Catalònica: Agregador de continguts digitals de Catalunya

Està a:  › Dades de registre

$\Sigma_1$-definability at higher cardinals: Thin sets, almost disjoint families and long well-orders

Identificadors del recurs

2050-5094

https://hdl.handle.net/2445/220654

758203

Procedència

(Dipòsit Digital de la Universitat de Barcelona)

Fitxa

Títol:: $\Sigma_1$-definability at higher cardinals: Thin sets, almost disjoint families and long well-orders
Matèria:: Teoria de conjunts; Lògica matemàtica; Set theory; Mathematical logic
Descripció:: 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 $\omega_1$.
Font:: Articles publicats en revistes (Matemàtiques i Informàtica)
Idioma:: English
Relació:: Reproducció del document publicat a: https://doi.org/10.1017/fms.2023.102; 2023, vol. 11; https://doi.org/10.1017/fms.2023.102
Autor/Productor:: Lücke, Philipp; Müller, Sandra
Drets:: cc-by (c) Lücke, P. et al., 2023; http://creativecommons.org/licenses/by/4.0/; info:eu-repo/semantics/openAccess
Data:: 2025-04-28T06:38:22Z; 2023-11-17; 2025-04-28T06:38:23Z
Tipus de recurs:: info:eu-repo/semantics/article; info:eu-repo/semantics/publishedVersion
Format:: 36 p.; application/pdf

oai_dc

Descarregar XML

<?xml version="1.0" encoding="UTF-8" ?>

< oai_dc:dc schemaLocation =" http://www.openarchives.org/OAI/2.0/oai_dc/ http://www.openarchives.org/OAI/2.0/oai_dc.xsd " >
1. < dc:title > $\Sigma_1$-definability at higher cardinals: Thin sets, almost disjoint families and long well-orders </ dc:title >
2. < dc:creator > Lücke, Philipp </ dc:creator >
3. < dc:creator > Müller, Sandra </ dc:creator >
4. < dc:subject > Teoria de conjunts </ dc:subject >
5. < dc:subject > Lògica matemàtica </ dc:subject >
6. < dc:subject > Set theory </ dc:subject >
7. < dc:subject > Mathematical logic </ dc:subject >
8. < dc: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 $\omega_1$. </ dc:description >
9. < dc:date > 2025-04-28T06:38:22Z </ dc:date >
10. < dc:date > 2025-04-28T06:38:22Z </ dc:date >
11. < dc:date > 2023-11-17 </ dc:date >
12. < dc:date > 2025-04-28T06:38:23Z </ dc:date >
13. < dc:type > info:eu-repo/semantics/article </ dc:type >
14. < dc:type > info:eu-repo/semantics/publishedVersion </ dc:type >
15. < dc:identifier > 2050-5094 </ dc:identifier >
16. < dc:identifier > https://hdl.handle.net/2445/220654 </ dc:identifier >
17. < dc:identifier > 758203 </ dc:identifier >
18. < dc:language > eng </ dc:language >
19. < dc:relation > Reproducció del document publicat a: https://doi.org/10.1017/fms.2023.102 </ dc:relation >
20. < dc:relation > 2023, vol. 11 </ dc:relation >
21. < dc:relation > https://doi.org/10.1017/fms.2023.102 </ dc:relation >
22. < dc:rights > cc-by (c) Lücke, P. et al., 2023 </ dc:rights >
23. < dc:rights > http://creativecommons.org/licenses/by/4.0/ </ dc:rights >
24. < dc:rights > info:eu-repo/semantics/openAccess </ dc:rights >
25. < dc:format > 36 p. </ dc:format >
26. < dc:format > application/pdf </ dc:format >
27. < dc:source > Articles publicats en revistes (Matemàtiques i Informàtica) </ dc:source >
</ oai_dc:dc >

