The Arbitrary Continuum: Forcing the Spectrum of Cardinal Structures Article Swipe
Revista, Zen
,
MATH, 10
·
YOU?
·
· 2025
· Open Access
·
· DOI: https://doi.org/10.5281/zenodo.17834486
YOU?
·
· 2025
· Open Access
·
· DOI: https://doi.org/10.5281/zenodo.17834486
This paper explores the profound implications of Paul Cohen's forcing technique for understanding the continuum hypothesis and, more broadly, the spectrum of cardinal structures in set theory. The continuum, the cardinality of the real numbers, is shown to be highly arbitrary, its size not uniquely determined by the standard axioms of Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). We delve into how forcing allows for the construction of set-theoretic universes where the continuum can take on various infinite cardinal values, demonstrating its independence from ZFC. Furthermore, we examine how forcing can manipulate other cardinal arithmetic properties, such as the generalized continuum hypothesis and cofinalities, thereby illustrating the rich and complex landscape of possible set-theoretic realities. The work highlights forcing as an indispensable tool for exploring the limits of ZFC and the inherent flexibility in cardinal assignments.
Related Topics
Concepts
Forcing (mathematics)
Mathematics
Axiom
Cardinality (data modeling)
Set theory
Continuum hypothesis
Set (abstract data type)
Spectrum (functional analysis)
Pure mathematics
Axiom independence
Cardinal number (linguistics)
Independence (probability theory)
Calculus (dental)
Discrete mathematics
Zermelo–Fraenkel set theory
Axiom of choice
Regular cardinal
Flexibility (engineering)
Infinitesimal
Metadata
- Type
- article
- Landing Page
- https://doi.org/10.5281/zenodo.17834486
- OA Status
- green
- OpenAlex ID
- https://openalex.org/W7109036422
All OpenAlex metadata
Raw OpenAlex JSON
- OpenAlex ID
-
https://openalex.org/W7109036422Canonical identifier for this work in OpenAlex
- DOI
-
https://doi.org/10.5281/zenodo.17834486Digital Object Identifier
- Title
-
The Arbitrary Continuum: Forcing the Spectrum of Cardinal StructuresWork title
- Type
-
articleOpenAlex work type
- Publication year
-
2025Year of publication
- Publication date
-
2025-12-05Full publication date if available
- Authors
-
Revista, Zen, MATH, 10List of authors in order
- Landing page
-
https://doi.org/10.5281/zenodo.17834486Publisher landing page
- Open access
-
YesWhether a free full text is available
- OA status
-
greenOpen access status per OpenAlex
- OA URL
-
https://doi.org/10.5281/zenodo.17834486Direct OA link when available
- Concepts
-
Forcing (mathematics), Mathematics, Axiom, Cardinality (data modeling), Set theory, Continuum hypothesis, Set (abstract data type), Spectrum (functional analysis), Pure mathematics, Axiom independence, Cardinal number (linguistics), Independence (probability theory), Calculus (dental), Discrete mathematics, Zermelo–Fraenkel set theory, Axiom of choice, Regular cardinal, Flexibility (engineering), InfinitesimalTop concepts (fields/topics) attached by OpenAlex
- Cited by
-
0Total citation count in OpenAlex
Full payload
| id | https://openalex.org/W7109036422 |
|---|---|
| doi | https://doi.org/10.5281/zenodo.17834486 |
| ids.doi | https://doi.org/10.5281/zenodo.17834486 |
| ids.openalex | https://openalex.org/W7109036422 |
| fwci | 0.0 |
| type | article |
| title | The Arbitrary Continuum: Forcing the Spectrum of Cardinal Structures |
| biblio.issue | |
| biblio.volume | |
| biblio.last_page | |
| biblio.first_page | |
| topics[0].id | https://openalex.org/T11151 |
| topics[0].field.id | https://openalex.org/fields/26 |
| topics[0].field.display_name | Mathematics |
| topics[0].score | 0.8221143484115601 |
| topics[0].domain.id | https://openalex.org/domains/3 |
| topics[0].domain.display_name | Physical Sciences |
| topics[0].subfield.id | https://openalex.org/subfields/2608 |
| topics[0].subfield.display_name | Geometry and Topology |
| topics[0].display_name | Advanced Topology and Set Theory |
| topics[1].id | https://openalex.org/T13400 |
| topics[1].field.id | https://openalex.org/fields/26 |
