{"doi":"10.2307/2272341","title":"A result concerning cardinalities of ultraproducts","abstract":"<jats:p>The cardinality problem for ultraproducts is as follows: Given an ultrafilter <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200065403_inline1\"/> over a set <jats:italic>I</jats:italic> and cardinals α<jats:sub><jats:italic>i</jats:italic></jats:sub>, <jats:italic>i</jats:italic> ∈ <jats:italic>I</jats:italic>, what is the cardinality of the ultraproduct <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200065403_inline2\"/>? Although many special results are known, several problems remain open (see [5] for a survey). For example, consider a uniform ultrafilter <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200065403_inline1\"/> over a set <jats:italic>I</jats:italic> of power <jats:italic>κ</jats:italic> (uniform means that all elements of <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200065403_inline1\"/> have power <jats:italic>κ</jats:italic>). It is open whether every countably incomplete <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200065403_inline1\"/> has the property that, for all infinite α, the ultra-power <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200065403_inline3\"/> has power <jats:italic>α<jats:sup>κ</jats:sup></jats:italic>. However, it is shown in [4] that certain countably incomplete <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200065403_inline1\"/>, namely the <jats:italic>κ</jats:italic>-regular <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200065403_inline1\"/>, have this property.</jats:p><jats:p>This paper is about another cardinality property of ultrafilters which was introduced by Eklof [1] to study ultraproducts of abelian groups. It is open whether every countably incomplete ultrafilter has the Eklof property. We shall show that certain countably incomplete ultrafilters, the <jats:italic>κ</jats:italic>-good ultrafilters, do have this property. The <jats:italic>κ</jats:italic>-good ultrafilters are important in model theory because they are exactly the ultrafilters <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200065403_inline1\"/> such that every ultraproduct modulo <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200065403_inline1\"/> is <jats:italic>κ</jats:italic>-saturated (see [5]).</jats:p><jats:p>Let <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S0022481200065403_inline1\"/> be an ultrafilter on a set <jats:italic>I</jats:italic>. Let α<jats:sub><jats:italic>i, n</jats:italic></jats:sub>, <jats:italic>i</jats:italic> ∈ <jats:italic>I</jats:italic>, <jats:italic>n</jats:italic> ∈ ω, be cardinals and α<jats:sub><jats:italic>i, n</jats:italic></jats:sub>, ≥ α<jats:sub><jats:italic>i, m</jats:italic></jats:sub> if <jats:italic>n</jats:italic> &lt; <jats:italic>m</jats:italic>. Let</jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" orientation=\"portrait\" mime-subtype=\"gif\" mimetype=\"image\" position=\"float\" xlink:type=\"simple\" xlink:href=\"S0022481200065403_eqnU1\"/></jats:disp-formula>.</jats:p><jats:p>Then <jats:italic>ρ<jats:sub>n</jats:sub></jats:italic> are nonincreasing and therefore there is some <jats:italic>m</jats:italic> and <jats:italic>ρ</jats:italic> such that <jats:italic>ρ<jats:sub>n</jats:sub></jats:italic> = <jats:italic>ρ</jats:italic> if <jats:italic>n</jats:italic> ≥ <jats:italic>m</jats:italic>. We call <jats:italic>ρ</jats:italic> the <jats:italic>eventual value</jats:italic> (abbreviated ev val) of <jats:italic>ρ<jats:sub>n</jats:sub></jats:italic>.</jats:p>","journal":"Journal of Symbolic Logic","year":1974,"id":37536,"datarank":0.7972844389081711,"base_score":1.9459101490553132,"endowment":1.9459101490553132,"self_citation_contribution":0.29188652235829704,"citation_network_contribution":0.5053979165498741,"self_endowment_contribution":0.29188652235829704,"citer_contribution":0.5053979165498741,"corpus_percentile":null,"corpus_rank":null,"citation_count":6,"citer_count":6,"citers_with_citation_signal":3,"citers_with_endowment":3,"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":187916,"name":"Karel Prikry","orcid":null,"position":1,"is_corresponding":false},{"id":187915,"name":"H. Jerome Keisler","orcid":null,"position":0,"is_corresponding":false}],"reference_count":0,"raw_metadata":{"has_enrichment":true,"base_score":1.9459101490553132,"endowment":1.9459101490553132,"datacite_reuse_total":0,"file_count":0,"downloads":0,"views":0,"has_version_chain":false,"is_dataset":false,"is_oa":false,"pmid":"24523987","pmcid":null,"openalex_id":"https://openalex.org/W2147918975","authors":[],"funders":[],"total_grants":0,"fwci":0.0,"citation_percentile":0.27697016,"influential_citations":0,"citation_trend":[{"year":2014,"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/S0022481200065403","host_type":"publisher"},{"url":"https://doi.org/10.2307/2272341","host_type":"journal"},{"url":"http://projecteuclid.org/euclid.jsl/1183738946","host_type":"repository"}],"fields_of_study":["Advanced Topology and Set Theory","Rings, Modules, and Algebras","Mathematical and Theoretical Analysis","Computer Science","Mathematics"],"mesh_terms":[],"keywords":["Ultrafilter","Ultraproduct","Cardinality (data modeling)","Power set","Mathematics","Countable set","Property (philosophy)","Discrete mathematics","Combinatorics","Regular cardinal","Set (abstract data type)","Cofinality","Power (physics)","Uncountable set","Computer science","Philosophy"],"sdg_mappings":[],"linked_datasets":[],"clinical_trials":[],"software_tools":[],"database_accessions":[],"source":"live","citation_network_status":"fetched"},"created_at":"2026-06-10T19:50:43.993520Z","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":[]}