New pages

(newest | oldest) View (newer 50 | older 50) (20 | 50 | 100 | 250 | 500)

• 06:01, 28 June 2022 ‎Successor of Ordinal Smaller than Limit Ordinal is also Smaller (hist | edit) ‎[654 bytes] ‎Prime.mover (talk | contribs) (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...")
• 05:22, 28 June 2022 ‎Exists Ordinal Greater than Set of Ordinals (hist | edit) ‎[1,961 bytes] ‎Prime.mover (talk | contribs) (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...")
• 05:06, 28 June 2022 ‎Equality of Successors implies Equality of Ordinals (hist | edit) ‎[976 bytes] ‎Prime.mover (talk | contribs) (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...")
• 22:16, 27 June 2022 ‎Ordinal Membership is Transitive (hist | edit) ‎[1,023 bytes] ‎Prime.mover (talk | contribs) (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...")
• 19:24, 27 June 2022 ‎Ordinal Membership is Trichotomy/Corollary (hist | edit) ‎[735 bytes] ‎Prime.mover (talk | contribs) (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...")
• 19:22, 27 June 2022 ‎Conditional Entropy Decreases if More Given/Corollary (hist | edit) ‎[803 bytes] ‎Usagiop (talk | contribs) (Created page with "== Corollary to Editing Conditional Entropy Decreases if More Given == Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $\AA, \DD \subseteq \Sigma$ be finite sub-$\sigma$-algebras. Then: <onlyinclude> :$\map H \AA \ge \map H {\AA \mid \DD} $ </onlyinclude> where: :$\map H \cdot$ denotes the entropy :$\map H {\cdot \mid \cdot}$...")
• 19:18, 27 June 2022 ‎Conditional Entropy Decreases if More Given (hist | edit) ‎[842 bytes] ‎Usagiop (talk | contribs) (Created page with "== Theorem == Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $\AA, \CC, \DD \subseteq \Sigma$ be finite sub-$\sigma$-algebras. Then: <onlyinclude> :$\CC \subseteq \DD \implies \map H {\AA \mid \CC} \ge \map H {\AA \mid \DD}$ </onlyinclude> where: :$\map H {\cdot \mid \cdot}$ denotes the conditional entropy == Proof...")
• 19:11, 27 June 2022 ‎Ordinal Membership is Trichotomy/Proof 1 (hist | edit) ‎[930 bytes] ‎Prime.mover (talk | contribs) (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...")
• 19:03, 27 June 2022 ‎Equivalence of Definitions of Artinian Module (hist | edit) ‎[1,041 bytes] ‎Usagiop (talk | contribs) (Created page with "== Theorem == {{TFAE|def = Artinian Module}} === Definition 1 === {{Definition:Artinian Module/Definition 1}} === Definition 2 === {{Definition:Artinian Module/Definition 2}} == Proof == === Definition 1 iff Definition 2 === Let $D$ be the set of all submodules of $M$. We shall show that: :Definition:Descending Chain Condition/Module...")
• 19:01, 27 June 2022 ‎Ordinal equals its Initial Segment (hist | edit) ‎[1,093 bytes] ‎Prime.mover (talk | contribs) (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: :$\forall \alpha,...")
• 19:00, 27 June 2022 ‎Ordinal Membership is Trichotomy/Proof 2 (hist | edit) ‎[525 bytes] ‎Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Ordinal Membership is Trichotomy}} == Proof == <onlyinclude> By Relation between Two Ordinals, it follows that: :$\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....")
• 05:33, 27 June 2022 ‎Well-Ordering of Class of All Ordinals under Subset Relation (hist | edit) ‎[1,776 bytes] ‎Prime.mover (talk | contribs) (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...")
• 23:05, 26 June 2022 ‎Nilpotent Element is Contained in Prime Ideals (hist | edit) ‎[852 bytes] ‎Usagiop (talk | contribs) (Created page with "== Theorem == Let $R$ be a commutative ring with unity. Let $\mathfrak p \subset R$ be a prime ideal of $R$. Let $x\in R$ be a nilpotent element. Then $x\in\mathfrak p$. == Proof == {{proofWanted}} Category:Prime Ideals of Rings")
• 14:12, 26 June 2022 ‎Class of All Ordinals is Proper Class/Proof 2 (hist | edit) ‎[550 bytes] ‎Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Class of All Ordinals is Proper Class}} == Proof == <onlyinclude> {{AimForCont}} $\On$ is a set. Then from the Burali-Forti Paradox, a contradiction could be deduced. Hence by Proof by Contradiction, $\On$ cannot be a set. Thus $\On$ is a class that is not a set. Hence $\On$ is a proper class....")
• 12:13, 26 June 2022 ‎Class of All Ordinals is Proper Class/Proof 1 (hist | edit) ‎[693 bytes] ‎Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Class of All Ordinals is Proper Class}} == Proof == <onlyinclude> We have that Successor Mapping on Ordinals is Strictly Progressing. The result follows from Superinductive Class under Strictly Progressing Mapping is not Set. {{qed}} </onlyinclude> == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Successor Mapping on Ordinals is...")
• 12:10, 26 June 2022 ‎Class of All Ordinals is Proper Class (hist | edit) ‎[550 bytes] ‎Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $\On$ denote the class of all ordinals. Then $\On$ is a proper class. That is, $\On$ is not a set. </onlyinclude> == Proof 1 == {{:Class of All Ordinals is Proper Class/Proof 1}} == Proof 2 == {{:Class of All Ordinals is Proper Class/Proof 2}} Category...")