edm

Descarregar XML

<?xml version="1.0" encoding="UTF-8" ?>

< rdf:RDF schemaLocation =" http://www.w3.org/1999/02/22-rdf-syntax-ns# http://www.europeana.eu/schemas/edm/EDM.xsd " >
1. < edm:ProvidedCHO about =" https://catalonica.bnc.cat/catalonicahub/lod/oai:diposit.ub.edu:2445_--_220654#ent0 " >
  1. < dc:creator > Lücke, Philipp </ dc:creator >
  2. < dc:creator > Müller, Sandra </ dc:creator >
  3. < dc:date > 2025-04-28T06:38:22Z </ dc:date >
  4. < dc:date > 2025-04-28T06:38:22Z </ dc:date >
  5. < dc:date > 2023-11-17 </ dc:date >
  6. < dc:date > 2025-04-28T06:38:23Z </ dc:date >
  7. < dc: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 $\omega_1$. </ dc:description >
  8. < dc:identifier > 2050-5094 </ dc:identifier >
  9. < dc:identifier > https://hdl.handle.net/2445/220654 </ dc:identifier >
  10. < dc:identifier > 758203 </ dc:identifier >
  11. < dc:language > eng </ dc:language >
  12. < dc:relation > Reproducció del document publicat a: https://doi.org/10.1017/fms.2023.102 </ dc:relation >
  13. < dc:relation > 2023, vol. 11 </ dc:relation >
  14. < dc:relation > https://doi.org/10.1017/fms.2023.102 </ dc:relation >
  15. < dc:rights > cc-by (c) Lücke, P. et al., 2023 </ dc:rights >
  16. < dc:rights > http://creativecommons.org/licenses/by/4.0/ </ dc:rights >
  17. < dc:rights > info:eu-repo/semantics/openAccess </ dc:rights >
  18. < dc:source > Articles publicats en revistes (Matemàtiques i Informàtica) </ dc:source >
  19. < dc:subject > Teoria de conjunts </ dc:subject >
  20. < dc:subject > Lògica matemàtica </ dc:subject >
  21. < dc:subject > Set theory </ dc:subject >
  22. < dc:subject > Mathematical logic </ dc:subject >
  23. < dc:title > $\Sigma_1$-definability at higher cardinals: Thin sets, almost disjoint families and long well-orders </ dc:title >
  24. < dc:type > info:eu-repo/semantics/article </ dc:type >
  25. < dc:type > info:eu-repo/semantics/publishedVersion </ dc:type >
  26. < edm:type > TEXT </ edm:type >
  </ edm:ProvidedCHO >
2. < ore:Aggregation about =" https://catalonica.bnc.cat/catalonicahub/lod/oai:diposit.ub.edu:2445_--_220654#ent1 " >
  1. < edm:aggregatedCHO resource =" https://catalonica.bnc.cat/catalonicahub/lod/oai:diposit.ub.edu:2445_--_220654#ent0 " />
  2. < edm:dataProvider > Dipòsit Digital de la Universitat de Barcelona </ edm:dataProvider >
  3. < edm:isShownAt resource =" https://hdl.handle.net/2445/220654 " />
  4. < edm:isShownBy resource =" http://diposit.ub.edu/dspace/bitstream/2445/220654/1/892974.pdf " />
  5. < edm:provider > Catalònica </ edm:provider >
  6. < edm:rights resource =" http://creativecommons.org/licenses/by/4.0/ " />
  </ ore:Aggregation >
</ rdf:RDF >

marc

Descarregar XML

<?xml version="1.0" encoding="UTF-8" ?>

< record schemaLocation =" http://www.loc.gov/MARC21/slim http://www.loc.gov/standards/marcxml/schema/MARC21slim.xsd " >
1. < leader > 00925njm 22002777a 4500 </ leader >
2. < datafield ind1 =" " ind2 =" " tag =" 042 " >
  1. < subfield code =" a " > dc </ subfield >
  </ datafield >
3. < datafield ind1 =" " ind2 =" " tag =" 720 " >
  1. < subfield code =" a " > Lücke, Philipp </ subfield >
  2. < subfield code =" e " > author </ subfield >
  </ datafield >
4. < datafield ind1 =" " ind2 =" " tag =" 720 " >
  1. < subfield code =" a " > Müller, Sandra </ subfield >
  2. < subfield code =" e " > author </ subfield >
  </ datafield >
5. < datafield ind1 =" " ind2 =" " tag =" 260 " >
  1. < subfield code =" c " > 2023-11-17 </ subfield >
  </ datafield >
6. < datafield ind1 =" " ind2 =" " tag =" 520 " >
  1. < subfield code =" a " > 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 $\omega_1$. </ subfield >
  </ datafield >
7. < datafield ind1 =" 8 " ind2 =" " tag =" 024 " >
  1. < subfield code =" a " > 2050-5094 </ subfield >
  </ datafield >
8. < datafield ind1 =" 8 " ind2 =" " tag =" 024 " >
  1. < subfield code =" a " > https://hdl.handle.net/2445/220654 </ subfield >
  </ datafield >
9. < datafield ind1 =" 8 " ind2 =" " tag =" 024 " >
  1. < subfield code =" a " > 758203 </ subfield >
  </ datafield >
10. < datafield ind1 =" 0 " ind2 =" 0 " tag =" 245 " >
  1. < subfield code =" a " > $\Sigma_1$-definability at higher cardinals: Thin sets, almost disjoint families and long well-orders </ subfield >
  </ datafield >
</ record >