| topics[1].field.display_name | Mathematics |
| topics[1].score | 0.038910795003175735 |
| topics[1].domain.id | https://openalex.org/domains/3 |
| topics[1].domain.display_name | Physical Sciences |
| topics[1].subfield.id | https://openalex.org/subfields/2610 |
| topics[1].subfield.display_name | Mathematical Physics |
| topics[1].display_name | Mathematical and Theoretical Analysis |
| topics[2].id | https://openalex.org/T12002 |
| topics[2].field.id | https://openalex.org/fields/17 |
| topics[2].field.display_name | Computer Science |
| topics[2].score | 0.023227499797940254 |
| topics[2].domain.id | https://openalex.org/domains/3 |
| topics[2].domain.display_name | Physical Sciences |
| topics[2].subfield.id | https://openalex.org/subfields/1703 |
| topics[2].subfield.display_name | Computational Theory and Mathematics |
| topics[2].display_name | Computability, Logic, AI Algorithms |
| is_xpac | False |
| apc_list | |
| apc_paid | |
| concepts[0].id | https://openalex.org/C197115733 |
| concepts[0].level | 2 |
| concepts[0].score | 0.8711010813713074 |
| concepts[0].wikidata | https://www.wikidata.org/wiki/Q1003136 |
| concepts[0].display_name | Forcing (mathematics) |
| concepts[1].id | https://openalex.org/C33923547 |
| concepts[1].level | 0 |
| concepts[1].score | 0.7425732016563416 |
| concepts[1].wikidata | https://www.wikidata.org/wiki/Q395 |
| concepts[1].display_name | Mathematics |
| concepts[2].id | https://openalex.org/C167729594 |
| concepts[2].level | 2 |
| concepts[2].score | 0.6759580969810486 |
| concepts[2].wikidata | https://www.wikidata.org/wiki/Q17736 |
| concepts[2].display_name | Axiom |
| concepts[3].id | https://openalex.org/C87117476 |
| concepts[3].level | 2 |
| concepts[3].score | 0.5445177555084229 |
| concepts[3].wikidata | https://www.wikidata.org/wiki/Q362383 |
| concepts[3].display_name | Cardinality (data modeling) |
| concepts[4].id | https://openalex.org/C153046414 |
| concepts[4].level | 3 |
| concepts[4].score | 0.45523640513420105 |
| concepts[4].wikidata | https://www.wikidata.org/wiki/Q12482 |
| concepts[4].display_name | Set theory |
| concepts[5].id | https://openalex.org/C96488702 |
| concepts[5].level | 2 |
| concepts[5].score | 0.40264737606048584 |
| concepts[5].wikidata | https://www.wikidata.org/wiki/Q208416 |
| concepts[5].display_name | Continuum hypothesis |
| concepts[6].id | https://openalex.org/C177264268 |
| concepts[6].level | 2 |
| concepts[6].score | 0.38491594791412354 |
| concepts[6].wikidata | https://www.wikidata.org/wiki/Q1514741 |
| concepts[6].display_name | Set (abstract data type) |
| concepts[7].id | https://openalex.org/C156778621 |
| concepts[7].level | 2 |
| concepts[7].score | 0.3783668875694275 |
| concepts[7].wikidata | https://www.wikidata.org/wiki/Q1365748 |
| concepts[7].display_name | Spectrum (functional analysis) |
| concepts[8].id | https://openalex.org/C202444582 |
| concepts[8].level | 1 |
| concepts[8].score | 0.37222036719322205 |
| concepts[8].wikidata | https://www.wikidata.org/wiki/Q837863 |
| concepts[8].display_name | Pure mathematics |
| concepts[9].id | https://openalex.org/C170088968 |
| concepts[9].level | 3 |
| concepts[9].score | 0.36901482939720154 |
| concepts[9].wikidata | https://www.wikidata.org/wiki/Q4830547 |
| concepts[9].display_name | Axiom independence |
| concepts[10].id | https://openalex.org/C2778408900 |
| concepts[10].level | 2 |
| concepts[10].score | 0.35202255845069885 |
| concepts[10].wikidata | https://www.wikidata.org/wiki/Q1329258 |
| concepts[10].display_name | Cardinal number (linguistics) |
| concepts[11].id | https://openalex.org/C35651441 |
| concepts[11].level | 2 |
| concepts[11].score | 0.351072758436203 |
| concepts[11].wikidata | https://www.wikidata.org/wiki/Q625303 |
| concepts[11].display_name | Independence (probability theory) |
| concepts[12].id | https://openalex.org/C2777686260 |
| concepts[12].level | 2 |
| concepts[12].score | 0.3263154923915863 |
| concepts[12].wikidata | https://www.wikidata.org/wiki/Q144037 |
| concepts[12].display_name | Calculus (dental) |
