{"id":"https://openalex.org/W2027586194","doi":"https://doi.org/10.1002/malq.200710073","title":"Unions and the axiom of choice","display_name":"Unions and the axiom of choice","publication_year":2008,"publication_date":"2008-11-04","ids":{"openalex":"https://openalex.org/W2027586194","doi":"https://doi.org/10.1002/malq.200710073","mag":"2027586194"},"language":"en","primary_location":{"id":"doi:10.1002/malq.200710073","is_oa":true,"landing_page_url":"https://doi.org/10.1002/malq.200710073","pdf_url":"https://onlinelibrary.wiley.com/doi/pdfdirect/10.1002/malq.200710073","source":{"id":"https://openalex.org/S172351290","display_name":"Mathematical logic quarterly","issn_l":"0942-5616","issn":["0942-5616","1521-3870"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematical Logic Quarterly","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://onlinelibrary.wiley.com/doi/pdfdirect/10.1002/malq.200710073","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5028328758","display_name":"Omar De la Cruz","orcid":"https://orcid.org/0000-0001-9489-4647"},"institutions":[{"id":"https://openalex.org/I40347166","display_name":"University of Chicago","ror":"https://ror.org/024mw5h28","country_code":"US","type":"education","lineage":["https://openalex.org/I40347166"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Omar De la Cruz","raw_affiliation_strings":["Department of Statistics, University of Chicago, 5734 S. University Avenue, Chicago, IL 60637, USA","Department of Statistics , University of Chicago , 5734 S. University Avenue, Chicago, IL 60637, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Statistics, University of Chicago, 5734 S. University Avenue, Chicago, IL 60637, USA","institution_ids":["https://openalex.org/I40347166"]},{"raw_affiliation_string":"Department of Statistics , University of Chicago , 5734 S. University Avenue, Chicago, IL 60637, USA","institution_ids":["https://openalex.org/I40347166"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101476392","display_name":"Eric Hall","orcid":"https://orcid.org/0000-0001-6159-7494"},"institutions":[{"id":"https://openalex.org/I75421653","display_name":"University of Missouri\u2013Kansas City","ror":"https://ror.org/01w0d5g70","country_code":"US","type":"education","lineage":["https://openalex.org/I75421653"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Eric J. Hall","raw_affiliation_strings":["Department of Mathematics and Statistics, University of Missouri, Kansas City, MO 64110, USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics, University of Missouri, Kansas City, MO 64110, USA","institution_ids":["https://openalex.org/I75421653"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5108590548","display_name":"Paul C. Howard","orcid":null},"institutions":[{"id":"https://openalex.org/I138216421","display_name":"Eastern Michigan University","ror":"https://ror.org/02ehshm78","country_code":"US","type":"education","lineage":["https://openalex.org/I138216421"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Paul Howard","raw_affiliation_strings":["Department of Mathematics, Eastern Michigan University, Ypsilanti, MI 48197, USA","Department of Mathematics Eastern Michigan University Ypsilanti, MI 48197 USA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Eastern Michigan University, Ypsilanti, MI 48197, USA","institution_ids":["https://openalex.org/I138216421"]},{"raw_affiliation_string":"Department of Mathematics Eastern Michigan University Ypsilanti, MI 48197 USA","institution_ids":["https://openalex.org/I138216421"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5028285276","display_name":"Kyriakos Keremedis","orcid":"https://orcid.org/0000-0001-8453-3477"},"institutions":[{"id":"https://openalex.org/I98805295","display_name":"University of the Aegean","ror":"https://ror.org/03zsp3p94","country_code":"GR","type":"education","lineage":["https://openalex.org/I98805295"]}],"countries":["GR"],"is_corresponding":false,"raw_author_name":"Kyriakos Keremedis","raw_affiliation_strings":["Department of Mathematics, University of the Aegean, Karlovassi 83200, Samos, Greece","Department of Mathematics; University of the Aegean; Karlovassi 83200 Samos Greece"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of the Aegean, Karlovassi 83200, Samos, Greece","institution_ids":["https://openalex.org/I98805295"]},{"raw_affiliation_string":"Department of Mathematics; University of the Aegean; Karlovassi 83200 Samos Greece","institution_ids":["https://openalex.org/I98805295"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5073538536","display_name":"Jean E. Rubin","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Jean E. Rubin","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":5,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":{"value":2630,"currency":"USD","value_usd":2630},"apc_paid":null,"fwci":0.3238,"has_fulltext":false,"cited_by_count":12,"citation_normalized_percentile":{"value":0.70994847,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"54","issue":"6","first_page":"652","last_page":"665"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10258","display_name":"Philosophy and Theoretical Science","score":0.9749000072479248,"subfield":{"id":"https://openalex.org/subfields/3205","display_name":"Experimental and Cognitive Psychology"},"field":{"id":"https://openalex.org/fields/32","display_name":"Psychology"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},"topics":[{"id":"https://openalex.org/T10258","display_name":"Philosophy and Theoretical Science","score":0.9749000072479248,"subfield":{"id":"https://openalex.org/subfields/3205","display_name":"Experimental and Cognitive