{"id":"https://openalex.org/W7221970280","doi":"https://doi.org/10.48550/arxiv.2610.19283","title":"What David Teaches Goliath: An Inevitable Extension of ZFC","display_name":"What David Teaches Goliath: An Inevitable Extension of ZFC","publication_year":2026,"publication_date":"2026-10-13","ids":{"openalex":"https://openalex.org/W7221970280","doi":"https://doi.org/10.48550/arxiv.2610.19283"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.19283","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.19283","pdf_url":"https://arxiv.org/pdf/2610.19283","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2610.19283","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5035885312","display_name":"Ali Enayat","orcid":"https://orcid.org/0000-0003-0372-3354","observed_orcids":["https://orcid.org/0000-0003-0372-3354"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Enayat, Ali","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5035885312"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.9990000128746033,"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"}},"topics":[{"id":"https://openalex.org/T11151","display_name":"Advanced Topology and Set Theory","score":0.9990000128746033,"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/T13400","display_name":"Mathematical and Theoretical Analysis","score":9.999999747378752e-05,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T10126","display_name":"Logic, programming, and type systems","score":9.999999747378752e-05,"subfield":{"id":"https://openalex.org/subfields/1712","display_name":"Software"},"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/mahlo-cardinals","display_name":"Mahlo cardinals","score":0.9010000228881836},{"id":"https://openalex.org/keywords/elementary-embeddings","display_name":"elementary embeddings","score":0.42399999499320984},{"id":"https://openalex.org/keywords/models-of-arithmetic","display_name":"models of arithmetic","score":0.6859999895095825},{"id":"https://openalex.org/keywords/tree-property","display_name":"tree property","score":0.6190000176429749},{"id":"https://openalex.org/keywords/indiscernibles","display_name":"indiscernibles","score":0.9139999747276306},{"id":"https://openalex.org/keywords/hereditarily-finite-sets","display_name":"hereditarily finite sets","score":0.8240000009536743}],"concepts":[{"id":"https://openalex.org/C2778029271","wikidata":"https://www.wikidata.org/wiki/Q5421931","display_name":"Extension (predicate logic)","level":2,"score":0.8751000165939331},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.7541999816894531},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7008000016212463},{"id":"https://openalex.org/C153046414","wikidata":"https://www.wikidata.org/wiki/Q12482","display_name":"Set theory","level":3,"score":0.6279000043869019},{"id":"https://openalex.org/C84999194","wikidata":"https://www.wikidata.org/wiki/Q1","display_name":"Universe","level":2,"score":0.5698000192642212},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.5509999990463257},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.5170000195503235},{"id":"https://openalex.org/C77618280","wikidata":"https://www.wikidata.org/wiki/Q1155772","display_name":"Scheme (mathematics)","level":2,"score":0.4489000141620636},{"id":"https://openalex.org/C2780432614","wikidata":"https://www.wikidata.org/wiki/Q864213","display_name":"Conservative extension","level":2,"score":0.41769999265670776},{"id":"https://openalex.org/C47030870","wikidata":"https://www.wikidata.org/wiki/Q467606","display_name":"Model theory","level":2,"score":0.4146000146865845},{"id":"https://openalex.org/C184264201","wikidata":"https://www.wikidata.org/wiki/Q21199","display_name":"Natural number","level":2,"score":0.4090999960899353},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3970000147819519},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.36970001459121704},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.36570000648498535},{"id":"https://openalex.org/C199879729","wikidata":"https://www.wikidata.org/wiki/Q205140","display_name":"Infinite