique Σ12-statement with this feature in the sense that any other Σ12-statement with this feature is Ω-equivalent to CH over ZFC. It is natural to look for other strengthenings of ZFC that have an even greater degree of Ω-completeness. For example, one can ask for recursively enumerable axioms A such that relative to large cardinal axioms ZFC + A is Ω-complete for all of third-order arithmetic. Going further, for each specifiable segment Vλ of the universe of sets (for example, one might take Vλ to be the least level that satisfies there is a proper class of huge cardinals), one can ask for recursively enumerable axioms A such that relative to large cardinal axioms ZFC + A is Ω-complete for the theory of Vλ. If such theories exist, extend one another, and are unique in the sense that any other such theory B with the same level of Ω-completeness as A is actually Ω-equivalent to A over ZFC, then this would show that there is a unique Ω-complete picture of the successive fragments of the universe of sets and it would make for a very strong case for axioms complementing large cardinal axioms. In this paper we show that uniqueness must fail. In particular, we show that if there is one such theory that Ω-implies CH then there is another that Ω-implies ¬-CH.