<div class="csl-bib-body">
<div class="csl-entry">Lücke, P., & Müller, S. (2023). Σ₁-definability at higher cardinals: Thin sets, almost disjoint families and long well-orders. <i>Forum of Mathematics, Sigma</i>, <i>11</i>, Article e103. https://doi.org/10.1017/fms.2023.102</div>
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dc.identifier.issn
2050-5094
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dc.identifier.uri
http://hdl.handle.net/20.500.12708/193090
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dc.description.abstract
Given an uncountable cardinal κ, we consider the question of whether subsets of the power set of κ that are usually constructed with the help of the axiom of choice are definable by Σ₁-formulas that only use the cardinal κ and sets of hereditary cardinality less than κ as parameters. For limits of measurable cardinals, we prove a perfect set theorem for sets definable in this way and use it to generalize two classical nondefinability results to higher cardinals. First, we show that a classical result of Mathias on the complexity of maximal almost disjoint families of sets of natural numbers can be generalized to measurable limits of measurables. Second, we prove that for a limit of countably many measurable cardinals, the existence of a simply definable well-ordering of subsets of κ of length at least κ⁺ implies the existence of a projective well-ordering of the reals. In addition, we determine the exact consistency strength of the nonexistence of Σ₁-definitions of certain objects at singular strong limit cardinals. Finally, we show that both large cardinal assumptions and forcing axioms cause analogs of these statements to hold at the first uncountable cardinal ω₁.
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dc.description.sponsorship
FWF - Österr. Wissenschaftsfonds
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dc.description.sponsorship
FWF - Österr. Wissenschaftsfonds
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dc.description.sponsorship
FWF - Österr. Wissenschaftsfonds
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dc.language.iso
en
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dc.publisher
Cambridge University Press
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dc.relation.ispartof
Forum of Mathematics, Sigma
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dc.rights.uri
http://creativecommons.org/licenses/by/4.0/
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dc.subject
Σ₁-definability
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dc.subject
large cardinals
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dc.subject
iterated ultrapowers
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dc.subject
perfect subsets
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dc.subject
almost disjoint families
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dc.subject
definable well-orders
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dc.title
Σ₁-definability at higher cardinals: Thin sets, almost disjoint families and long well-orders