mets

Descarregar XML

<?xml version="1.0" encoding="UTF-8" ?>

< mets ID =" DSpace_ITEM_2445-220654 " OBJID =" hdl:2445/220654 " PROFILE =" DSpace METS SIP Profile 1.0 " TYPE =" DSpace ITEM " schemaLocation =" http://www.loc.gov/METS/ http://www.loc.gov/standards/mets/mets.xsd " >
1. < metsHdr CREATEDATE =" 2025-06-11T13:48:51Z " >
  1. < agent ROLE =" CUSTODIAN " TYPE =" ORGANIZATION " >
    1. < name > Dipòsit Digital de la Universitat de Barcelona </ name >
    </ agent >
  </ metsHdr >
2. < dmdSec ID =" DMD_2445_220654 " >
  1. < mdWrap MDTYPE =" MODS " >
    1. < xmlData schemaLocation =" http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-1.xsd " >
      1. < mods:mods schemaLocation =" http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-1.xsd " >
        
        < mods:name >
        
        < mods:role >
        
        < mods:roleTerm type =" text " > author </ mods:roleTerm >
        
        </ mods:role >
        
        < mods:namePart > Lücke, Philipp </ mods:namePart >
        
        </ mods:name >
        
        < mods:name >
        
        < mods:role >
        
        < mods:roleTerm type =" text " > author </ mods:roleTerm >
        
        </ mods:role >
        
        < mods:namePart > Müller, Sandra </ mods:namePart >
        
        </ mods:name >
        
        < mods:extension >
        
        < mods:dateAccessioned encoding =" iso8601 " > 2025-04-28T06:38:22Z </ mods:dateAccessioned >
        
        </ mods:extension >
        
        < mods:extension >
        
        < mods:dateAvailable encoding =" iso8601 " > 2025-04-28T06:38:22Z </ mods:dateAvailable >
        
        </ mods:extension >
        
        < mods:originInfo >
        
        < mods:dateIssued encoding =" iso8601 " > 2023-11-17 </ mods:dateIssued >
        
        </ mods:originInfo >
        
        < mods:identifier type =" issn " > 2050-5094 </ mods:identifier >
        
        < mods:identifier type =" uri " > https://hdl.handle.net/2445/220654 </ mods:identifier >
        
        < mods:identifier type =" idgrec " > 758203 </ mods:identifier >
        
        < mods:abstract > 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 $\omega_1$. </ mods:abstract >
        
        < mods:language >
        
        < mods:languageTerm authority =" rfc3066 " > eng </ mods:languageTerm >
        
        </ mods:language >
        
        < mods:accessCondition type =" useAndReproduction " > cc-by (c) Lücke, P. et al., 2023 </ mods:accessCondition >
        
        < mods:titleInfo >
        
        < mods:title > $\Sigma_1$-definability at higher cardinals: Thin sets, almost disjoint families and long well-orders </ mods:title >
        
