Inter-model reflection principles
We introduce and consider the inner-model reflection principle, which asserts that whenever a statement 𝜑(𝑎) in the first-order language of set theory is true in the set-theoretic universe V, then it is also true in a proper inner model 𝑊⊊𝑉. A stronger principle, the ground-model reflection principl...
| Main Authors: | , , , , , |
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| Format: | Journal article |
| Language: | English |
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Springer
2019
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| _version_ | 1797076660387840000 |
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| author | Barton, N Caicedo, AE Fuchs, G Hamkins, JD Reitz, J Schindler, R |
| author_facet | Barton, N Caicedo, AE Fuchs, G Hamkins, JD Reitz, J Schindler, R |
| author_sort | Barton, N |
| collection | OXFORD |
| description | We introduce and consider the inner-model reflection principle, which asserts that whenever a statement 𝜑(𝑎) in the first-order language of set theory is true in the set-theoretic universe V, then it is also true in a proper inner model 𝑊⊊𝑉. A stronger principle, the ground-model reflection principle, asserts that any such 𝜑(𝑎) true in V is also true in some non-trivial ground model of the universe with respect to set forcing. These principles each express a form of width reflection in contrast to the usual height reflection of the Lévy–Montague reflection theorem. They are each equiconsistent with ZFC and indeed Π2-conservative over ZFC, being forceable by class forcing while preserving any desired rank-initial segment of the universe. Furthermore, the inner-model reflection principle is a consequence of the existence of sufficient large cardinals, and lightface formulations of the reflection principles follow from the maximality principle MP and from the inner-model hypothesis IMH. We also consider some questions concerning the expressibility of the principles. |
| first_indexed | 2024-03-07T00:06:54Z |
| format | Journal article |
| id | oxford-uuid:77d6ab3a-9d39-4b9b-a2bd-aee4ad88fc9a |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T00:06:54Z |
| publishDate | 2019 |
| publisher | Springer |
| record_format | dspace |
| spelling | oxford-uuid:77d6ab3a-9d39-4b9b-a2bd-aee4ad88fc9a2022-03-26T20:26:51ZInter-model reflection principlesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:77d6ab3a-9d39-4b9b-a2bd-aee4ad88fc9aEnglishSymplectic ElementsSpringer2019Barton, NCaicedo, AEFuchs, GHamkins, JDReitz, JSchindler, RWe introduce and consider the inner-model reflection principle, which asserts that whenever a statement 𝜑(𝑎) in the first-order language of set theory is true in the set-theoretic universe V, then it is also true in a proper inner model 𝑊⊊𝑉. A stronger principle, the ground-model reflection principle, asserts that any such 𝜑(𝑎) true in V is also true in some non-trivial ground model of the universe with respect to set forcing. These principles each express a form of width reflection in contrast to the usual height reflection of the Lévy–Montague reflection theorem. They are each equiconsistent with ZFC and indeed Π2-conservative over ZFC, being forceable by class forcing while preserving any desired rank-initial segment of the universe. Furthermore, the inner-model reflection principle is a consequence of the existence of sufficient large cardinals, and lightface formulations of the reflection principles follow from the maximality principle MP and from the inner-model hypothesis IMH. We also consider some questions concerning the expressibility of the principles. |
| spellingShingle | Barton, N Caicedo, AE Fuchs, G Hamkins, JD Reitz, J Schindler, R Inter-model reflection principles |
| title | Inter-model reflection principles |
| title_full | Inter-model reflection principles |
| title_fullStr | Inter-model reflection principles |
| title_full_unstemmed | Inter-model reflection principles |
| title_short | Inter-model reflection principles |
| title_sort | inter model reflection principles |
| work_keys_str_mv | AT bartonn intermodelreflectionprinciples AT caicedoae intermodelreflectionprinciples AT fuchsg intermodelreflectionprinciples AT hamkinsjd intermodelreflectionprinciples AT reitzj intermodelreflectionprinciples AT schindlerr intermodelreflectionprinciples |