| concepts[13].id | https://openalex.org/C118615104 |
| concepts[13].level | 1 |
| concepts[13].score | 0.3247554898262024 |
| concepts[13].wikidata | https://www.wikidata.org/wiki/Q121416 |
| concepts[13].display_name | Discrete mathematics |
| concepts[14].id | https://openalex.org/C51460 |
| concepts[14].level | 5 |
| concepts[14].score | 0.29364222288131714 |
| concepts[14].wikidata | https://www.wikidata.org/wiki/Q191849 |
| concepts[14].display_name | Zermelo–Fraenkel set theory |
| concepts[15].id | https://openalex.org/C151797676 |
| concepts[15].level | 4 |
| concepts[15].score | 0.2892475724220276 |
| concepts[15].wikidata | https://www.wikidata.org/wiki/Q179692 |
| concepts[15].display_name | Axiom of choice |
| concepts[16].id | https://openalex.org/C89507440 |
| concepts[16].level | 2 |
| concepts[16].score | 0.2672179937362671 |
| concepts[16].wikidata | https://www.wikidata.org/wiki/Q1193137 |
| concepts[16].display_name | Regular cardinal |
| concepts[17].id | https://openalex.org/C2780598303 |
| concepts[17].level | 2 |
| concepts[17].score | 0.2647360861301422 |
| concepts[17].wikidata | https://www.wikidata.org/wiki/Q65921492 |
| concepts[17].display_name | Flexibility (engineering) |
| concepts[18].id | https://openalex.org/C91229774 |
| concepts[18].level | 2 |
| concepts[18].score | 0.255767285823822 |
| concepts[18].wikidata | https://www.wikidata.org/wiki/Q193885 |
| concepts[18].display_name | Infinitesimal |
| keywords[0].id | https://openalex.org/keywords/forcing |
| keywords[0].score | 0.8711010813713074 |
| keywords[0].display_name | Forcing (mathematics) |
| keywords[1].id | https://openalex.org/keywords/axiom |
| keywords[1].score | 0.6759580969810486 |
| keywords[1].display_name | Axiom |
| keywords[2].id | https://openalex.org/keywords/cardinality |
| keywords[2].score | 0.5445177555084229 |
| keywords[2].display_name | Cardinality (data modeling) |
| keywords[3].id | https://openalex.org/keywords/set-theory |
| keywords[3].score | 0.45523640513420105 |
| keywords[3].display_name | Set theory |
| keywords[4].id | https://openalex.org/keywords/continuum-hypothesis |
| keywords[4].score | 0.40264737606048584 |
| keywords[4].display_name | Continuum hypothesis |
| keywords[5].id | https://openalex.org/keywords/set |
| keywords[5].score | 0.38491594791412354 |
| keywords[5].display_name | Set (abstract data type) |
| keywords[6].id | https://openalex.org/keywords/spectrum |
| keywords[6].score | 0.3783668875694275 |
| keywords[6].display_name | Spectrum (functional analysis) |
| keywords[7].id | https://openalex.org/keywords/axiom-independence |
| keywords[7].score | 0.36901482939720154 |
| keywords[7].display_name | Axiom independence |
| language | |
| locations[0].id | doi:10.5281/zenodo.17834486 |
| locations[0].is_oa | True |
| locations[0].source.id | https://openalex.org/S4306400562 |
| locations[0].source.issn | |
| locations[0].source.type | repository |
| locations[0].source.is_oa | True |
| locations[0].source.issn_l | |
| locations[0].source.is_core | False |
| locations[0].source.is_in_doaj | False |
| locations[0].source.display_name | Zenodo (CERN European Organization for Nuclear Research) |
| locations[0].source.host_organization | https://openalex.org/I67311998 |
| locations[0].source.host_organization_name | European Organization for Nuclear Research |
| locations[0].source.host_organization_lineage | https://openalex.org/I67311998 |
| locations[0].license | cc-by |
| locations[0].pdf_url | |
| locations[0].version | |
| locations[0].raw_type | article-journal |
| locations[0].license_id | https://openalex.org/licenses/cc-by |
| locations[0].is_accepted | False |
| locations[0].is_published | |
| locations[0].raw_source_name | |
| locations[0].landing_page_url | https://doi.org/10.5281/zenodo.17834486 |
| indexed_in | datacite |
| authorships[0].author.id | |
| authorships[0].author.orcid | |
| authorships[0].author.display_name | Revista, Zen |
| authorships[0].author_position | first |
| authorships[0].raw_author_name | Revista, Zen |
| authorships[0].is_corresponding | True |
| authorships[1].author.id | |
