{"id":"https://openalex.org/W2989860728","doi":"https://doi.org/10.3390/axioms11070351","title":"Does Set Theory Really Ground Arithmetic Truth?","display_name":"Does Set Theory Really Ground Arithmetic Truth?","publication_year":2022,"publication_date":"2022-07-21","ids":{"openalex":"https://openalex.org/W2989860728","doi":"https://doi.org/10.3390/axioms11070351","mag":"2989860728"},"language":"en","primary_location":{"id":"doi:10.3390/axioms11070351","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms11070351","pdf_url":"https://www.mdpi.com/2075-1680/11/7/351/pdf?version=1658403050","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2075-1680/11/7/351/pdf?version=1658403050","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5058326973","display_name":"Alfredo Roque Freire","orcid":"https://orcid.org/0000-0003-0132-355X"},"institutions":[{"id":"https://openalex.org/I60858718","display_name":"University of Aveiro","ror":"https://ror.org/00nt41z93","country_code":"PT","type":"education","lineage":["https://openalex.org/I60858718"]}],"countries":["PT"],"is_corresponding":true,"raw_author_name":"Alfredo Roque Freire","raw_affiliation_strings":["Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal","institution_ids":["https://openalex.org/I60858718"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5058326973"],"corresponding_institution_ids":["https://openalex.org/I60858718"],"apc_list":{"value":1400,"currency":"CHF","value_usd":1515},"apc_paid":{"value":1400,"currency":"CHF","value_usd":1515},"fwci":0.3844,"has_fulltext":true,"cited_by_count":2,"citation_normalized_percentile":{"value":0.62128528,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":95,"max":96},"biblio":{"volume":"11","issue":"7","first_page":"351","last_page":"351"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10258","display_name":"Philosophy and Theoretical Science","score":0.9954000115394592,"subfield":{"id":"https://openalex.org/subfields/3205","display_name":"Experimental and Cognitive Psychology"},"field":{"id":"https://openalex.org/fields/32","display_name":"Psychology"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},"topics":[{"id":"https://openalex.org/T10258","display_name":"Philosophy and Theoretical Science","score":0.9954000115394592,"subfield":{"id":"https://openalex.org/subfields/3205","display_name":"Experimental and Cognitive Psychology"},"field":{"id":"https://openalex.org/fields/32","display_name":"Psychology"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.995199978351593,"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/T12170","display_name":"History and Theory of Mathematics","score":0.9865000247955322,"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/uncountable-set","display_name":"Uncountable set","score":0.7339447140693665},{"id":"https://openalex.org/keywords/interpretation","display_name":"Interpretation (philosophy)","score":0.7089614868164062},{"id":"https://openalex.org/keywords/axiom","display_name":"Axiom","score":0.6941599249839783},{"id":"https://openalex.org/keywords/countable-set","display_name":"Countable set","score":0.6906996965408325},{"id":"https://openalex.org/keywords/set","display_name":"Set (abstract data type)","score":0.5948078036308289},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5629993081092834},{"id":"https://openalex.org/keywords/arithmetic","display_name":"Arithmetic","score":0.5525949001312256},{"id":"https://openalex.org/keywords/g\u00f6dels-incompleteness-theorems","display_name":"G\u00f6del's incompleteness theorems","score":0.5300530791282654},{"id":"https://openalex.org/keywords/set-theory","display_name":"Set theory","score":0.4771130383014679},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.41812625527381897},{"id":"https://openalex.org/keywords/regular-cardinal","display_name":"Regular cardinal","score":0.41026702523231506},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.2916485369205475},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.15023043751716614}],"concepts":[{"id":"https://openalex.org/C142399903","wikidata":"https://www.wikidata.org/wiki/Q1128796","display_name":"Uncountable set","level":3,"score":0.7339447140693665},{"id":"https://openalex.org/C527412718","wikidata":"https://www.wikidata.org/wiki/Q855395","display_name":"Interpretation (philosophy)","level":2,"score":0.7089614868164062},{"id":"https://openalex.org/C167729594","wikidata":"https://www.wikidata.org/wiki/Q17736","display_name":"Axiom","level":2,"score":0.6941599249839783},{"id":"https://openalex.org/C110729354","wikidata":"https://www.wikidata.org/wiki/Q185478","display_name":"Countable