{"id":"https://openalex.org/W2035687351","doi":"https://doi.org/10.1002/malq.19920380142","title":"A NOTATION SYSTEM FOR ORDINAL USING \u03a8\u2010FUNCTIONS ON INACCESSIBLE MAHLO NUMBERS","display_name":"A NOTATION SYSTEM FOR ORDINAL USING \u03a8\u2010FUNCTIONS ON INACCESSIBLE MAHLO NUMBERS","publication_year":1992,"publication_date":"1992-01-01","ids":{"openalex":"https://openalex.org/W2035687351","doi":"https://doi.org/10.1002/malq.19920380142","mag":"2035687351"},"language":"en","primary_location":{"id":"doi:10.1002/malq.19920380142","is_oa":false,"landing_page_url":"https://doi.org/10.1002/malq.19920380142","pdf_url":null,"source":{"id":"https://openalex.org/S172351290","display_name":"Mathematical logic quarterly","issn_l":"0044-3050","issn":["0044-3050","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":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5085697482","display_name":"Helmut Pfeiffer","orcid":null},"institutions":[{"id":"https://openalex.org/I119449181","display_name":"Technische Universit\u00e4t Ilmenau","ror":"https://ror.org/01weqhp73","country_code":"DE","type":"education","lineage":["https://openalex.org/I119449181"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Helmut Pfeiffer","raw_affiliation_strings":["Ilmenauweg 12 W-3016 Seelze 7 Germany","Ilmenauweg 12 W\u20103016 Seelze 7 Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Ilmenauweg 12 W-3016 Seelze 7 Germany","institution_ids":["https://openalex.org/I119449181"]},{"raw_affiliation_string":"Ilmenauweg 12 W\u20103016 Seelze 7 Germany","institution_ids":["https://openalex.org/I119449181"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5085697482","display_name":"Helmut Pfeiffer","orcid":null},"institutions":[{"id":"https://openalex.org/I119449181","display_name":"Technische Universit\u00e4t Ilmenau","ror":"https://ror.org/01weqhp73","country_code":"DE","type":"education","lineage":["https://openalex.org/I119449181"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"H. Pfeiffer","raw_affiliation_strings":["Ilmenauweg 12 W-3016 Seelze 7 Germany","Ilmenauweg 12 W\u20103016 Seelze 7 Germany"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Ilmenauweg 12 W-3016 Seelze 7 Germany","institution_ids":["https://openalex.org/I119449181"]},{"raw_affiliation_string":"Ilmenauweg 12 W\u20103016 Seelze 7 Germany","institution_ids":["https://openalex.org/I119449181"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I119449181"],"apc_list":{"value":2630,"currency":"USD","value_usd":2630},"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.17276667,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"38","issue":"1","first_page":"431","last_page":"456"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11727","display_name":"Advanced Algebra and Logic","score":0.9940000176429749,"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":0.9940000176429749,"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/T11010","display_name":"Logic, Reasoning, and Knowledge","score":0.9789999723434448,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/T10126","display_name":"Logic, programming, and type systems","score":0.9646999835968018,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7120280265808105},{"id":"https://openalex.org/keywords/hierarchy","display_name":"Hierarchy","score":0.7084237933158875},{"id":"https://openalex.org/keywords/recursion","display_name":"Recursion (computer science)","score":0.6381793022155762},{"id":"https://openalex.org/keywords/notation","display_name":"Notation","score":0.6121504902839661},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4724857211112976},{"id":"https://openalex.org/keywords/function","display_name":"Function (biology)","score":0.4291646480560303},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.34109365940093994},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.33535340428352356},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.30104872584342957},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.16667300462722778},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.15940862894058228}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7120280265808105},{"id":"https://openalex.org/C31170391","wikidata":"https://www.wikidata.org/wiki/Q188619","display_name":"Hierarchy","level":2,"score":0.7084237933158875},{"id":"https://openalex.org/C168773036","wikidata":"https://www.wikidata.org/wiki/Q264164","display_name":"Recursion (computer science)","level":2,"score":0.6381793022155762},{"id":"https://openalex.org/C45357846","wikidata":"https://www.wikidata.org/wiki/Q2001982","display_name":"Notation","level":2,"score":0.6121504902839661},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4724857211112976},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.4291646480560303},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.34109365940093994},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.33535340428352356},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.30104872584342957},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.16667300462722778},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.15940862894058228},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.0},{"id":"https://openalex.org/C34447519","wikidata":"https://www.wikidata.org/wiki/Q179522","display_name":"Market economy","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1002/malq.19920380142","is_oa":false,"landing_page_url":"https://doi.org/10.1002/malq.19920380142","pdf_url":null,"source":{"id":"https://openalex.org/S172351290","display_name":"Mathematical logic quarterly","issn_l":"0044-3050","issn":["0044-3050","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":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":6,"referenced_works":["https://openalex.org/W1574937548","https://openalex.org/W2034547948","https://openalex.org/W2066878363","https://openalex.org/W2074842064","https://openalex.org/W2077219408","https://openalex.org/W2315675929"],"related_works":["https://openalex.org/W2504004674","https://openalex.org/W1976679120","https://openalex.org/W2377519369","https://openalex.org/W3202311233","https://openalex.org/W2024638892","https://openalex.org/W4281614748","https://openalex.org/W2498744856","https://openalex.org/W763418848","https://openalex.org/W32024214","https://openalex.org/W2765883196"],"abstract_inverted_index":{"Abstract":[0],"G.":[1],"J\u00e4ger":[2,74],"gave":[3],"in":[4,33],"Arch.":[5],"Math.":[6],"Logik":[7],"Grundlagenforsch.":[8],"24":[9],"(1984),":[10],"49\u201062,":[11],"a":[12,17,20,60,116,136,144,164,171],"recursive":[13],"notation":[14],"system":[15,42,174],"on":[16,163],"basis":[18],"of":[19,23,63,111,146,157,167,175],"hierarchy":[21,61],"I\u03b1\u00df":[22],"\u03b1\u2010inaccessible":[24],"regular":[25,91,101,117,126],"ordinals":[26,45,102,147],"using":[27],"collapsing":[28,129,142],"functions":[29,130],"following":[30],"W.":[31],"Buchholz":[32],"Ann.":[34],"Pure":[35],"Appl.":[36],"Logic":[37],"32":[38],"(1986),":[39],"195\u2010207.":[40],"J\u00e4ger's":[41],"stops,":[43],"when":[44],"\u03b1":[46,50],"with":[47],"I\u03b10":[48],"=":[49,122],"enter.":[51],"This":[52],"border":[53],"is":[54,69,79,94,115,148,159],"now":[55],"overcome":[56],"by":[57,161],"introducing":[58],"additional":[59],"J\u03b1\u00df":[62],"weakly":[64],"inaccessible":[65],"Mahlo":[66,98,107],"numbers,":[67],"which":[68],"defined":[70,95],"similarly":[71],"to":[72,150,186],"the":[73,112,141,155],"hierarchy.":[75],"An":[76],"ordinal":[77,108,176],"\u03bc":[78,87,89,109],"called":[80],"Mahlo,":[81],"if":[82],"every":[83,106],"normal\u2010function":[84],"f":[85],":":[86],"\u2192":[88],"has":[90],"fixpoints.":[92],"Collapsing":[93],"for":[96,105],"both":[97],"and":[99,152,154],"simply":[100],"such":[103,119],"that":[104,120],"out":[110],"J\u2010hierarchy":[113],"\u03a8\u03bc\u03b1":[114],"\u03c3":[118,127,151],"I\u03c30":[121],"\u03c3.":[123],"For":[124],"these":[125,168],"again":[128],"\u03a8\u03c3":[131],"are":[132],"defined.":[133],"To":[134],"get":[135],"proper":[137],"systematical":[138],"order":[139],"into":[140],"procedure,":[143],"pair":[145],"associated":[149],"\u03b1,":[153],"definition":[156],"\u03a8\u03c3\u03b1":[158],"given":[160],"recursion":[162],"suitable":[165],"well\u2010ordering":[166],"pairs.":[169],"Thus":[170],"fairly":[172],"large":[173],"notations":[177],"can":[178],"be":[179],"established.":[180],"It":[181],"seems":[182],"rather":[183],"straightforward,":[184],"how":[185],"extend":[187],"this":[188],"setting":[189],"further.":[190]},"counts_by_year":[],"updated_date":"2026-06-23T06:36:01.041984","created_date":"2025-10-10T00:00:00"}
