{"id":14966287,"url":"https://github.com/choukh/set-theory","last_synced_at":"2025-10-25T16:30:47.036Z","repository":{"id":143285360,"uuid":"262253972","full_name":"choukh/Set-Theory","owner":"choukh","description":"A formalization of the textbook Elements of Set Theory","archived":false,"fork":false,"pushed_at":"2021-09-30T05:32:20.000Z","size":3978,"stargazers_count":59,"open_issues_count":0,"forks_count":4,"subscribers_count":2,"default_branch":"master","last_synced_at":"2024-10-30T01:43:37.556Z","etag":null,"topics":["coq","formal-languages","math","set-theory","theorem-proving"],"latest_commit_sha":null,"homepage":"","language":"Coq","has_issues":true,"has_wiki":null,"has_pages":null,"mirror_url":null,"source_name":null,"license":"mit","status":null,"scm":"git","pull_requests_enabled":true,"icon_url":"https://github.com/choukh.png","metadata":{"files":{"readme":"README.md","changelog":null,"contributing":null,"funding":null,"license":"LICENSE","code_of_conduct":null,"threat_model":null,"audit":null,"citation":null,"codeowners":null,"security":null,"support":null,"governance":null,"roadmap":null,"authors":null}},"created_at":"2020-05-08T07:17:20.000Z","updated_at":"2024-05-23T16:38:01.000Z","dependencies_parsed_at":"2023-08-01T03:01:29.695Z","dependency_job_id":null,"html_url":"https://github.com/choukh/Set-Theory","commit_stats":null,"previous_names":[],"tags_count":0,"template":false,"template_full_name":null,"repository_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/choukh%2FSet-Theory","tags_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/choukh%2FSet-Theory/tags","releases_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/choukh%2FSet-Theory/releases","manifests_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/choukh%2FSet-Theory/manifests","owner_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners/choukh","download_url":"https://codeload.github.com/choukh/Set-Theory/tar.gz/refs/heads/master","host":{"name":"GitHub","url":"https://github.com","kind":"github","repositories_count":238174122,"owners_count":19428629,"icon_url":"https://github.com/github.png","version":null,"created_at":"2022-05-30T11:31:42.601Z","updated_at":"2022-07-04T15:15:14.044Z","host_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub","repositories_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories","repository_names_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repository_names","owners_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners"}},"keywords":["coq","formal-languages","math","set-theory","theorem-proving"],"created_at":"2024-09-24T13:36:09.693Z","updated_at":"2025-10-25T16:30:46.623Z","avatar_url":"https://github.com/choukh.png","language":"Coq","funding_links":[],"categories":[],"sub_categories":[],"readme":"[中文](./README.zh-CN.md) 👈\n\n# Set-Theory\n\nThis project is a Coq formalization of the textbook Elements of Set Theory - Herbert B. Enderton. It is basically written in the order of the textbook, without considering modularity. It is suitable as an aid to the learning of set theory, not as a general mathematical library.\n\n## Requirement\n```\nCoq 8.13.2\n```\n\n## Build\n```\nmake\n```\n\n## Meta.v\n- Law of excluded middle\n- Church's iota operator\n- Informative excluded middle\n- Decidable inhabitance of type\n\n## ZFC0.v\n- Axiom of extensionality\n- Axiom of empty set\n- Axiom of union\n- Axiom of power set\n- Axiom schema of replacement\n\n## ZFC1.v\n- Pair\n- Singleton\n- Binary union\n- Union of a family of sets\n\n## ZFC2.v\n- Set comprehension\n- Intersaction, binary intersaction\n- Ordered pair\n- Cartesian product\n\n## ZFC3.v\n- Axiom of infinity\n- Axiom of choice\n\n## EST2.v\n- Complement\n- Proper subset\n- Algebra of sets\n\n## EST3_1.v\n- Relation, function\n- Inverse, composition\n\n## EST3_2.v\n- Injection, surjection, bijection\n- Left inverse and right inverse of function\n- Restriction, image\n- Function space\n- Infinite Cartesian product\n\n## EST3_3.v\n- Binary