set","level":2,"score":0.6906996965408325},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.5948078036308289},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5629993081092834},{"id":"https://openalex.org/C94375191","wikidata":"https://www.wikidata.org/wiki/Q11205","display_name":"Arithmetic","level":1,"score":0.5525949001312256},{"id":"https://openalex.org/C180937797","wikidata":"https://www.wikidata.org/wiki/Q200787","display_name":"G\u00f6del's incompleteness theorems","level":3,"score":0.5300530791282654},{"id":"https://openalex.org/C153046414","wikidata":"https://www.wikidata.org/wiki/Q12482","display_name":"Set theory","level":3,"score":0.4771130383014679},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.41812625527381897},{"id":"https://openalex.org/C89507440","wikidata":"https://www.wikidata.org/wiki/Q1193137","display_name":"Regular cardinal","level":2,"score":0.41026702523231506},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2916485369205475},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.15023043751716614},{"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/C2781302003","wikidata":"https://www.wikidata.org/wiki/Q3294091","display_name":"G\u00f6del","level":2,"score":0.0},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0}],"mesh":[],"locations_count":5,"locations":[{"id":"doi:10.3390/axioms11070351","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms11070351","pdf_url":"https://www.mdpi.com/2075-1680/11/7/351/pdf?version=1658403050","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:1db55ff0758e44d68740eeee55c7da6c","is_oa":true,"landing_page_url":"https://doaj.org/article/1db55ff0758e44d68740eeee55c7da6c","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Axioms, Vol 11, Iss 7, p 351 (2022)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/2075-1680/11/7/351/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/axioms11070351","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Axioms; Volume 11; Issue 7; Pages: 351","raw_type":"Text"},{"id":"pmh:oai:philarchive.org/rec/FREDST-2","is_oa":false,"landing_page_url":"https://philarchive.org/rec/FREDST-2","pdf_url":null,"source":{"id":"https://openalex.org/S4306402130","display_name":"PhilPapers (PhilPapers Foundation)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"info:eu-repo/semantics/article"},{"id":"pmh:oai:philpapers.org/rec/FREDST-2","is_oa":false,"landing_page_url":"https://philpapers.org/rec/FREDST-2","pdf_url":null,"source":{"id":"https://openalex.org/S4306402131","display_name":"PhilPapers (PhilPapers Foundation)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"info:eu-repo/semantics/article"}],"best_oa_location":{"id":"doi:10.3390/axioms11070351","is_oa":true,"landing_page_url":"https://doi.org/10.3390/axioms11070351","pdf_url":"https://www.mdpi.com/2075-1680/11/7/351/pdf?version=1658403050","source":{"id":"https://openalex.org/S4210173132","display_name":"Axioms","issn_l":"2075-1680","issn":["2075-1680"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Axioms","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2590986966","display_name":"Center for Research and Development in Mathematics and Applications","funder_award_id":"UIDB/04106/2020","funder_id":"https://openalex.org/F4320334779","funder_display_name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia"},{"id":"https://openalex.org/G3069120617","display_name":null,"funder_award_id":"04106","funder_id":"https://openalex.org/F4320334779","funder_display_name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia"},{"id":"https://openalex.org/G3187269428","display_name":null,"funder_award_id":"UIDB/04106/2020 and UIDP/04106/2020","funder_id":"https://openalex.org/F4320334779","funder_display_name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia"},{"id":"https://openalex.org/G3461949579","display_name":null,"funder_award_id":"UIDB/04106/2020","funder_id":"https://openalex.org/F4320329951","funder_display_name":"Center for Research and Development in Mathematics and Applications"},{"id":"https://openalex.org/G4397684180","display_name":null,"funder_award_id":"UIDB/04106/2020 and UIDP/04106/2020","funder_id":"https://openalex.org/F4320329951","funder_display_name":"Center for Research and Development in Mathematics and Applications"},{"id":"https://openalex.org/G7394643644","display_name":"Center for Research and Development in Mathematics and Applications","funder_award_id":"UIDP/04106/2020","funder_id":"https://openalex.org/F4320334779","funder_display_name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia"}],"funders":[{"id":"https://openalex.org/F4320329951","display_name":"Center