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- 06:01, 28 June 2022 diff hist +654 N Successor of Ordinal Smaller than Limit Ordinal is also Smaller Created page with "== Theorem == Let $\On$ denote the class of all ordinals. Let $\lambda \in \On$ be a limit ordinal. Then: :$\forall \alpha \in \On: \alpha < \lambda \implies \alpha^+ < \lambda$ == Proof == {{ProofWanted}} == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Exists Ordinal Greater than Set..." current
- 05:23, 28 June 2022 diff hist +43 m Zero is Smallest Ordinal current
- 05:23, 28 June 2022 diff hist +38 m Minimal Element of an Ordinal current
- 05:22, 28 June 2022 diff hist +1,961 N Exists Ordinal Greater than Set of Ordinals Created page with "== Theorem == Let $S$ be a set of ordinals. Then there exists an ordinal greater than every element of $S$: :If $S$ contains a greatest ordinal $\alpha$, then $\alpha^+$ is greater than every element of $S$ :If $S$ does not contain a greatest ordinal, then $\bigcup S$ is greater than every Def..." current
- 05:06, 28 June 2022 diff hist +976 N Equality of Successors implies Equality of Ordinals Created page with "== Theorem == Let $\On$ denote the class of all ordinals. Then: :$\forall \alpha, \beta \in \On: \alpha^+ = \beta^+ \implies \alpha = \beta$ == Proof == From Class of All Ordinals is Well-Ordered by Subset Relation: :$\alpha^+$ is the immediate successor of $\alpha$ :$\beta^+$ is the immediate successor of $\beta$ and no two Definition:Di..." current
- 22:20, 27 June 2022 diff hist −35 m Successor is Less than Successor whoops, no current Tag: Manual revert
- 22:20, 27 June 2022 diff hist −45 m Equality of Successors current Tag: Manual revert
- 22:20, 27 June 2022 diff hist +45 m Equality of Successors Tag: Reverted
- 22:19, 27 June 2022 diff hist +35 Successor is Less than Successor Tag: Reverted
- 22:16, 27 June 2022 diff hist +1,023 N Ordinal Membership is Transitive Created page with "== Theorem == Let $\On$ denote the class of all ordinals. Then: :$\forall \alpha, \beta, \gamma \in \On: \paren {\alpha \in \beta} \land \paren {\beta \in \gamma} \implies \alpha \in \gamma$ == Proof == By Strict Ordering of Ordinals is Equivalent to Membership Relation the statement to be proved is equivalent to: :$\forall \alpha, \beta, \gamma \in \On: \paren {\alpha \subsetneqq \beta} \la..." current
- 22:07, 27 June 2022 diff hist +156 Talk:Class of All Ordinals is Ordinal current
- 20:17, 27 June 2022 diff hist +16 m Ordinal equals its Initial Segment current
- 20:17, 27 June 2022 diff hist +8 m Definition:Usual Ordering of Ordinals current
- 20:17, 27 June 2022 diff hist 0 m Strict Ordering of Ordinals is Equivalent to Membership Relation Prime.mover moved page Strict Ordering of Ordinals is Equivalent to Elementhood to Strict Ordering of Ordinals is Equivalent to Membership Relation current
- 19:25, 27 June 2022 diff hist +2 Ordinal Membership is Trichotomy/Corollary current
- 19:24, 27 June 2022 diff hist +733 N Ordinal Membership is Trichotomy/Corollary Created page with "== Corollary to Ordinal Membership is Trichotomy == <onlyinclude> Let $\alpha$ be an ordinal. Let $x, y \in \alpha$ such that $x \ne y$. Then either: :$x \in y$ or: $y \in x$ </onlyinclude> == Proof == We have that Element of Ordinal is Ordinal. The result then follows directly from Ordinal Membership is Trichotomy {{qed}} == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Mel..."
- 19:18, 27 June 2022 diff hist +114 Ordinal Membership is Trichotomy current
- 19:17, 27 June 2022 diff hist −33 m User:StarTower/Structure Notes/Definition Pages/Heading Two Statistics/Plan of Use/Sourceless Pages current
- 19:15, 27 June 2022 diff hist +3 Equivalence of Definitions of Artinian Module current
- 19:15, 27 June 2022 diff hist +69 N Category:Artinian Modules Created page with "{{SubjectCategory|Artinian Module}} Category:Examples of Modules" current
- 19:15, 27 June 2022 diff hist +64 N Category:Definitions/Artinian Modules Created page with "{{DefinitionCategory|def = Artinian Module|Examples of Modules}}" current
- 19:14, 27 June 2022 diff hist +27 Definition:Artinian Module current
- 19:14, 27 June 2022 diff hist +96 Definition:Artinian Module
- 19:11, 27 June 2022 diff hist +930 N Ordinal Membership is Trichotomy/Proof 1 Created page with "== Theorem == {{:Ordinal Membership is Trichotomy}} == Proof == <onlyinclude> From Class of All Ordinals is Well-Ordered by Subset Relation, $\On$ is a nest. Hence: :$\forall \alpha, \beta \in \On: \paren {\alpha \subsetneqq \beta} \lor \paren {\beta \subsetneqq \alpha} \lor \paren {\alpha = \beta}$ As Ordinal is Transitive, this is equivalent to: :$\forall \alpha, \beta \in \On: \paren {\alpha \in \beta} \lor \paren {\beta..." current
- 19:08, 27 June 2022 diff hist +12 Definition:Usual Ordering of Ordinals
- 19:06, 27 June 2022 diff hist −130 Ordinal Membership is Trichotomy
- 19:01, 27 June 2022 diff hist 0 m Strict Ordering of Ordinals is Equivalent to Membership Relation
- 19:01, 27 June 2022 diff hist +1,077 N Ordinal equals its Initial Segment Created page with "== Theorem == Let $\On$ denote the class of all ordinals. Let $<$ denote the (strict) usual ordering of $\On$. Let $\alpha$ be an ordinal. Then $\alpha$ is equal to its own initial segment: :$\alpha = \set {\beta \in \On: \beta < \alpha}$ == Proof == From Strict Ordering of Ordinals is Equivalent to Elementhood: :$\forall \alpha,..."
