{"id":"https://openalex.org/W2154759565","doi":"https://doi.org/10.2307/2275697","title":"A modified sentence unprovable in <b>PA</b>","display_name":"A modified sentence unprovable in <b>PA</b>","publication_year":1994,"publication_date":"1994-12-01","ids":{"openalex":"https://openalex.org/W2154759565","doi":"https://doi.org/10.2307/2275697","mag":"2154759565"},"language":"en","primary_location":{"id":"doi:10.2307/2275697","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2275697","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/A5110937376","display_name":"L. Gordeev","orcid":null},"institutions":[{"id":"https://openalex.org/I8087733","display_name":"University of T\u00fcbingen","ror":"https://ror.org/03a1kwz48","country_code":"DE","type":"education","lineage":["https://openalex.org/I8087733"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"L. Gordeev","raw_affiliation_strings":["WSI fur Informatik, Mathematische Logik, Universitat Tubingen, D-72076 Tubingen, Germany, E-mail:"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"WSI fur Informatik, Mathematische Logik, Universitat Tubingen, D-72076 Tubingen, Germany, E-mail:","institution_ids":["https://openalex.org/I8087733"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5110937376"],"corresponding_institution_ids":["https://openalex.org/I8087733"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":4,"citation_normalized_percentile":{"value":0.23928535,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"59","issue":"4","first_page":"1154","last_page":"1157"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11567","display_name":"semigroups and automata theory","score":0.9975000023841858,"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/T11567","display_name":"semigroups and automata theory","score":0.9975000023841858,"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/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.9861000180244446,"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/T10126","display_name":"Logic, programming, and type systems","score":0.9843000173568726,"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/combinatorics","display_name":"Combinatorics","score":0.7359892129898071},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7037723660469055},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.5765484571456909},{"id":"https://openalex.org/keywords/sentence","display_name":"Sentence","score":0.5169005393981934},{"id":"https://openalex.org/keywords/cardinality","display_name":"Cardinality (data modeling)","score":0.5017142295837402},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.4594356417655945},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.11390399932861328},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.10001283884048462},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.07221034169197083}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7359892129898071},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7037723660469055},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.5765484571456909},{"id":"https://openalex.org/C2777530160","wikidata":"https://www.wikidata.org/wiki/Q41796","display_name":"Sentence","level":2,"score":0.5169005393981934},{"id":"https://openalex.org/C87117476","wikidata":"https://www.wikidata.org/wiki/Q362383","display_name":"Cardinality (data modeling)","level":2,"score":0.5017142295837402},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4594356417655945},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.11390399932861328},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.10001283884048462},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.07221034169197083},{"id":"https://openalex.org/C124101348","wikidata":"https://www.wikidata.org/wiki/Q172491","display_name":"Data mining","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.2307/2275697","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2275697","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"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":7,"referenced_works":["https://openalex.org/W6616102","https://openalex.org/W19637469","https://openalex.org/W1597834884","https://openalex.org/W2040462567","https://openalex.org/W2081739832","https://openalex.org/W2082844458","https://openalex.org/W3039529476"],"related_works":["https://openalex.org/W2002177687","https://openalex.org/W2058438338","https://openalex.org/W2019471580","https://openalex.org/W2168299207","https://openalex.org/W4224920876","https://openalex.org/W2941284322","https://openalex.org/W2017309389","https://openalex.org/W2485961789","https://openalex.org/W24519920","https://openalex.org/W2087778462"],"abstract_inverted_index":{"H.":[0,149],"Friedman":[1],"proved":[2],"that":[3,49,85,129],"the":[4,15,22,32,146,162,167,182,236,239],"following":[5],"sentence":[6,123,157],"SWQO(":[7,103,113,197],")":[8,104,114,212,234],"is":[9,44,88,98,105,115,127,175,235],"true":[10],"but":[11],"not":[12,106],"provable":[13,107],"in":[14,108,171,177,196],"theory":[16],"ATR":[17,96],"0":[18,42,53,97],"(cf.":[19],"[S];":[20],"\u2254":[21,31],"set":[23],"of":[24,34,130,141,238],"finite":[25,136,139],"rooted":[26],"trees,":[27],"\u2223":[28,30,69,184,187],"T":[29,35,52,55,58,67,86,92],"cardinality":[33],"\u2208":[36,60],").":[37,242],"For":[38],"every":[39],"k":[40,71,189,214],"&gt;":[41],"there":[43,77],"an":[45,220],"n":[46,59,65,83],"so":[47],"large":[48],"for":[50,219],"any":[51],",":[54,102,110,227],"1":[56],",\u2026,":[57],"with":[61,148],"\u2200":[62],"i":[63,68,74,79,87,186,192,211,218],"\u2264":[64,70,82,188,213],"(\u2223":[66],"\u00b7":[72,190,217],"(":[73,191],"+":[75,193,215],"1))":[76],"are":[78],"&lt;":[80,229],"j":[81,93],"such":[84],"homeomorphically":[89],"embeddable":[90],"into":[91],".":[94,120,205],"Since":[95],"stronger":[99,117],"than":[100,118],"PA":[101,109,119,131],"either.":[111],"Furthermore,":[112],"proof-theoretically":[116],"A":[121],"similar":[122],"whose":[124],"proof-theoretical":[125],"strength":[126],"exactly":[128],"was":[132,158],"obtained":[133],"by":[134,138,160],"replacing":[135],"trees":[137],"sequences":[140],"bounded":[142],"natural":[143],"numbers":[144],"under":[145],"embeddability":[147],"Friedman's":[150],"asymmetrical":[151],"gap-condition":[152,165],"(see":[153,166],"[SS]).":[154],"The":[155],"latter":[156],"modified":[159],"using":[161],"weaker":[163],"symmetrical":[164],"corresponding":[168],"particular":[169],"case":[170],"[Gl]).":[172],"Further":[173],"improvement":[174],"indicated":[176],"[G2,":[178],"Corollary":[179],"4.5].":[180],"Namely,":[181],"assertion":[183],"X":[185,210,233,241],"1)":[194],"as":[195],"J":[198],"),":[199],"above,":[200],"can":[201],"be":[202],"weakened":[203],"to":[204,208],"(in":[206],"fact,":[207],"wt(":[209,232],"\u03b5":[216,223],"arbitrary":[221],"fixed":[222],"=":[224],"l":[225,228],"\u22121":[226],"0,":[230],"where":[231],"weight":[237],"\u201cword\u201d":[240]},"counts_by_year":[],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}