for Research and Development in Mathematics and Applications","ror":null},{"id":"https://openalex.org/F4320334779","display_name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","ror":"https://ror.org/00snfqn58"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2989860728.pdf","grobid_xml":"https://content.openalex.org/works/W2989860728.grobid-xml"},"referenced_works_count":30,"referenced_works":["https://openalex.org/W131761759","https://openalex.org/W1561920671","https://openalex.org/W1587096194","https://openalex.org/W1971643033","https://openalex.org/W1972672801","https://openalex.org/W1988764612","https://openalex.org/W1988802190","https://openalex.org/W2007983321","https://openalex.org/W2082963564","https://openalex.org/W2098563247","https://openalex.org/W2114108968","https://openalex.org/W2114804254","https://openalex.org/W2118110454","https://openalex.org/W2126232520","https://openalex.org/W2333976070","https://openalex.org/W2483494328","https://openalex.org/W2555836453","https://openalex.org/W2735263090","https://openalex.org/W2783791283","https://openalex.org/W2800655773","https://openalex.org/W2954410128","https://openalex.org/W2978528525","https://openalex.org/W3095143979","https://openalex.org/W3101337583","https://openalex.org/W6633691677","https://openalex.org/W6635020914","https://openalex.org/W6643271893","https://openalex.org/W6740775194","https://openalex.org/W6755520377","https://openalex.org/W6922399809"],"related_works":["https://openalex.org/W3019662020","https://openalex.org/W2058498852","https://openalex.org/W2100222222","https://openalex.org/W2512721618","https://openalex.org/W2600675026","https://openalex.org/W3139954723","https://openalex.org/W4242301765","https://openalex.org/W2036283644","https://openalex.org/W2480446702","https://openalex.org/W4232994235"],"abstract_inverted_index":{"We":[0,65,121,146,153,182,201,220],"consider":[1,154],"the":[2,15,32,48,59,62,69,76,96,107,150,205,237],"foundational":[3],"relation":[4],"between":[5],"arithmetic":[6,19,25,72,86,132,210],"and":[7,38,168],"set":[8,80,89,108,198],"theory.":[9,81],"Our":[10,27],"goal":[11],"is":[12,29,101,129,186,211,234],"to":[13,30,39,54,174,188],"criticize":[14],"construction":[16],"of":[17,35,58,61,71,84,119,158,191,207,209],"standard":[18,117],"models":[20,41,118],"as":[21,42],"providing":[22],"grounds":[23],"for":[24,161,166],"truth.":[26],"method":[28],"emphasize":[31],"incomplete":[33,49],"picture":[34,50],"both":[36],"theories":[37,90],"treat":[40],"their":[43],"syntactical":[44],"counterparts.":[45],"Insisting":[46],"on":[47],"will":[51],"allow":[52],"us":[53],"argue":[55,222],"in":[56,78,136,180,195,216],"favor":[57],"revisability":[60],"standard-model":[63],"interpretation.":[64],"start":[66],"briefly":[67],"characterizing":[68],"expansion":[70,100],"\u2018truth\u2019":[73],"provided":[74],"by":[75,148],"interpretation":[77,135,179],"a":[79,123,130,196],"Interpreted":[82],"versions":[83],"an":[85],"theory":[87,199],"into":[88],"generally":[91],"have":[92,189],"more":[93],"theorems":[94],"than":[95],"original.":[97],"This":[98],"theorem":[99],"not":[102,143,193],"complete":[103],"however.":[104],"Using":[105],"this,":[106],"theoretic":[109],"multiversalist":[110,125],"concludes":[111],"that":[112,184,204,223,227],"there":[113,128],"are":[114,172,218],"multiple":[115],"legitimate":[116],"arithmetic.":[120],"suggest":[122],"different":[124],"conclusion:":[126],"while":[127,213],"single":[131],"structure,":[133],"its":[134],"each":[137],"universe":[138],"may":[139],"vary":[140],"or":[141],"even":[142],"be":[144,175],"possible.":[145],"continue":[147],"defining":[149],"coordination":[151,228],"problem.":[152],"two":[155],"independent":[156],"communities":[157],"mathematicians":[159],"responsible":[160],"deciding":[162],"over":[163],"new":[164],"axioms":[165],"ZF":[167],"PA.":[169],"How":[170],"likely":[171],"they":[173],"coordinated":[176],"regarding":[177],"PA\u2019s":[178],"ZF?":[181],"prove":[183],"it":[185,233],"possible":[187],"extensions":[190,208,215],"PA":[192],"interpretable":[194,214],"given":[197],"ST.":[200],"further":[202],"show":[203],"number":[206],"uncountable,":[212],"ST":[217],"countable.":[219],"finally":[221],"this":[224],"fact":[225],"suggests":[226],"can":[229],"only":[230],"work":[231],"if":[232],"assumed":[235],"from":[236],"start.":[238]},"counts_by_year":[{"year":2025,"cited_by_count":2}],"updated_date":"2026-06-02T09:04:35.204637","created_date":"2025-10-10T00:00:00"}
