{"id":"https://openalex.org/W1980572514","doi":"https://doi.org/10.2307/2266507","title":"On natural numbers, integers, and rationals","display_name":"On natural numbers, integers, and rationals","publication_year":1949,"publication_date":"1949-06-23","ids":{"openalex":"https://openalex.org/W1980572514","doi":"https://doi.org/10.2307/2266507","mag":"1980572514"},"language":"en","primary_location":{"id":"doi:10.2307/2266507","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2266507","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/A5086895461","display_name":"Frederic B. Fitch","orcid":null},"institutions":[{"id":"https://openalex.org/I32971472","display_name":"Yale University","ror":"https://ror.org/03v76x132","country_code":"US","type":"education","lineage":["https://openalex.org/I32971472"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Frederic B. Fitch","raw_affiliation_strings":["Yale University","Yale University  ;"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Yale University","institution_ids":["https://openalex.org/I32971472"]},{"raw_affiliation_string":"Yale University  ;","institution_ids":["https://openalex.org/I32971472"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5086895461"],"corresponding_institution_ids":["https://openalex.org/I32971472"],"apc_list":{"value":3550,"currency":"USD","value_usd":3550},"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":2,"citation_normalized_percentile":{"value":0.07111936,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"14","issue":"2","first_page":"81","last_page":"84"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10622","display_name":"Quantum Mechanics and Applications","score":0.661300003528595,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10622","display_name":"Quantum Mechanics and Applications","score":0.661300003528595,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10126","display_name":"Logic, programming, and type systems","score":0.5796999931335449,"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/T12170","display_name":"History and Theory of Mathematics","score":0.5641999840736389,"subfield":{"id":"https://openalex.org/subfields/2614","display_name":"Theoretical Computer Science"},"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/relation","display_name":"Relation (database)","score":0.85184645652771},{"id":"https://openalex.org/keywords/natural-number","display_name":"Natural number","score":0.8191462755203247},{"id":"https://openalex.org/keywords/rational-number","display_name":"Rational number","score":0.7235409021377563},{"id":"https://openalex.org/keywords/number-theory","display_name":"Number theory","score":0.6664971709251404},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6659426093101501},{"id":"https://openalex.org/keywords/square","display_name":"Square (algebra)","score":0.5575237274169922},{"id":"https://openalex.org/keywords/natural","display_name":"Natural (archaeology)","score":0.546583354473114},{"id":"https://openalex.org/keywords/simple","display_name":"Simple (philosophy)","score":0.5158897638320923},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.31915247440338135},{"id":"https://openalex.org/keywords/epistemology","display_name":"Epistemology","score":0.2234433889389038},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.17721086740493774},{"id":"https://openalex.org/keywords/philosophy","display_name":"Philosophy","score":0.08782586455345154}],"concepts":[{"id":"https://openalex.org/C25343380","wikidata":"https://www.wikidata.org/wiki/Q277521","display_name":"Relation (database)","level":2,"score":0.85184645652771},{"id":"https://openalex.org/C184264201","wikidata":"https://www.wikidata.org/wiki/Q21199","display_name":"Natural number","level":2,"score":0.8191462755203247},{"id":"https://openalex.org/C46875033","wikidata":"https://www.wikidata.org/wiki/Q1244890","display_name":"Rational number","level":2,"score":0.7235409021377563},{"id":"https://openalex.org/C169654258","wikidata":"https://www.wikidata.org/wiki/Q12479","display_name":"Number theory","level":2,"score":0.6664971709251404},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6659426093101501},{"id":"https://openalex.org/C135692309","wikidata":"https://www.wikidata.org/wiki/Q111124","display_name":"Square (algebra)","level":2,"score":0.5575237274169922},{"id":"https://openalex.org/C2776608160","wikidata":"https://www.wikidata.org/wiki/Q4785462","display_name":"Natural (archaeology)","level":2,"score":0.546583354473114},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.5158897638320923},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.31915247440338135},{"id":"https://openalex.org/C111472728","wikidata":"https://www.wikidata.org/wiki/Q9471","display_name":"Epistemology","level":1,"score":0.2234433889389038},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.17721086740493774},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.08782586455345154},{"id":"https://openalex.org/C166957645","wikidata":"https://www.wikidata.org/wiki/Q23498","display_name":"Archaeology","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/C95457728","wikidata":"https://www.wikidata.org/wiki/Q309","display_name":"History","level":0,"score":0.0},{"id":"https://openalex.org/C77088390","wikidata":"https://www.wikidata.org/wiki/Q8513","display_name":"Database","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.2307/2266507","is_oa":false,"landing_page_url":"https://doi.org/10.2307/2266507","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/1183730615","is_oa":false,"landing_page_url":"http://projecteuclid.org/euclid.jsl/1183730615","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":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W1964628587","https://openalex.org/W2508096977","https://openalex.org/W4321514130","https://openalex.org/W2009640904","https://openalex.org/W4297778831","https://openalex.org/W3144606069","https://openalex.org/W1972495134","https://openalex.org/W4230071522","https://openalex.org/W2601782347","https://openalex.org/W2509885431"],"abstract_inverted_index":{"A":[0],"theory":[1,12,49,102],"of":[2,21,34,40,50,66,76,81,86,95,100,116,120,139,143,162,165],"natural":[3,106,129,152],"numbers":[4],"will":[5,13,37,53],"be":[6,15,38,54,111,134,157],"outlined":[7],"in":[8],"what":[9],"follows.":[10],"This":[11],"also":[14],"extended":[16],"to":[17,110,124,133,147,156,169],"give":[18],"an":[19],"account":[20],"positive":[22,27],"and":[23,26,28,42,68,89,127,150,172],"negative":[24,29],"integers":[25],"rational":[30],"numbers.":[31],"The":[32,97],"system":[33],"logic":[35],"used":[36],"that":[39,56,104],"Whitehead":[41],"Russell's":[43],"Principia":[44],"mathematica":[45],"with":[46,61,69],"the":[47,57,62,73,79,84,90,105,114,117,128,137,140,151,160,163],"simple":[48],"types.":[51],"It":[52],"assumed":[55],"reader":[58],"is":[59,103,109,132,155],"familiar":[60],"more":[63],"elementary":[64],"properties":[65],"relations":[67],"such":[70],"notions":[71],"as":[72,113,136,159],"relative":[74],"product":[75],"two":[77],"relations,":[78],"square":[80,164],"a":[82,87,121,144,166],"relation,":[83,88],"cube":[85],"various":[91],"other":[92],"whole-number":[93],"powers":[94],"relations.":[96],"guiding":[98],"principle":[99],"this":[101],"number":[107,130,153],"zero":[108],"regarded":[112,135,158],"relation":[115,122,138,145,161,167],"zevoth":[118],"power":[119,142],"\u0410":[123,125,146,148,168,170],"itself,":[126,149,171],"1":[131],"first":[141],"2":[154],"so":[173],"on.":[174]},"counts_by_year":[],"updated_date":"2026-08-21T09:56:20.448147","created_date":"2025-10-10T00:00:00"}
