{"id":"https://openalex.org/W2029975206","doi":"https://doi.org/10.2307/2271895","title":"Natural models and Ackermann-type set theories","display_name":"Natural models and Ackermann-type set theories","publication_year":1975,"publication_date":"1975-06-01","ids":{"openalex":"https://openalex.org/W2029975206","doi":"https://doi.org/10.2307/2271895","mag":"2029975206"},"language":"en","primary_location":{"id":"doi:10.2307/2271895","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2271895","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/A5019337960","display_name":"John Lake","orcid":null},"institutions":[{"id":"https://openalex.org/I28257850","display_name":"London South Bank University","ror":"https://ror.org/02vwnat91","country_code":"GB","type":"education","lineage":["https://openalex.org/I28257850"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"John Lake","raw_affiliation_strings":["Polytechnic of the South Bank, London SE1 OAA, England"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Polytechnic of the South Bank, London SE1 OAA, England","institution_ids":["https://openalex.org/I28257850"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5019337960"],"corresponding_institution_ids":["https://openalex.org/I28257850"],"apc_list":{"value":3550,"currency":"USD","value_usd":3550},"apc_paid":null,"fwci":0.5324,"has_fulltext":false,"cited_by_count":4,"citation_normalized_percentile":{"value":0.62885572,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"40","issue":"2","first_page":"151","last_page":"158"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11727","display_name":"Advanced Algebra and Logic","score":0.9919000267982483,"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":0.9919000267982483,"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":0.9890000224113464,"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/T11010","display_name":"Logic, Reasoning, and Knowledge","score":0.9753000140190125,"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/ackermann-function","display_name":"Ackermann function","score":0.9598758220672607},{"id":"https://openalex.org/keywords/notation","display_name":"Notation","score":0.6784696578979492},{"id":"https://openalex.org/keywords/axiom","display_name":"Axiom","score":0.6380196213722229},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6244286298751831},{"id":"https://openalex.org/keywords/natural-number","display_name":"Natural number","score":0.5632358193397522},{"id":"https://openalex.org/keywords/axiom-of-choice","display_name":"Axiom of choice","score":0.48692309856414795},{"id":"https://openalex.org/keywords/set-theory","display_name":"Set theory","score":0.4810647964477539},{"id":"https://openalex.org/keywords/type","display_name":"Type (biology)","score":0.4453568756580353},{"id":"https://openalex.org/keywords/type-theory","display_name":"Type theory","score":0.44483184814453125},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.443086177110672},{"id":"https://openalex.org/keywords/pascal","display_name":"Pascal (unit)","score":0.4378964900970459},{"id":"https://openalex.org/keywords/predicate","display_name":"Predicate (mathematical logic)","score":0.431770920753479},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4310012757778168},{"id":"https://openalex.org/keywords/class","display_name":"Class (philosophy)","score":0.42903196811676025},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.37815046310424805},{"id":"https://openalex.org/keywords/calculus","display_name":"Calculus (dental)","score":0.3302762508392334},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.2297375202178955},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.22670474648475647},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.18086117506027222},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.10770672559738159}],"concepts":[{"id":"https://openalex.org/C33436860","wikidata":"https://www.wikidata.org/wiki/Q341835","display_name":"Ackermann function","level":3,"score":0.9598758220672607},{"id":"https://openalex.org/C45357846","wikidata":"https://www.wikidata.org/wiki/Q2001982","display_name":"Notation","level":2,"score":0.6784696578979492},{"id":"https://openalex.org/C167729594","wikidata":"https://www.wikidata.org/wiki/Q17736","display_name":"Axiom","level":2,"score":0.6380196213722229},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6244286298751831},{"id":"https://openalex.org/C184264201","wikidata":"https://www.wikidata.org/wiki/Q21199","display_name":"Natural number","level":2,"score":0.5632358193397522},{"id":"https://openalex.org/C151797676","wikidata":"https://www.wikidata.org/wiki/Q179692","display_name":"Axiom of choice","level":4,"score":0.48692309856414795},{"id":"https://openalex.org/C153046414","wikidata":"https://www.wikidata.org/wiki/Q12482","display_name":"Set theory","level":3,"score":0.4810647964477539},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.4453568756580353},{"id":"https://openalex.org/C93682546","wikidata":"https://www.wikidata.org/wiki/Q1056428","display_name":"Type