{"id":"https://openalex.org/W2123679842","doi":"https://doi.org/10.2307/2274188","title":"Infinite subscripts from infinite exponents","display_name":"Infinite subscripts from infinite exponents","publication_year":1984,"publication_date":"1984-06-01","ids":{"openalex":"https://openalex.org/W2123679842","doi":"https://doi.org/10.2307/2274188","mag":"2123679842"},"language":"en","primary_location":{"id":"doi:10.2307/2274188","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2274188","pdf_url":null,"source":{"id":"https://openalex.org/S177675236","display_name":"Journal of Symbolic Logic","issn_l":"0022-4812","issn":["0022-4812","1943-5886"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Symbolic Logic","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5001636447","display_name":"James E. Baumgartner","orcid":"https://orcid.org/0000-0002-9589-3504"},"institutions":[{"id":"https://openalex.org/I107672454","display_name":"Dartmouth College","ror":"https://ror.org/049s0rh22","country_code":"US","type":"education","lineage":["https://openalex.org/I107672454"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"James E. Baumgartner","raw_affiliation_strings":["Dartmouth College, Hanover, New Hampshire 03755","Dartmouth College Hanover, New Hampshire 03755"],"affiliations":[{"raw_affiliation_string":"Dartmouth College, Hanover, New Hampshire 03755","institution_ids":["https://openalex.org/I107672454"]},{"raw_affiliation_string":"Dartmouth College Hanover, New Hampshire 03755","institution_ids":["https://openalex.org/I107672454"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5111402073","display_name":"James M. Henle","orcid":null},"institutions":[{"id":"https://openalex.org/I202524275","display_name":"Smith College","ror":"https://ror.org/0497crr92","country_code":"US","type":"education","lineage":["https://openalex.org/I202524275"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"James M. Henle","raw_affiliation_strings":["Smith College, Northampton, Massachusetts 01063","Smith College, Northampton, Massachusetts, 01063"],"affiliations":[{"raw_affiliation_string":"Smith College, Northampton, Massachusetts 01063","institution_ids":["https://openalex.org/I202524275"]},{"raw_affiliation_string":"Smith College, Northampton, Massachusetts, 01063","institution_ids":["https://openalex.org/I202524275"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5001636447"],"corresponding_institution_ids":["https://openalex.org/I107672454"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.27130002,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"49","issue":"2","first_page":"558","last_page":"562"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11727","display_name":"Advanced Algebra and Logic","score":0.9962999820709229,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11727","display_name":"Advanced Algebra and Logic","score":0.9962999820709229,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.9872999787330627,"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/T13400","display_name":"Mathematical and Theoretical Analysis","score":0.9778000116348267,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7355077266693115},{"id":"https://openalex.org/keywords/exponent","display_name":"Exponent","score":0.7351023554801941},{"id":"https://openalex.org/keywords/axiom-of-choice","display_name":"Axiom of choice","score":0.6883217096328735},{"id":"https://openalex.org/keywords/partition","display_name":"Partition (number theory)","score":0.6506211757659912},{"id":"https://openalex.org/keywords/homogeneous","display_name":"Homogeneous","score":0.5425096750259399},{"id":"https://openalex.org/keywords/axiom","display_name":"Axiom","score":0.5380304455757141},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.48188120126724243},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.46168580651283264},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.32616958022117615},{"id":"https://openalex.org/keywords/set-theory","display_name":"Set theory","score":0.15635472536087036},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.1270604431629181},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.08789780735969543}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7355077266693115},{"id":"https://openalex.org/C2780388253","wikidata":"https://www.wikidata.org/wiki/Q5421508","display_name":"Exponent","level":2,"score":0.7351023554801941},{"id":"https://openalex.org/C151797676","wikidata":"https://www.wikidata.org/wiki/Q179692","display_name":"Axiom of choice","level":4,"score":0.6883217096328735},{"id":"https://openalex.org/C42812","wikidata":"https://www.wikidata.org/wiki/Q1082910","display_name":"Partition (number