        </ mods:titleInfo >
        
        < mods:genre > info:eu-repo/semantics/article </ mods:genre >
        
        </ mods:mods >
      </ xmlData >
    </ mdWrap >
  </ dmdSec >
3. < amdSec ID =" TMD_2445_220654 " >
  1. < rightsMD ID =" RIG_2445_220654 " >
    1. < mdWrap MDTYPE =" OTHER " MIMETYPE =" text/plain " OTHERMDTYPE =" DSpaceDepositLicense " >
      1. < binData > 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 </ binData >
      </ mdWrap >
    </ rightsMD >
  </ amdSec >
4. < amdSec ID =" FO_2445_220654_1 " >
  1. < techMD ID =" TECH_O_2445_220654_1 " >
    1. < mdWrap MDTYPE =" PREMIS " >
      1. < xmlData schemaLocation =" http://www.loc.gov/standards/premis http://www.loc.gov/standards/premis/PREMIS-v1-0.xsd " >
        
        < premis:premis >
        
        < premis:object >
        
        < premis:objectIdentifier >
        
        < premis:objectIdentifierType > URL </ premis:objectIdentifierType >
        
        < premis:objectIdentifierValue > http://diposit.ub.edu/dspace/bitstream/2445/220654/1/892974.pdf </ premis:objectIdentifierValue >
        
        </ premis:objectIdentifier >
        
        < premis:objectCategory > File </ premis:objectCategory >
        
        < premis:objectCharacteristics >
        
        < premis:fixity >
        
        < premis:messageDigestAlgorithm > MD5 </ premis:messageDigestAlgorithm >
        
        < premis:messageDigest > e8c67ca0f628e48d68ecdfd0416d4b72 </ premis:messageDigest >
        
        </ premis:fixity >
        
        < premis:size > 478202 </ premis:size >
        
        < premis:format >
        
        < premis:formatDesignation >
        
        < premis:formatName > application/pdf </ premis:formatName >
        
        </ premis:formatDesignation >
        
        </ premis:format >
        
        </ premis:objectCharacteristics >
        
        < premis:originalName > 892974.pdf </ premis:originalName >
        
        </ premis:object >
        
        </ premis:premis >
        
        </ xmlData >
      </ mdWrap >
    </ techMD >
  </ amdSec >
5. < amdSec ID =" FT_2445_220654_6 " >
  1. < techMD ID =" TECH_T_2445_220654_6 " >
    1. < mdWrap MDTYPE =" PREMIS " >
      1. < xmlData schemaLocation =" http://www.loc.gov/standards/premis http://www.loc.gov/standards/premis/PREMIS-v1-0.xsd " >
        
        < premis:premis >
        
        < premis:object >
        
        < premis:objectIdentifier >
        
        < premis:objectIdentifierType > URL </ premis:objectIdentifierType >
        
        < premis:objectIdentifierValue > http://diposit.ub.edu/dspace/bitstream/2445/220654/6/892974.pdf.txt </ premis:objectIdentifierValue >
        
        </ premis:objectIdentifier >
        
        < premis:objectCategory > File </ premis:objectCategory >
        
        < premis:objectCharacteristics >
        
        < premis:fixity >
        
        < premis:messageDigestAlgorithm > MD5 </ premis:messageDigestAlgorithm >
        
        < premis:messageDigest > 93e884476d179f64eafaea0ea34d079e </ premis:messageDigest >
        
