<div class="csl-bib-body">
<div class="csl-entry">Schlutzenberg, F. S. (2025, January). <i>Mouse sets and correctness in L(R)</i> [Conference Presentation]. Set Theory (Workshop 2503), Oberwolfach, Germany. http://hdl.handle.net/20.500.12708/224895</div>
</div>
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dc.identifier.uri
http://hdl.handle.net/20.500.12708/224895
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dc.description.abstract
Work in ZF + AD + V = L(R). Given a nice countable set X
of reals, it is natural to ask whether X is a mouse set: whether there is
a mouse M such that X = R ∩ M, where R is the set of real numbers.
Also,, how correct is X (how elementary is X in R, for example)?
For example, the set of all OD reals is a mouse set (see [9]).
In this talk, we consider more local versions of ordinal definability. At the projective level, R ∩ M_n is exactly
the set of reals which are ∆^1_{n+2} in a countable ordinal, where M_n is the canonical
proper class mouse with n Woodins. Also, M_{2n} is
Σ^1_{2n+2}-correct but not Σ^1_{2n+3}-correct, and M_{2n+1} is also Σ^1_{2n+2}-correct but not Σ^1_{2n+3}-correct.
Further, defining M_0=L and M_{−1} = L_{ω_1^{ck}}, the analogous results hold for these models, except that R ∩ L_{ω_1^{ck}} is the set of ∆^1_1 reals (in no parameters).) Moreover, writing <^{M_n} for the usual wellorder of M_n, <^{M_n} ↾ R^{M_n} is (∆^1_{n+2})^{M_n}-definable.
Although M = L_{ω_1^{ck}} is not Σ^1_1-correct, it can define Σ^1_1 truth, via "anti-correctness", due to Spector-Gandy and Ville:
– (Π^1_1)^V ↾ M is uniformly (Σ^1_1)^M , meaning that there is a recursive map f, with f(ϕ) = ψ_ϕ,
sending Π^1_1 formulas ϕ to Σ^1_1 formulas ψ_ϕ, such that for all x ∈ R ∩ M, we have ϕ(x) ⇐⇒ M models ψ_ϕ(x).
– Similarly, (Π^1_1)^M is uniformly (Σ^1_1)^V ↾ M .
Higher up, (Σ^1_3)^V ↾ L is not definable over L, and in fact, there are ∆^1_3 reals which are
not in L. But the anti-correctness for L_{ω_1^{ck}} has an analogue for M1 and Π^1_3.
We will discuss various new results along these lines at levels of the Levy hierarchy through L(R).
en
dc.description.sponsorship
FWF - Österr. Wissenschaftsfonds
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dc.language.iso
en
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dc.subject
mouse
en
dc.subject
ordinal definability
en
dc.subject
L(R)
en
dc.title
Mouse sets and correctness in L(R)
en
dc.type
Presentation
en
dc.type
Vortrag
de
dc.relation.grantno
Y1498
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dc.type.category
Conference Presentation
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tuw.project.title
Determiniertheit und Woodin Limes von Woodin Kardinalzahlen
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tuw.researchTopic.id
I1
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tuw.researchTopic.name
Logic and Computation
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tuw.researchTopic.value
100
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tuw.publication.orgunit
E104-08 - Forschungsbereich Mengenlehre
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tuw.event.name
Set Theory (Workshop 2503)
en
dc.description.sponsorshipexternal
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)
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dc.relation.grantnoexternal
EXC 2044-390685587
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tuw.event.startdate
12-01-2025
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tuw.event.enddate
17-01-2025
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tuw.event.online
On Site
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tuw.event.type
Event for scientific audience
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tuw.event.place
Oberwolfach
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tuw.event.country
DE
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tuw.event.institution
MFO
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tuw.event.presenter
Schlutzenberg, Farmer Salamander
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wb.sciencebranch
Mathematik
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wb.sciencebranch.oefos
1010
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wb.sciencebranch.value
100
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item.openairetype
conference paper not in proceedings
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item.openairecristype
http://purl.org/coar/resource_type/c_18cp
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item.cerifentitytype
Publications
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item.languageiso639-1
en
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item.grantfulltext
none
-
item.fulltext
no Fulltext
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crisitem.author.dept
E104-08 - Forschungsbereich Mengenlehre
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crisitem.author.parentorg
E104 - Institut für Diskrete Mathematik und Geometrie