theory)","level":2,"score":0.6506211757659912},{"id":"https://openalex.org/C66882249","wikidata":"https://www.wikidata.org/wiki/Q169336","display_name":"Homogeneous","level":2,"score":0.5425096750259399},{"id":"https://openalex.org/C167729594","wikidata":"https://www.wikidata.org/wiki/Q17736","display_name":"Axiom","level":2,"score":0.5380304455757141},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.48188120126724243},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.46168580651283264},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.32616958022117615},{"id":"https://openalex.org/C153046414","wikidata":"https://www.wikidata.org/wiki/Q12482","display_name":"Set theory","level":3,"score":0.15635472536087036},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.1270604431629181},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.08789780735969543},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.2307/2274188","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2274188","pdf_url":null,"source":{"id":"https://openalex.org/S177675236","display_name":"Journal of Symbolic Logic","issn_l":"0022-4812","issn":["0022-4812","1943-5886"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310311721","host_organization_name":"Cambridge University Press","host_organization_lineage":["https://openalex.org/P4310311721","https://openalex.org/P4310311702"],"host_organization_lineage_names":["Cambridge University Press","University of Cambridge"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Symbolic Logic","raw_type":"journal-article"},{"id":"pmh:oai:CULeuclid:euclid.jsl/1183741556","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.jsl/1183741556","pdf_url":null,"source":{"id":"https://openalex.org/S4306400787","display_name":"Project Euclid (Cornell University)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":3,"referenced_works":["https://openalex.org/W2114195278","https://openalex.org/W2155993241","https://openalex.org/W2322754816"],"related_works":["https://openalex.org/W2031083694","https://openalex.org/W1504849543","https://openalex.org/W2467119940","https://openalex.org/W288388583","https://openalex.org/W3194077685","https://openalex.org/W423103167","https://openalex.org/W178365807","https://openalex.org/W2045975809","https://openalex.org/W3112295066","https://openalex.org/W2373903136"],"abstract_inverted_index":{"Definition.":[0],"For":[1,20,231,239],"ordinals":[2],"\u03b1":[3,9,18,21,50,159,164,172],"\u2264":[4,163],"\u03ba":[5,24,34,44,83,86,97,167,170,178,191],",":[6,37,165],"=":[7,10,17,60],"[X]":[8,49],"{p":[11],"\u2286":[12],"X":[13,41,52],"\u2223":[14],"ot":[15],"(p)":[16],"}.":[19],"\u2264,":[22],"\u03ba,":[23],"a":[25],"cardinal,":[26],"holds":[27],"iff":[28,89],"for":[29,56,74,90,118,174],"every":[30],"partition":[31,244],"F:":[32],"[\u039a]":[33,43],"\u2192":[35,84,168],"A":[36,59],"there":[38],"is":[39,53,64,102,108,140],"an":[40,144],"\u2208":[42],"with":[45,120,208],"F":[46],"constant":[47],"on":[48],".":[51,98,179,192,201],"called":[54],"homogeneous":[55],"F.":[57],"When":[58],"2":[61,88,162,199],"the":[62,71,125,131,184,206,216,219,232,240],"subscript":[63,185],"omitted.":[65],"It":[66,139],"has":[67],"been":[68],"known":[69,111],"since":[70],"sixties":[72],"that":[73,143],"finite":[75,235],"exponents":[76],"all":[77,91,175],"such":[78],"properties":[79,129],"are":[80,137,153],"equivalent,":[81],"i.e.,":[82],"(":[85,169],")":[87,171],"n":[92],"&lt;":[93,96,177,190],"\u03c9,":[94],"\u03b2":[95],"The":[99,115],"infinite":[100,145,151],"case":[101],"much":[103],"more":[104],"difficult,":[105],"though":[106],"it":[107],"not":[109],"actually":[110],"to":[112,198,218],"be":[113,148,187],"different.":[114],"usual":[116],"methods":[117],"dealing":[119],"partitions":[121],"fail,":[122],"and":[123,214,246],"as":[124],"weakest":[126],"of":[127,133,211,221,234],"these":[128],"violates":[130],"Axiom":[132],"Choice,":[134],"many":[135],"techniques":[136],"unavailable.":[138],"extremely":[141],"unlikely":[142],"exponent":[146],"can":[147,186],"increased,":[149],"but":[150],"subscripts":[152],"generally":[154],"possible.":[155],"Theorem.":[156],"If":[157],"\u2225":[158,160],"\u00b7":[161],"then":[166],"implies":[173],"\u03bb":[176,189,200],"In":[180,193,202],"\u00a71,":[181],"we":[182,195,204,227],"show":[183],"any":[188],"\u00a72":[194],"extend":[196],"this":[197],"\u00a73":[203],"discuss":[205],"possibilities":[207],"limited":[209],"amounts":[210],"well-ordered":[212],"choice,":[213,247],"apply":[215],"theorem":[217],"problem":[220],"obliging":[222],"ordinals.":[223],"Except":[224],"in":[225],"\u00a73,":[226],"assume":[228],"no":[229],"choice.":[230],"story":[233],"exponents,":[236],"see":[237,248],"[1].":[238],"relationship":[241],"between":[242],"infinite-exponent":[243],"relations":[245],"[4].":[249]},"counts_by_year":[],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}