set","level":3,"score":0.35929998755455017},{"id":"https://openalex.org/C144237770","wikidata":"https://www.wikidata.org/wiki/Q747534","display_name":"Mathematical economics","level":1,"score":0.3555000126361847},{"id":"https://openalex.org/C113174947","wikidata":"https://www.wikidata.org/wiki/Q2859736","display_name":"Tree (set theory)","level":2,"score":0.34540000557899475},{"id":"https://openalex.org/C2776608160","wikidata":"https://www.wikidata.org/wiki/Q4785462","display_name":"Natural (archaeology)","level":2,"score":0.31949999928474426},{"id":"https://openalex.org/C102565666","wikidata":"https://www.wikidata.org/wiki/Q1378301","display_name":"Universal set","level":3,"score":0.2653999924659729},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.25839999318122864}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.19283","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.19283","pdf_url":"https://arxiv.org/pdf/2610.19283","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2610.19283","doi":"https://doi.org/10.48550/arxiv.2610.19283","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.19283","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.19283","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.19283","pdf_url":"https://arxiv.org/pdf/2610.19283","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7221970280.pdf","grobid_xml":"https://content.openalex.org/works/W7221970280.grobid-xml"},"referenced_works_count":15,"referenced_works":["https://openalex.org/W142428960","https://openalex.org/W1984346100","https://openalex.org/W2007983321","https://openalex.org/W2034208453","https://openalex.org/W2053454736","https://openalex.org/W2073171092","https://openalex.org/W2083282082","https://openalex.org/W2093587077","https://openalex.org/W2110407978","https://openalex.org/W2530997533","https://openalex.org/W3075633002","https://openalex.org/W4393270728","https://openalex.org/W4399991943","https://openalex.org/W6903663104","https://openalex.org/W7130339751"],"related_works":["https://openalex.org/W957204474","https://openalex.org/W8863226","https://openalex.org/W2070815955","https://openalex.org/W980890407","https://openalex.org/W3075633002","https://openalex.org/W2034208453","https://openalex.org/W7096875307","https://openalex.org/W3207758492","https://openalex.org/W2969256124","https://openalex.org/W4388275141"],"abstract_inverted_index":{"We":[0],"examine":[1],"the":[2,6,17,36,49,54,77,87,113,120],"metamathematical":[3,110],"significance":[4],"of":[5,19,39,51,63,67,74,76,80,102,115,123],"Levy":[7],"scheme":[8],"$\u039b$,":[9],"which":[10,48,96],"asserts,":[11],"for":[12],"each":[13],"natural":[14],"number":[15],"$n$,":[16],"existence":[18],"an":[20,99],"$n$-Mahlo":[21],"cardinal":[22],"$\u03ba$":[23],"such":[24],"that":[25],"$V_\u03ba\\prec_{\u03a3_n}V$.":[26],"Several":[27],"initially":[28],"unrelated":[29],"constructions":[30],"single":[31],"out":[32],"$\\mathsf{ZFC}+\u039b$:":[33],"comparisons":[34],"with":[35,59],"model":[37],"theory":[38,70,83],"arithmetic":[40],"and":[41,65],"finite":[42,117],"set":[43,57,69,82],"theory;":[44],"class":[45,50,62],"theories":[46,58],"in":[47,71,95],"ordinals":[52],"has":[53],"tree":[55],"property;":[56],"a":[60,92],"proper":[61],"indiscernibles;":[64],"interpretations":[66],"ordinary":[68],"Cantorian":[72],"parts":[73],"models":[75],"extension":[78,101],"\\mathsf{NFUA}":[79],"Quine--Jensen":[81],"$\\mathsf{NFU}$.":[84],"Taken":[85],"together,":[86],"results":[88,111],"surveyed":[89],"here":[90],"provide":[91],"precise":[93],"sense":[94],"$\\mathsf{ZFC}+\u039b$":[97],"is":[98],"inevitable":[100],"$\\mathsf{ZFC}$":[103],"if":[104],"one":[105],"attempts":[106],"to":[107,119],"generalize":[108],"certain":[109],"about":[112],"microcosm":[114],"hereditarily":[116],"sets":[118],"entire":[121],"macrocosm":[122],"sets.":[124]},"counts_by_year":[],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2026-10-16T00:00:00"}