theory","level":3,"score":0.44483184814453125},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.443086177110672},{"id":"https://openalex.org/C75608658","wikidata":"https://www.wikidata.org/wiki/Q44395","display_name":"Pascal (unit)","level":2,"score":0.4378964900970459},{"id":"https://openalex.org/C140146324","wikidata":"https://www.wikidata.org/wiki/Q1144319","display_name":"Predicate (mathematical logic)","level":2,"score":0.431770920753479},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4310012757778168},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.42903196811676025},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.37815046310424805},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.3302762508392334},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2297375202178955},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.22670474648475647},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.18086117506027222},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.10770672559738159},{"id":"https://openalex.org/C207467116","wikidata":"https://www.wikidata.org/wiki/Q4385666","display_name":"Inverse","level":2,"score":0.0},{"id":"https://openalex.org/C18903297","wikidata":"https://www.wikidata.org/wiki/Q7150","display_name":"Ecology","level":1,"score":0.0},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.0},{"id":"https://openalex.org/C199343813","wikidata":"https://www.wikidata.org/wiki/Q12128","display_name":"Dentistry","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/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","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}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.2307/2271895","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2271895","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/1183739376","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.jsl/1183739376","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":7,"referenced_works":["https://openalex.org/W772296680","https://openalex.org/W1587347525","https://openalex.org/W2001852972","https://openalex.org/W2023283996","https://openalex.org/W2138180936","https://openalex.org/W2171239532","https://openalex.org/W2609118749"],"related_works":["https://openalex.org/W2467005568","https://openalex.org/W4242944805","https://openalex.org/W2505434214","https://openalex.org/W624501779","https://openalex.org/W4383737506","https://openalex.org/W2612506836","https://openalex.org/W4322716113","https://openalex.org/W614204383","https://openalex.org/W2029404646","https://openalex.org/W2029975206"],"abstract_inverted_index":{"Our":[0],"results":[1,17,196],"concern":[2],"the":[3,19,40,57,64,90,95,101,114,120,150,163,187,195],"natural":[4],"models":[5],"of":[6,21,59,92,98,116,126,152,165,168,194],"Ackermann-type":[7],"set":[8,27,164],"theories,":[9],"but":[10,202],"they":[11],"can":[12],"also":[13],"be":[14],"viewed":[15],"as":[16],"about":[18,200],"definability":[20],"ordinals":[22],"in":[23,32,39,106,113,146,173,212],"certain":[24],"sets.":[25,61],"Ackermann's":[26],"theory":[28,121],"A":[29,118],"was":[30],"introduced":[31],"[1]":[33],"and":[34,51,69,80,108,181],"it":[35],"is":[36,119,129],"now":[37],"formulated":[38],"first":[41,178],"order":[42,179],"predicate":[43],"calculus":[44],"with":[45],"identity,":[46],"using":[47,176],"\u2208":[48],"for":[49,56,73,86,162,191],"membership":[50],"an":[52,192],"individual":[53],"constant":[54],"V":[55,79],"class":[58],"all":[60,100],"We":[62,185],"use":[63,159],"letters":[65,83],"\u03d5,":[66],"\u03c7,":[67],"\u03b8,":[68],"\u03c7":[70],"to":[71,84,189],"stand":[72,85],"formulae":[74],"which":[75,170,197,208],"do":[76],"not":[77,111],"contain":[78],"capital":[81],"Greek":[82],"any":[87],"formulae.":[88],"Then,":[89],"axioms":[91],"A*":[93,122],"are":[94,104,140,171,198,209],"universal":[96],"closures":[97],"where":[99],"free":[102],"variables":[103,141],"shown":[105],"A4":[107],"z":[109],"does":[110],"occur":[112],"\u0398":[115],"A2.":[117],"\u2212":[123],"A5.":[124],"Most":[125],"our":[127],"notation":[128,151],"standard":[130],"(for":[131],"instance,":[132],"\u03b1,":[133],"\u03b2,":[134],"\u03b3,":[135],"\u03b4,":[136],"\u03ba,":[137],"\u03bb,":[138],"\u03be":[139],"ranging":[142],"over":[143],"ordinals)":[144],"and,":[145],"general,":[147],"we":[148,158,203],"follow":[149],"[7].":[153],"When":[154],"x":[155],"\u2286":[156],"R\u03b1,":[157],"Df(R\u03b1,":[160],"x)":[161],"those":[166,206],"elements":[167],"R\u03b1":[169],"definable":[172],"\u3008R\u03b1,":[174],"\u2208\u3009,":[175],"a":[177],"\u2208-formula":[180],"parameters":[182],"from":[183],"x.":[184],"refer":[186],"reader":[188],"[7]":[190],"outline":[193],"known":[199],"A,":[201],"shall":[204],"summarise":[205],"facts":[207],"frequently":[210],"used":[211],"this":[213],"paper.":[214]},"counts_by_year":[],"updated_date":"2026-08-21T09:56:20.448147","created_date":"2025-10-10T00:00:00"}