        </ premis:fixity >
        
        < premis:size > 147633 </ premis:size >
        
        < premis:format >
        
        < premis:formatDesignation >
        
        < premis:formatName > text/plain </ premis:formatName >
        
        </ premis:formatDesignation >
        
        </ premis:format >
        
        </ premis:objectCharacteristics >
        
        < premis:originalName > 892974.pdf.txt </ premis:originalName >
        
        </ premis:object >
        
        </ premis:premis >
        
        </ xmlData >
      </ mdWrap >
    </ techMD >
  </ amdSec >
6. < fileSec >
  1. < fileGrp USE =" ORIGINAL " >
    1. < file ADMID =" FO_2445_220654_1 " CHECKSUM =" e8c67ca0f628e48d68ecdfd0416d4b72 " CHECKSUMTYPE =" MD5 " GROUPID =" GROUP_BITSTREAM_2445_220654_1 " ID =" BITSTREAM_ORIGINAL_2445_220654_1 " MIMETYPE =" application/pdf " SEQ =" 1 " SIZE =" 478202 " >
      1. < FLocat LOCTYPE =" URL " href =" http://diposit.ub.edu/dspace/bitstream/2445/220654/1/892974.pdf " type =" simple " />
      </ file >
    </ fileGrp >
  2. < fileGrp USE =" TEXT " >
    1. < file ADMID =" FT_2445_220654_6 " CHECKSUM =" 93e884476d179f64eafaea0ea34d079e " CHECKSUMTYPE =" MD5 " GROUPID =" GROUP_BITSTREAM_2445_220654_6 " ID =" BITSTREAM_TEXT_2445_220654_6 " MIMETYPE =" text/plain " SEQ =" 6 " SIZE =" 147633 " >
      1. < FLocat LOCTYPE =" URL " href =" http://diposit.ub.edu/dspace/bitstream/2445/220654/6/892974.pdf.txt " type =" simple " />
      </ file >
    </ fileGrp >
  </ fileSec >
7. < structMap LABEL =" DSpace Object " TYPE =" LOGICAL " >
  1. < div ADMID =" DMD_2445_220654 " TYPE =" DSpace Object Contents " >
    1. < div TYPE =" DSpace BITSTREAM " >
      1. < fptr FILEID =" BITSTREAM_ORIGINAL_2445_220654_1 " />
      </ div >
    </ div >
  </ structMap >
</ mets >

mods

Descarregar XML

<?xml version="1.0" encoding="UTF-8" ?>

< mods:mods schemaLocation =" http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-1.xsd " >
1. < mods:name >
  1. < mods:namePart > Lücke, Philipp </ mods:namePart >
  </ mods:name >
2. < mods:name >
  1. < mods:namePart > Müller, Sandra </ mods:namePart >
  </ mods:name >
3. < mods:extension >
  1. < mods:dateAvailable encoding =" iso8601 " > 2025-04-28T06:38:22Z </ mods:dateAvailable >
  </ mods:extension >
4. < mods:extension >
  1. < mods:dateAccessioned encoding =" iso8601 " > 2025-04-28T06:38:22Z </ mods:dateAccessioned >
  </ mods:extension >
5. < mods:originInfo >
  1. < mods:dateIssued encoding =" iso8601 " > 2023-11-17 </ mods:dateIssued >
  </ mods:originInfo >
6. < mods:identifier type =" issn " > 2050-5094 </ mods:identifier >
7. < mods:identifier type =" uri " > https://hdl.handle.net/2445/220654 </ mods:identifier >
8. < mods:identifier type =" idgrec " > 758203 </ mods:identifier >
9. < mods:abstract > 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 $\omega_1$. </ mods:abstract >
10. < mods:language >
  1. < mods:languageTerm > eng </ mods:languageTerm >
  </ mods:language >
11. < mods:accessCondition type =" useAndReproduction " > http://creativecommons.org/licenses/by/4.0/ </ mods:accessCondition >
12. < mods:accessCondition type =" useAndReproduction " > info:eu-repo/semantics/openAccess </ mods:accessCondition >
13. < mods:accessCondition type =" useAndReproduction " > cc-by (c) Lücke, P. et al., 2023 </ mods:accessCondition >
14. < mods:titleInfo >
  1. < mods:title > $\Sigma_1$-definability at higher cardinals: Thin sets, almost disjoint families and long well-orders </ mods:title >
  </ mods:titleInfo >
15. < mods:genre > info:eu-repo/semantics/article </ mods:genre >
16. < mods:genre > info:eu-repo/semantics/publishedVersion </ mods:genre >
</ mods:mods >

