$\mathbf {\Sigma }_1$ -definability at higher cardinals: Thin sets, almost disjoint families and long well-orders
Given an uncountable cardinal $\kappa $ , we consider the question of whether subsets of the power set of $\kappa $ that are usually constructed with the help of the axiom of choice are definable by $\Sigma _1$ -formulas that only use the cardinal $\kappa $ and sets of here...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2023-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509423001020/type/journal_article |
| _version_ | 1797614806510862336 |
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| author | Philipp Lücke Sandra Müller |
| author_facet | Philipp Lücke Sandra Müller |
| author_sort | Philipp Lücke |
| collection | DOAJ |
| description | Given an uncountable cardinal
$\kappa $
, we consider the question of whether subsets of the power set of
$\kappa $
that are usually constructed with the help of the axiom of choice are definable by
$\Sigma _1$
-formulas that only use the cardinal
$\kappa $
and sets of hereditary cardinality less than
$\kappa $
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
$\kappa $
of length at least
$\kappa ^+$
implies the existence of a projective well-ordering of the reals. In addition, we determine the exact consistency strength of the nonexistence of
$\Sigma _1$
-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
$\omega _1$
. |
| first_indexed | 2024-03-11T07:17:27Z |
| format | Article |
| id | doaj.art-33fe8aa5c3064e40b0e30565843d3ccf |
| institution | Directory Open Access Journal |
| issn | 2050-5094 |
| language | English |
| last_indexed | 2024-03-11T07:17:27Z |
| publishDate | 2023-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj.art-33fe8aa5c3064e40b0e30565843d3ccf2023-11-17T08:11:57ZengCambridge University PressForum of Mathematics, Sigma2050-50942023-01-011110.1017/fms.2023.102$\mathbf {\Sigma }_1$ -definability at higher cardinals: Thin sets, almost disjoint families and long well-ordersPhilipp Lücke0https://orcid.org/0000-0001-8746-5887Sandra Müller1https://orcid.org/0000-0002-7224-187XFachbereich Mathematik, Universität Hamburg, Bundesstraße 55, Hamburg, 20146, Germany; E-mail: Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Gran Via de les Corts Catalanes 585, Barcelona, 08007, SpainInstitut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstraße 8-10/104, Wien, 1040, Austria; E-mail:Given an uncountable cardinal $\kappa $ , we consider the question of whether subsets of the power set of $\kappa $ that are usually constructed with the help of the axiom of choice are definable by $\Sigma _1$ -formulas that only use the cardinal $\kappa $ and sets of hereditary cardinality less than $\kappa $ 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 $\kappa $ of length at least $\kappa ^+$ implies the existence of a projective well-ordering of the reals. In addition, we determine the exact consistency strength of the nonexistence of $\Sigma _1$ -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 $\omega _1$ .https://www.cambridge.org/core/product/identifier/S2050509423001020/type/journal_article03E4703E3503E4503E55 |
| spellingShingle | Philipp Lücke Sandra Müller $\mathbf {\Sigma }_1$ -definability at higher cardinals: Thin sets, almost disjoint families and long well-orders Forum of Mathematics, Sigma 03E47 03E35 03E45 03E55 |
| title | $\mathbf {\Sigma }_1$ -definability at higher cardinals: Thin sets, almost disjoint families and long well-orders |
| title_full | $\mathbf {\Sigma }_1$ -definability at higher cardinals: Thin sets, almost disjoint families and long well-orders |
| title_fullStr | $\mathbf {\Sigma }_1$ -definability at higher cardinals: Thin sets, almost disjoint families and long well-orders |
| title_full_unstemmed | $\mathbf {\Sigma }_1$ -definability at higher cardinals: Thin sets, almost disjoint families and long well-orders |
| title_short | $\mathbf {\Sigma }_1$ -definability at higher cardinals: Thin sets, almost disjoint families and long well-orders |
| title_sort | mathbf sigma 1 definability at higher cardinals thin sets almost disjoint families and long well orders |
| topic | 03E47 03E35 03E45 03E55 |
| url | https://www.cambridge.org/core/product/identifier/S2050509423001020/type/journal_article |
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