- 19:00, 27 June 2022 diff hist +525 N Ordinal Membership is Trichotomy/Proof 2 Created page with "== Theorem == {{:Ordinal Membership is Trichotomy}} == Proof == <onlyinclude> By Relation between Two Ordinals, it follows that: :$\paren {\alpha = \beta} \lor \paren {\alpha \subset \beta} \lor \paren {\beta \subset \alpha}$ By Transitive Set is Proper Subset of Ordinal iff Element of Ordinal, the result follows. {{qed}} </onlyinclude> == Sources == * {{BookReference|Introduction to Axiomatic Set Theory|1971|Gaisi Takeuti|author2 = Wilson M. Zaring}}: $\S 7...." current
- 19:00, 27 June 2022 diff hist +84 N Category:Ordinal Membership is Trichotomy Created page with "{{SubjectCategory|result = Ordinal Membership is Trichotomy}} Category:Ordinals" current
- 19:00, 27 June 2022 diff hist +44 Ordinal Membership is Trichotomy
- 18:39, 27 June 2022 diff hist +1,545 N Strict Ordering of Ordinals is Equivalent to Membership Relation Created page with "== Theorem == Let $\On$ denote the class of all ordinals. Let $<$ denote the (strict) usual ordering of $\On$. Then: :$\forall \alpha, \beta \in \On: \alpha < \beta \iff \alpha \in \beta$ == Proof == === Necessary Condition === Let $\alpha \in \beta$. Then from Ordinal is Transitive: :$\alpha \subseteq \beta$ But if $\alpha = \beta$ we would have $\alpha \in \alpha$. This is cont..."
- 18:14, 27 June 2022 diff hist +825 N Definition:Usual Ordering of Ordinals Created page with "== Definition == Let $\On$ denote the class of all ordinals. The '''usual ordering''' $\le$ on $\On$ is the subset relation: :$a \le b \iff a \subseteq b$ and its corresponding strict version: :$a < b \iff a \subsetneqq b$ == Also see == * Well-Ordering of Class of All Ordinals under Subset Relation, demonstrating that $\le$ is a well-ordering. == Sources == * {..."
- 18:06, 27 June 2022 diff hist +15 Definition:Artinian Module/Definition 1
- 18:06, 27 June 2022 diff hist +15 Definition:Artinian Module/Definition 2
- 18:05, 27 June 2022 diff hist +15 m Definition:Minimal Condition/Ordered Set current
- 18:05, 27 June 2022 diff hist +14 m Definition:Minimal Condition/Subset Ordering current
- 18:04, 27 June 2022 diff hist +14 Definition:Noetherian Module/Definition 3 current
- 18:04, 27 June 2022 diff hist +14 m Definition:Minimal Condition/Submodule Ordering current
- 18:03, 27 June 2022 diff hist +56 N Definition:Maximal Condition Prime.mover moved page Definition:Maximal Condition to Definition:Maximal Condition on Submodules current Tag: New redirect
- 18:03, 27 June 2022 diff hist 0 m Definition:Maximal Condition on Submodules Prime.mover moved page Definition:Maximal Condition to Definition:Maximal Condition on Submodules current
- 18:03, 27 June 2022 diff hist +151 Definition:Maximal Condition on Submodules
- 17:59, 27 June 2022 diff hist +153 Definition:Minimal Condition/Submodule Ordering
- 17:57, 27 June 2022 diff hist +1 m Definition:Artinian Module/Definition 2 Undo revision 582938 by Usagiop (talk) Tag: Undo
- 05:49, 27 June 2022 diff hist +13 m Book:Raymond M. Smullyan/Set Theory and the Continuum Problem/Revised Edition current
- 05:46, 27 June 2022 diff hist +98 Nilpotent Element is Contained in Prime Ideals Clarified flow current
- 05:38, 27 June 2022 diff hist +47 m Well-Ordering of Class of All Ordinals under Subset Relation current
- 05:33, 27 June 2022 diff hist +1,729 N Well-Ordering of Class of All Ordinals under Subset Relation Created page with "== Theorem == Let $\On$ denote the class of all ordinals. $\On$ is well-ordered by the subset relation such that the following $3$ conditions hold: {{begin-axiom}} {{axiom | n = 1 | lc= the smallest ordinal is $0$ }} {{axiom | n = 2 | lc= for $\alpha \in \On$, the Definition:Immedi..."
- 05:28, 27 June 2022 diff hist −4 m G-Tower is Well-Ordered under Subset Relation/Corollary current
- 05:27, 27 June 2022 diff hist −12 m G-Tower is Well-Ordered under Subset Relation/Union of Limit Elements current