Psychology"},"field":{"id":"https://openalex.org/fields/32","display_name":"Psychology"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.9480000138282776,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11477","display_name":"Epistemology, Ethics, and Metaphysics","score":0.9470999836921692,"subfield":{"id":"https://openalex.org/subfields/1211","display_name":"Philosophy"},"field":{"id":"https://openalex.org/fields/12","display_name":"Arts and Humanities"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/countable-set","display_name":"Countable set","score":0.9495406746864319},{"id":"https://openalex.org/keywords/axiom-of-choice","display_name":"Axiom of choice","score":0.817703127861023},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7862101793289185},{"id":"https://openalex.org/keywords/axiom","display_name":"Axiom","score":0.7259663343429565},{"id":"https://openalex.org/keywords/assertion","display_name":"Assertion","score":0.5446426868438721},{"id":"https://openalex.org/keywords/choice-function","display_name":"Choice function","score":0.45004260540008545},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.36725854873657227},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.24151787161827087},{"id":"https://openalex.org/keywords/set-theory","display_name":"Set theory","score":0.2331485152244568},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.06454712152481079},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.03696480393409729}],"concepts":[{"id":"https://openalex.org/C110729354","wikidata":"https://www.wikidata.org/wiki/Q185478","display_name":"Countable set","level":2,"score":0.9495406746864319},{"id":"https://openalex.org/C151797676","wikidata":"https://www.wikidata.org/wiki/Q179692","display_name":"Axiom of choice","level":4,"score":0.817703127861023},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7862101793289185},{"id":"https://openalex.org/C167729594","wikidata":"https://www.wikidata.org/wiki/Q17736","display_name":"Axiom","level":2,"score":0.7259663343429565},{"id":"https://openalex.org/C40422974","wikidata":"https://www.wikidata.org/wiki/Q741248","display_name":"Assertion","level":2,"score":0.5446426868438721},{"id":"https://openalex.org/C20656852","wikidata":"https://www.wikidata.org/wiki/Q2513496","display_name":"Choice function","level":2,"score":0.45004260540008545},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.36725854873657227},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.24151787161827087},{"id":"https://openalex.org/C153046414","wikidata":"https://www.wikidata.org/wiki/Q12482","display_name":"Set theory","level":3,"score":0.2331485152244568},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.06454712152481079},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.03696480393409729},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1002/malq.200710073","is_oa":true,"landing_page_url":"https://doi.org/10.1002/malq.200710073","pdf_url":"https://onlinelibrary.wiley.com/doi/pdfdirect/10.1002/malq.200710073","source":{"id":"https://openalex.org/S172351290","display_name":"Mathematical logic quarterly","issn_l":"0942-5616","issn":["0942-5616","1521-3870"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematical Logic Quarterly","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1002/malq.200710073","is_oa":true,"landing_page_url":"https://doi.org/10.1002/malq.200710073","pdf_url":"https://onlinelibrary.wiley.com/doi/pdfdirect/10.1002/malq.200710073","source":{"id":"https://openalex.org/S172351290","display_name":"Mathematical logic quarterly","issn_l":"0942-5616","issn":["0942-5616","1521-3870"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320595","host_organization_name":"Wiley","host_organization_lineage":["https://openalex.org/P4310320595"],"host_organization_lineage_names":["Wiley"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematical Logic Quarterly","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2027586194.pdf"},"referenced_works_count":13,"referenced_works":["https://openalex.org/W281550026","https://openalex.org/W1541012876","https://openalex.org/W1608084331","https://openalex.org/W1991515818","https://openalex.org/W1996087038","https://openalex.org/W2013634817","https://openalex.org/W2033201776","https://openalex.org/W2040799636","https://openalex.org/W2047055229","https://openalex.org/W2055381384","https://openalex.org/W2080644513","https://openalex.org/W2103529109","https://openalex.org/W4232297660"],"related_works":["https://openalex.org/W4214831444","https://openalex.org/W2055381384","https://openalex.org/W4234951658","https://openalex.org/W2015215276","https://openalex.org/W2792089084","https://openalex.org/W2009720468","https://openalex.org/W2491940682","https://openalex.org/W3112295066","https://openalex.org/W2339451782","https://openalex.org/W2027586194"],"abstract_inverted_index":{"Abstract":[0],"We":[1],"study":[2],"statements":[3],"about":[4],"countable":[5,17,45,56,80,83],"and":[6,9,15,18],"well\u2010ordered":[7,19],"unions":[8],"their":[10],"relation":[11],"to":[12,16],"each":[13],"other":[14],"forms":[20],"of":[21,24,44,58,67,72,82],"the":[22,65,79],"axiom":[23,66],"choice.":[25],"Using":[26],"WO":[27,42,49,59,63,70,73],"as":[28],"an":[29],"abbreviation":[30],"for":[31,69],"\u201cwell\u2010orderable\u201d,":[32],"here":[33],"are":[34],"two":[35],"typical":[36],"results:":[37],"The":[38],"assertion":[39],"that":[40,54,78],"every":[41,55],"family":[43,57],"sets":[46,60,74,84],"has":[47,61],"a":[48,62],"union":[50,81],"does":[51,75],"not":[52,76],"imply":[53,77],"union;":[64],"choice":[68],"families":[71],"is":[85],"WO.":[86],"(\u00a9":[87],"2008":[88],"WILEY\u2010VCH":[89],"Verlag":[90],"GmbH":[91],"&amp;":[92],"Co.":[93],"KGaA,":[94],"Weinheim)":[95]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":1},{"year":2013,"cited_by_count":1}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}