| authorships[1].author.orcid | |
| authorships[1].author.display_name | MATH, 10 |
| authorships[1].author_position | last |
| authorships[1].raw_author_name | MATH, 10 |
| authorships[1].is_corresponding | False |
| has_content.pdf | False |
| has_content.grobid_xml | False |
| is_paratext | False |
| open_access.is_oa | True |
| open_access.oa_url | https://doi.org/10.5281/zenodo.17834486 |
| open_access.oa_status | green |
| open_access.any_repository_has_fulltext | False |
| created_date | 2025-12-06T00:00:00 |
| display_name | The Arbitrary Continuum: Forcing the Spectrum of Cardinal Structures |
| has_fulltext | False |
| is_retracted | False |
| updated_date | 2025-12-06T23:14:57.273132 |
| primary_topic.id | https://openalex.org/T11151 |
| primary_topic.field.id | https://openalex.org/fields/26 |
| primary_topic.field.display_name | Mathematics |
| primary_topic.score | 0.8221143484115601 |
| primary_topic.domain.id | https://openalex.org/domains/3 |
| primary_topic.domain.display_name | Physical Sciences |
| primary_topic.subfield.id | https://openalex.org/subfields/2608 |
| primary_topic.subfield.display_name | Geometry and Topology |
| primary_topic.display_name | Advanced Topology and Set Theory |
| cited_by_count | 0 |
| locations_count | 1 |
| best_oa_location.id | doi:10.5281/zenodo.17834486 |
| best_oa_location.is_oa | True |
| best_oa_location.source.id | https://openalex.org/S4306400562 |
| best_oa_location.source.issn | |
| best_oa_location.source.type | repository |
| best_oa_location.source.is_oa | True |
| best_oa_location.source.issn_l | |
| best_oa_location.source.is_core | False |
| best_oa_location.source.is_in_doaj | False |
| best_oa_location.source.display_name | Zenodo (CERN European Organization for Nuclear Research) |
| best_oa_location.source.host_organization | https://openalex.org/I67311998 |
| best_oa_location.source.host_organization_name | European Organization for Nuclear Research |
| best_oa_location.source.host_organization_lineage | https://openalex.org/I67311998 |
| best_oa_location.license | cc-by |
| best_oa_location.pdf_url | |
| best_oa_location.version | |
| best_oa_location.raw_type | article-journal |
| best_oa_location.license_id | https://openalex.org/licenses/cc-by |
| best_oa_location.is_accepted | False |
| best_oa_location.is_published | False |
| best_oa_location.raw_source_name | |
| best_oa_location.landing_page_url | https://doi.org/10.5281/zenodo.17834486 |
| primary_location.id | doi:10.5281/zenodo.17834486 |
| primary_location.is_oa | True |
| primary_location.source.id | https://openalex.org/S4306400562 |
| primary_location.source.issn | |
| primary_location.source.type | repository |
| primary_location.source.is_oa | True |
| primary_location.source.issn_l | |
| primary_location.source.is_core | False |
| primary_location.source.is_in_doaj | False |
| primary_location.source.display_name | Zenodo (CERN European Organization for Nuclear Research) |
| primary_location.source.host_organization | https://openalex.org/I67311998 |
| primary_location.source.host_organization_name | European Organization for Nuclear Research |
| primary_location.source.host_organization_lineage | https://openalex.org/I67311998 |
| primary_location.license | cc-by |
| primary_location.pdf_url | |
| primary_location.version | |
| primary_location.raw_type | article-journal |
| primary_location.license_id | https://openalex.org/licenses/cc-by |
| primary_location.is_accepted | False |
| primary_location.is_published | False |
| primary_location.raw_source_name | |
| primary_location.landing_page_url | https://doi.org/10.5281/zenodo.17834486 |
| publication_date | 2025-12-05 |
| publication_year | 2025 |
| referenced_works_count | 0 |
| abstract_inverted_index.We | 60 |
| abstract_inverted_index.an | 122 |
| abstract_inverted_index.as | 99, 121 |
| abstract_inverted_index.be | 38 |
| abstract_inverted_index.by | 46 |
| abstract_inverted_index.in | 24, 135 |
| abstract_inverted_index.is | 35 |
| abstract_inverted_index.of | 6, 21, 31, 50, 57, 69, 113, 129 |
| abstract_inverted_index.on | 77 |