relation\n- Equivalence relation, equivalence class, quotient set\n- Trichotomy, linear order\n\n## EST4_1.v\n- Natural number\n- Induction principle\n- Transitive set\n- Peano structure\n- Recursion theorem\n\n## EST4_2.v\n- Embedding of type-theoretic nat\n- Natural number arithmetic: addition, multiplication, exponentiation\n\n## EST4_3.v\n- Linear ordering of ω\n- Well ordering of ω\n- Strong induction principle\n\n## EST5_1.v\n- Integer\n- Integer arithmetic: addition, additive inverse\n\n## EST5_2.v\n- Multiplication of integers\n- Order of integers\n- Embedding of the natural numbers\n\n## EST5_3.v\n- Rational number\n- Rational number arithmetic: addition, additive inverse, multiplication, multiplicative inverse\n\n## EST5_4.v\n- Order of rational numbers\n- Embedding of the integers\n- Algebra regarding to inverse\n\n## EST5_5.v\n- Real number (Dedekind cut)\n- Order of real numbers\n- Completeness of the real numbers\n- Real number arithmetic: addition, additive inverse\n\n## EST5_6.v\n- Absolute value of real number\n- Multiplication of non-negative real numbers\n- Multiplicative inverse of positive real number\n\n## EST5_6.v\n- Arithmetic of rational numbers: multiplication, multiplicative inverse\n- Embedding of the rational numbers\n- Density of the real numbers\n\n## EST6_1.v\n- Equinumerous\n- Cantor's theorem\n- Pigeonhole principle\n- Finite cardinal\n\n## EST6_2.v\n- Infinite cardinal\n- Cardinal arithmetic: addition, multiplication, exponentiation\n\n## EST6_3.v\n- Dominate\n- Schröder–Bernstein theorem\n- Order of cardinals\n- Aleph Zero\n\n## EST6_4.v\n- Systematic discussion on AC\n  - Uniformization\n  - Infinite Cartesian product of nonempty sets is nonempty\n  - Choice function\n  - Cardinal comparability\n  - Zorn's lemma\n  - Tukey's lemma\n  - Hausdorff maximal principle\n- Aleph Zero is the least infinite cardinal\n- Dedekind infinite\n- Infinite sum of cardinals\n- Infinite product of cardinals\n\n## EST6_5.v\n- Countable set\n  - Countable union of countable sets is countable\n\n## EST6_6.v\n- Algebra of infinite cardinals\n  - Cardinal multiplied by itself equals to itself\n  - Absortion law of cardinal addition and multiplication\n\n## EST7_1.v\n- Partial order, linear order\n- Minimal, minimum, maximal, maximum\n- Bound, supremum, infimum\n\n## EST7_2.v\n- Well order\n- Transfinite induction principle\n- Transfinite recursion theorem\n- Transitive closure of set\n\n## EST7_3.v\n- Order structure\n- Isomorphism\n- Epsilon image\n\n## EST7_4.v\n- Ordinal\n- Order of ordinals\n- Burali-Forti's paradox\n- Successor ordinal, limit ordinal\n- Transfinite induction schema on ordinals\n\n## EST7_5.v\n- Hartog's number\n- Equivalence among well order theorem, AC and Zorn's lemma\n- von Neumann cardinal assignment\n- Initial cardinal, successor cardinal\n\n## EST7_6.v\n- Transfinite recursion schema on ordinals\n- von Neumann universe\n- Rank\n- Axiom of regularity\n\n## EST8_1.v\n- Ordinal class\n- Ordinal operations\n  - Subclass separation\n  - Normal operation\n- Aleph number\n- Beth number\n\n## EST8_2.v\n- Properties of ordinal operations\n- Veblen fixed-point theorem\n  - Enumeration of fixed-point is normal operation\n  - There exist fixed-point of fixed-point\n\n## EST8_3.v\n- Order types\n- Addition of order types\n\n## EST8_4.v\n- Multiplication of order types\n- Laws of order type arithmetic\n\n## EST8_5.v\n- Order type arithmetic on well-ordered structure\n\n## EST8_6.v\n- Ordinal Arithmetic (defined as order type arithmetic)\n  - Addition, multiplication\n\n## EST8_7.v\n- Ordinal Arithmetic (defined by recursion)\n  - Addition, multiplication, exponentiation\n- Tetration, epsilon numbers\n\n## EX{n}.v\n- Solution to exercises of Chapter n\n","project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fchoukh%2Fset-theory","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Fchoukh%2Fset-theory","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fchoukh%2Fset-theory/lists"}