Thin Equivalence Relations in Scaled

We give a new proof via inner model theory that every thin Σ Jα(R) 1 equivalence relation is ∆ Jα(R) 1, where α begins a Σ1 gap and Σ Jα(R) 1 is closed under number quanti cation, assuming AD Jα(R). In the recent past several results were shown with inner model theory which had been previously prove...

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Main Author: Ralf Schindler
Other Authors: The Pennsylvania State University CiteSeerX Archives
Format: Text
Language:English
Published: 2009
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Online Access:http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.168.189
http://wwwmath.uni-muenster.de/math/inst/logik/org/staff/rds/Thin.pdf
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spelling ftciteseerx:oai:CiteSeerX.psu:10.1.1.168.189 2023-05-15T18:12:45+02:00 Thin Equivalence Relations in Scaled Ralf Schindler The Pennsylvania State University CiteSeerX Archives 2009 application/pdf http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.168.189 http://wwwmath.uni-muenster.de/math/inst/logik/org/staff/rds/Thin.pdf en eng http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.168.189 http://wwwmath.uni-muenster.de/math/inst/logik/org/staff/rds/Thin.pdf Metadata may be used without restrictions as long as the oai identifier remains attached to it. http://wwwmath.uni-muenster.de/math/inst/logik/org/staff/rds/Thin.pdf text 2009 ftciteseerx 2016-01-07T15:54:21Z We give a new proof via inner model theory that every thin Σ Jα(R) 1 equivalence relation is ∆ Jα(R) 1, where α begins a Σ1 gap and Σ Jα(R) 1 is closed under number quanti cation, assuming AD Jα(R). In the recent past several results were shown with inner model theory which had been previously proved by direct application of the axiom of determinacy. We show a result of Harrington and Sami [1] about thin equivalence relations with inner model theory from an improved determinacy assumption. Recall that an equivalence relation E is called thin if there is no perfect set of pairwise E inequivalent reals. Theorem 0.1. Let α ≥ 2 begin a Σ1 gap in L(R). Assume AD Jα(R). Also, setting Γ = Σ Jα(R) 1, assume Γ to be closed under number quanti cation, i.e., ∀ ω Γ ⊂ Γ. Let E be a thin Γ equivalence relation. Let N be an α-suitable mouse with a capturing term for the complete Γ set. Then E is ˘ Γ in any real coding N as a parameter. The notion of α suitable mice with capturing terms (which is due to Woodin), is described in our section 1 and in detail in [6]. Such α suitable mice are in a sense analogues of M # n which capture more complicated sets of reals than the projective sets. The pointclass Γ = Σ Jα(R) 1 as in the statement of Theorem 0.1 is Text sami Unknown
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description We give a new proof via inner model theory that every thin Σ Jα(R) 1 equivalence relation is ∆ Jα(R) 1, where α begins a Σ1 gap and Σ Jα(R) 1 is closed under number quanti cation, assuming AD Jα(R). In the recent past several results were shown with inner model theory which had been previously proved by direct application of the axiom of determinacy. We show a result of Harrington and Sami [1] about thin equivalence relations with inner model theory from an improved determinacy assumption. Recall that an equivalence relation E is called thin if there is no perfect set of pairwise E inequivalent reals. Theorem 0.1. Let α ≥ 2 begin a Σ1 gap in L(R). Assume AD Jα(R). Also, setting Γ = Σ Jα(R) 1, assume Γ to be closed under number quanti cation, i.e., ∀ ω Γ ⊂ Γ. Let E be a thin Γ equivalence relation. Let N be an α-suitable mouse with a capturing term for the complete Γ set. Then E is ˘ Γ in any real coding N as a parameter. The notion of α suitable mice with capturing terms (which is due to Woodin), is described in our section 1 and in detail in [6]. Such α suitable mice are in a sense analogues of M # n which capture more complicated sets of reals than the projective sets. The pointclass Γ = Σ Jα(R) 1 as in the statement of Theorem 0.1 is
author2 The Pennsylvania State University CiteSeerX Archives
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author Ralf Schindler
spellingShingle Ralf Schindler
Thin Equivalence Relations in Scaled
author_facet Ralf Schindler
author_sort Ralf Schindler
title Thin Equivalence Relations in Scaled
title_short Thin Equivalence Relations in Scaled
title_full Thin Equivalence Relations in Scaled
title_fullStr Thin Equivalence Relations in Scaled
title_full_unstemmed Thin Equivalence Relations in Scaled
title_sort thin equivalence relations in scaled
publishDate 2009
url http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.168.189
http://wwwmath.uni-muenster.de/math/inst/logik/org/staff/rds/Thin.pdf
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