Set-theoretic geology
Set-theoretic geology
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集合论地质学
DOI：
10.1016/j.apal.2014.11.004
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发表时间：
2011
期刊：
影响因子：
--
通讯作者：
Jonas Reitz
中科院分区：
文献类型：
--
作者：
G. Fuchs;J. Hamkins;Jonas Reitz
论域V的一个基是一个传递真类W <$V，使得W <$ZFC和V是通过在W上的集合强迫得到的，使得V= W [G]，对于某个W-一般滤子G <$P∈ W。模型V满足基公理GA，如果没有这样的W适当地包含在V中。模型W是V的基石，如果W是V的基并且满足基公理。V的幔是V的所有基的交集。V的类属幔是V的所有集迫扩张的所有基的交集。类属HOD，写作gHOD，是所有集迫扩张的所有HOD的交集。类属HOD始终是ZFC的模型，类属地幔始终是ZF的模型。每一个ZFC模型都是另一个ZFC模型的地幔和类属地幔。我们证明了这个定理，同时也控制最终模型的HOD，以及通用的HOD。反复地提取地幔，穿过内地幔，到达我们所说的外核，当所有的外层作用力都被剥离后，剩下的东西。许多基本问题仍然悬而未决。
A ground of the universe V is a transitive proper class W⊆ V, such that W⊨ ZFC and V is obtained by set forcing over W, so that V= W [G] for some W-generic filter G⊆ P∈ W. The model V satisfies the ground axiom GA if there are no such W properly contained in V. The model W is a bedrock of V if W is a ground of V and satisfies the ground axiom. The mantle of V is the intersection of all grounds of V. The generic mantle of V is the intersection of all grounds of all set-forcing extensions of V. The generic HOD, written gHOD, is the intersection of all HODs of all set-forcing extensions. The generic HOD is always a model of ZFC, and the generic mantle is always a model of ZF. Every model of ZFC is the mantle and generic mantle of another model of ZFC. We prove this theorem while also controlling the HOD of the final model, as well as the generic HOD. Iteratively taking the mantle penetrates down through the inner mantles to what we call the outer core, what remains when all outer layers of forcing have been stripped away. Many fundamental questions remain open.