{"doi":"10.2178/jsl/1107298524","title":"On relatively analytic and Borel subsets","abstract":"<jats:title>Abstract</jats:title><jats:p>Define <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200007337_inline1\" /> to be the smallest cardinality of a function <jats:italic>f</jats:italic>: <jats:italic>X→Y</jats:italic> with I, <jats:italic>X, Y</jats:italic>, ⊆ 2<jats:sup><jats:italic>ω</jats:italic></jats:sup> such that there is no Borel function <jats:italic>g</jats:italic> ⊇ <jats:italic>f</jats:italic>. In this paper we prove that it is relatively consistent with ZFC to have b &lt; <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200007337_inline1\" /> where b is, as usual, smallest cardinality of an unbounded family in Ω<jats:sup><jats:italic>ω</jats:italic></jats:sup>. This answers a question raised by Zapletal.</jats:p><jats:p>We also show that it is relatively consistent with ZFC that there exists <jats:italic>X</jats:italic> ⊆ 2<jats:sup><jats:italic>ω</jats:italic></jats:sup> such that the Borei order of <jats:italic>X</jats:italic> is bounded but there exists a relatively analytic subset of <jats:italic>X</jats:italic> which is not relatively coanalytic. This answers a question of Mauldin.</jats:p>","journal":"Journal of Symbolic Logic","year":2005,"id":604780,"datarank":0.35654572553030905,"base_score":1.3862943611198906,"endowment":1.3862943611198906,"self_citation_contribution":0.20794415416798362,"citation_network_contribution":0.1486015713623254,"self_endowment_contribution":0.20794415416798362,"citer_contribution":0.1486015713623254,"corpus_percentile":null,"corpus_rank":null,"citation_count":3,"citer_count":1,"citers_with_citation_signal":1,"citers_with_endowment":1,"datacite_reuse_total":0,"is_dataset":false,"is_dataset_confidence":null,"is_data_producer":false,"deposit_databanks":null,"is_oa":false,"file_count":0,"downloads":0,"has_version_chain":false,"published_date":null,"fair_score":null,"fair_percentile":null,"algorithm_id":"datarank_citation_only_1hop_v6","ranking_scope":"data_only","authors":[{"id":1551821,"name":"Arnold W. Miller","orcid":null,"position":0,"is_corresponding":false}],"reference_count":0,"raw_metadata":{"has_enrichment":true,"resolved":true,"title":"On relatively analytic and Borel subsets","abstract":"<jats:title>Abstract</jats:title><jats:p>Define <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200007337_inline1\" /> to be the smallest cardinality of a function <jats:italic>f</jats:italic>: <jats:italic>X→Y</jats:italic> with I, <jats:italic>X, Y</jats:italic>, ⊆ 2<jats:sup><jats:italic>ω</jats:italic></jats:sup> such that there is no Borel function <jats:italic>g</jats:italic> ⊇ <jats:italic>f</jats:italic>. In this paper we prove that it is relatively consistent with ZFC to have b &lt; <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200007337_inline1\" /> where b is, as usual, smallest cardinality of an unbounded family in Ω<jats:sup><jats:italic>ω</jats:italic></jats:sup>. This answers a question raised by Zapletal.</jats:p><jats:p>We also show that it is relatively consistent with ZFC that there exists <jats:italic>X</jats:italic> ⊆ 2<jats:sup><jats:italic>ω</jats:italic></jats:sup> such that the Borei order of <jats:italic>X</jats:italic> is bounded but there exists a relatively analytic subset of <jats:italic>X</jats:italic> which is not relatively coanalytic. This answers a question of Mauldin.</jats:p>","is_dataset_classified":null,"base_score":1.3862943611198906,"endowment":1.3862943611198906,"datacite_reuse_total":0,"file_count":0,"downloads":0,"views":0,"has_version_chain":false,"is_dataset":false,"is_oa":false,"pmid":"21097893","pmcid":null,"openalex_id":"https://openalex.org/W1970521282","authors":[],"funders":[],"total_grants":0,"fwci":0.8305,"citation_percentile":0.67308331,"influential_citations":0,"citation_trend":[{"year":2013,"count":1}],"oa_status":"closed","license":"https://www.cambridge.org/core/terms","oa_locations":[{"url":"https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022481200007337","host_type":"publisher"},{"url":"https://doi.org/10.2178/jsl/1107298524","host_type":"journal"},{"url":"http://projecteuclid.org/euclid.jsl/1107298524","host_type":"repository"},{"url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.641.3683","host_type":""}],"fields_of_study":["Advanced Topology and Set Theory","Limits and Structures in Graph Theory","Rings, Modules, and Algebras"],"mesh_terms":[],"keywords":["Cardinality (data modeling)","Mathematics","Bounded function","Combinatorics","Existential quantification","Function (biology)","Order (exchange)","Discrete mathematics","Mathematical analysis","Computer science"],"sdg_mappings":[],"linked_datasets":[],"clinical_trials":[],"software_tools":[],"database_accessions":[],"source":"live","citation_network_status":"fetched"},"created_at":"2026-07-30T00:53:50.890977Z","pmid":null,"pmcid":null,"fwci":null,"citation_percentile":null,"influential_citations":0,"oa_status":null,"license":null,"views":0,"total_file_size_bytes":0,"version_count":0,"fair_f":null,"fair_a":null,"fair_i":null,"fair_r":null,"fair_zscore":null,"fair_rationale":null,"fair_model":null,"fair_agent_version":null,"fair_fulltext_source":null,"fair_has_llm":null,"fair_computed_at":null,"clinical_trials":[],"software_tools":[],"db_accessions":[],"linked_datasets":[],"topics":[]}