| abstract_inverted_index.to | 37 |
| abstract_inverted_index.we | 88 |
| abstract_inverted_index.The | 27, 117 |
| abstract_inverted_index.ZFC | 130 |
| abstract_inverted_index.and | 104, 110, 131 |
| abstract_inverted_index.can | 75, 92 |
| abstract_inverted_index.for | 11, 66, 125 |
| abstract_inverted_index.how | 63, 90 |
| abstract_inverted_index.its | 41, 83 |
| abstract_inverted_index.not | 43 |
| abstract_inverted_index.set | 25, 52 |
| abstract_inverted_index.the | 3, 13, 19, 29, 32, 47, 55, 67, 73, 100, 108, 127, 132 |
| abstract_inverted_index.Paul | 7 |
| abstract_inverted_index.This | 0 |
| abstract_inverted_index.ZFC. | 86 |
| abstract_inverted_index.and, | 16 |
| abstract_inverted_index.from | 85 |
| abstract_inverted_index.into | 62 |
| abstract_inverted_index.more | 17 |
| abstract_inverted_index.real | 33 |
| abstract_inverted_index.rich | 109 |
| abstract_inverted_index.size | 42 |
| abstract_inverted_index.such | 98 |
| abstract_inverted_index.take | 76 |
| abstract_inverted_index.tool | 124 |
| abstract_inverted_index.with | 54 |
| abstract_inverted_index.work | 118 |
| abstract_inverted_index.Axiom | 56 |
| abstract_inverted_index.delve | 61 |
| abstract_inverted_index.other | 94 |
| abstract_inverted_index.paper | 1 |
| abstract_inverted_index.shown | 36 |
| abstract_inverted_index.where | 72 |
| abstract_inverted_index.(ZFC). | 59 |
| abstract_inverted_index.Choice | 58 |
| abstract_inverted_index.allows | 65 |
| abstract_inverted_index.axioms | 49 |
| abstract_inverted_index.highly | 39 |
| abstract_inverted_index.limits | 128 |
| abstract_inverted_index.theory | 53 |
| abstract_inverted_index.Cohen's | 8 |
| abstract_inverted_index.complex | 111 |
| abstract_inverted_index.examine | 89 |
| abstract_inverted_index.forcing | 9, 64, 91, 120 |
| abstract_inverted_index.theory. | 26 |
| abstract_inverted_index.thereby | 106 |
| abstract_inverted_index.values, | 81 |
| abstract_inverted_index.various | 78 |
| abstract_inverted_index.broadly, | 18 |
| abstract_inverted_index.cardinal | 22, 80, 95, 136 |
| abstract_inverted_index.explores | 2 |
| abstract_inverted_index.infinite | 79 |
| abstract_inverted_index.inherent | 133 |
| abstract_inverted_index.numbers, | 34 |
| abstract_inverted_index.possible | 114 |
| abstract_inverted_index.profound | 4 |
| abstract_inverted_index.spectrum | 20 |
| abstract_inverted_index.standard | 48 |
| abstract_inverted_index.uniquely | 44 |
| abstract_inverted_index.continuum | 14, 74, 102 |
| abstract_inverted_index.exploring | 126 |
| abstract_inverted_index.landscape | 112 |
| abstract_inverted_index.technique | 10 |
| abstract_inverted_index.universes | 71 |
| abstract_inverted_index.arbitrary, | 40 |
| abstract_inverted_index.arithmetic | 96 |
| abstract_inverted_index.continuum, | 28 |
| abstract_inverted_index.determined | 45 |
| abstract_inverted_index.highlights | 119 |
| abstract_inverted_index.hypothesis | 15, 103 |
| abstract_inverted_index.manipulate | 93 |
| abstract_inverted_index.realities. | 116 |
| abstract_inverted_index.structures | 23 |
| abstract_inverted_index.cardinality | 30 |
| abstract_inverted_index.flexibility | 134 |
| abstract_inverted_index.generalized | 101 |
| abstract_inverted_index.properties, | 97 |
| abstract_inverted_index.Furthermore, | 87 |
| abstract_inverted_index.assignments. | 137 |
| abstract_inverted_index.construction | 68 |
| abstract_inverted_index.illustrating | 107 |
| abstract_inverted_index.implications | 5 |
| abstract_inverted_index.independence | 84 |
| abstract_inverted_index.cofinalities, | 105 |
| abstract_inverted_index.demonstrating | 82 |
| abstract_inverted_index.indispensable | 123 |
| abstract_inverted_index.set-theoretic | 70, 115 |
| abstract_inverted_index.understanding | 12 |
| abstract_inverted_index.Zermelo-Fraenkel | 51 |
| cited_by_percentile_year | |
| countries_distinct_count | 0 |
| institutions_distinct_count | 2 |
| citation_normalized_percentile.value | 0.89099596 |
| citation_normalized_percentile.is_in_top_1_percent | False |
| citation_normalized_percentile.is_in_top_10_percent | True |