qdc

Descarregar XML

<?xml version="1.0" encoding="UTF-8" ?>

< qdc:qualifieddc schemaLocation =" http://purl.org/dc/elements/1.1/ http://dublincore.org/schemas/xmls/qdc/2006/01/06/dc.xsd http://purl.org/dc/terms/ http://dublincore.org/schemas/xmls/qdc/2006/01/06/dcterms.xsd http://dspace.org/qualifieddc/ http://www.ukoln.ac.uk/metadata/dcmi/xmlschema/qualifieddc.xsd " >
1. < dc:title > $\Sigma_1$-definability at higher cardinals: Thin sets, almost disjoint families and long well-orders </ dc:title >
2. < dc:creator > Lücke, Philipp </ dc:creator >
3. < dc:creator > Müller, Sandra </ dc:creator >
4. < dc:subject.classification > Teoria de conjunts </ dc:subject.classification >
5. < dc:subject.classification > Lògica matemàtica </ dc:subject.classification >
6. < dc:subject.other > Set theory </ dc:subject.other >
7. < dc:subject.other > Mathematical logic </ dc:subject.other >
8. < dcterms:abstract > 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 $\omega_1$. </ dcterms:abstract >
9. < dcterms:dateAccepted > 2025-04-28T06:38:22Z </ dcterms:dateAccepted >
10. < dcterms:available > 2025-04-28T06:38:22Z </ dcterms:available >
11. < dcterms:created > 2025-04-28T06:38:22Z </ dcterms:created >
12. < dcterms:issued > 2023-11-17 </ dcterms:issued >
13. < dc:type > info:eu-repo/semantics/article </ dc:type >
14. < dc:type > info:eu-repo/semantics/publishedVersion </ dc:type >
15. < dc:identifier > 2050-5094 </ dc:identifier >
16. < dc:identifier > https://hdl.handle.net/2445/220654 </ dc:identifier >
17. < dc:identifier > 758203 </ dc:identifier >
18. < dc:identifier.issn > 2050-5094 </ dc:identifier.issn >
19. < dc:language > eng </ dc:language >
20. < dc:relation > Reproducció del document publicat a: https://doi.org/10.1017/fms.2023.102 </ dc:relation >
21. < dc:relation > 2023, vol. 11 </ dc:relation >
22. < dc:relation > https://doi.org/10.1017/fms.2023.102 </ dc:relation >
23. < dc:rights > http://creativecommons.org/licenses/by/4.0/ </ dc:rights >
24. < dc:rights > info:eu-repo/semantics/openAccess </ dc:rights >
25. < dc:rights > cc-by (c) Lücke, P. et al., 2023 </ dc:rights >
26. < dc:source > Articles publicats en revistes (Matemàtiques i Informàtica) </ dc:source >
</ qdc:qualifieddc >

rdf

Descarregar XML

<?xml version="1.0" encoding="UTF-8" ?>

< rdf:RDF schemaLocation =" http://www.openarchives.org/OAI/2.0/rdf/ http://www.openarchives.org/OAI/2.0/rdf.xsd " >
1. < ow:Publication about =" oai:diposit.ub.edu:2445/220654 " >
  1. < dc:title > $\Sigma_1$-definability at higher cardinals: Thin sets, almost disjoint families and long well-orders </ dc:title >
  2. < dc:creator > Lücke, Philipp </ dc:creator >
  3. < dc:creator > Müller, Sandra </ dc:creator >
  4. < dc: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 $\omega_1$. </ dc:description >
  5. < dc:date > 2025-04-28T06:38:22Z </ dc:date >
  6. < dc:date > 2025-04-28T06:38:22Z </ dc:date >
  7. < dc:date > 2023-11-17 </ dc:date >
  8. < dc:date > 2025-04-28T06:38:23Z </ dc:date >
  9. < dc:type > info:eu-repo/semantics/article </ dc:type >
  10. < dc:type > info:eu-repo/semantics/publishedVersion </ dc:type >
  11. < dc:identifier > 2050-5094 </ dc:identifier >
  12. < dc:identifier > https://hdl.handle.net/2445/220654 </ dc:identifier >
  13. < dc:identifier > 758203 </ dc:identifier >
  14. < dc:language > eng </ dc:language >
  15. < dc:relation > Reproducció del document publicat a: https://doi.org/10.1017/fms.2023.102 </ dc:relation >
  16. < dc:relation > 2023, vol. 11 </ dc:relation >
  17. < dc:relation > https://doi.org/10.1017/fms.2023.102 </ dc:relation >
  18. < dc:rights > http://creativecommons.org/licenses/by/4.0/ </ dc:rights >
  19. < dc:rights > info:eu-repo/semantics/openAccess </ dc:rights >
  20. < dc:rights > cc-by (c) Lücke, P. et al., 2023 </ dc:rights >
  21. < dc:source > Articles publicats en revistes (Matemàtiques i Informàtica) </ dc:source >
  </ ow:Publication >
</ rdf:RDF >

Biblioteca de Catalunya Carrer de l'Hospital, 56. 08001 Barcelona Adreça electrònica: catalonica@bnc.cat Tlf.: +34 932 702 300

Repositori OAI