• 11:54, 26 June 2022 ‎Successor Mapping on Ordinals is Strictly Progressing (hist | edit) ‎[1,063 bytes] ‎Prime.mover (talk | contribs) (Created page with "== Theorem == Let $\On$ denote the class of all ordinals. Let $s$ denote the successor mapping on $\On$: :$\forall \alpha \in \On: \map s \alpha := \alpha \cup \set \alpha$ where $\alpha$ is an ordinal. Then $s$ is a strictly progressing mapping. == Proof == From Ordinal is Proper Subset of Successor: :$\alpha \subsetneqq \al...")
• 11:28, 26 June 2022 ‎Ordinal is Proper Subset of Successor (hist | edit) ‎[1,127 bytes] ‎Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $\alpha$ be an ordinal. Then: :$\alpha \subsetneqq \alpha^+$ where $\alpha^+$ denotes the successor set of $\alpha$. That is: :$\alpha$ is a proper subset of $\alpha^+$. </onlyinclude> == Proof == {{AimForCont}} $\alpha = \alpha^+$. By definition: :$\alpha^+ = \alpha \cup \set \alpha$ and so: :$\alpha \subseteq \alpha^+$ and: :$\alpha \in \alpha^+$ whi...")
• 09:06, 26 June 2022 ‎Union of Set of Ordinals is Ordinal/Corollary/Proof 2 (hist | edit) ‎[2,152 bytes] ‎Prime.mover (talk | contribs) (Created page with "== Corollary to Union of Set of Ordinals is Ordinal == {{:Union of Set of Ordinals is Ordinal/Corollary}} == Proof == <onlyinclude> {{begin-eqn}} {{eqn | l = x | o = \in | r = \bigcup_{z \mathop \in y} \map F z | c = }} {{eqn | ll= \leadsto | q = \exists z \in y | l = x | o = \in | r = \map F z | c = {{Defof|Set Union}} }} {{eqn | ll= \leadsto | l = x | o = \in | r = \On | c = {{hypothesis}} }}...")
• 09:03, 26 June 2022 ‎Union of Set of Ordinals is Ordinal/Corollary/Proof 1 (hist | edit) ‎[912 bytes] ‎Prime.mover (talk | contribs) (Created page with "== Corollary to Union of Set of Ordinals is Ordinal == {{:Union of Set of Ordinals is Ordinal/Corollary}} == Proof == <onlyinclude> By the Axiom of Replacement, $\map F y$ is a set. Thus by the Axiom of Unions, $\bigcup \map F y$ is a set. By Union of Set of Ordinals is Ordinal, $\bigcup \map F y$ is transitive. By the ...")
• 08:14, 26 June 2022 ‎Ordinal is not Element of Itself/Proof 3 (hist | edit) ‎[3,000 bytes] ‎Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Ordinal is not Element of Itself}} == Proof == <onlyinclude> Let $\On$ denote the class of all ordinals. From Class of All Ordinals is Minimally Superinductive over Successor Mapping, $\On$ is superinductive. Hence we can use the Principle of Superinduction. By Zero is Smallest Ordinal, $0$ is the smallest element of $\...")
• 07:47, 26 June 2022 ‎Successor Set of Ordinary Transitive Set is Ordinary (hist | edit) ‎[1,701 bytes] ‎Prime.mover (talk | contribs) (Created page with "== Theorem == Let $x$ be a transitive set which is also ordinary. Let $x^+$ denote the successor set of $x$: :$x^+ = x \cup \set x$ Then $x^+$ is also an ordinary set. == Proof == By definition of ordinary set: :$x \notin x$ {{AimForCont}} $x^+ \in x^+$. Then we have: {{begin-eqn}} {{eqn | l = x^+ | r = x \cup \set...")
• 07:29, 26 June 2022 ‎Empty Set is Ordinary (hist | edit) ‎[291 bytes] ‎Prime.mover (talk | contribs) (Created page with "== Theorem == The empty set is an ordinary set: :$\O \notin \O$ == Proof == By definition: :$\forall x: x \notin \O$ and so in particular: :$\O \notin \O$ Hence the result. {{qed}} Category:Empty Set Category:Ordinary Sets")
• 22:34, 25 June 2022 ‎Ordinal is Transitive/Proof 4 (hist | edit) ‎[1,277 bytes] ‎Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Ordinal is Transitive}} == Proof == <onlyinclude> Let $\alpha$ be an ordinal by Definition 4. {{:Definition:Ordinal/Definition 4}} The proof proceeds by the Principle of Superinduction. From Empty Class is Transitive we start with the fact that $0$ is transitive. {{qed|lemma}} Let $x$ be transitive. {{begin-eqn}} {{eqn | l...")
• 12:20, 25 June 2022 ‎Ordinal is Transitive/Proof 1 (hist | edit) ‎[339 bytes] ‎Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Ordinal is Transitive}} == Proof == <onlyinclude> Let $S$ be an ordinal by Definition 1: {{:Definition:Ordinal/Definition 1}} Thus $S$ is {{apriori}} transitive. {{qed}} </onlyinclude> Category:Ordinal is Transitive")
• 07:50, 25 June 2022 ‎Ordinal is Transitive/Proof 3 (hist | edit) ‎[393 bytes] ‎Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Ordinal is Transitive}} == Proof == <onlyinclude> Let $S$ be an ordinal by Definition 3. By definition: :$\forall a \in S: a = S_a \subseteq S$ where $S_a$ denotes the initial segment of $S$ determined by $a$. That is, $S$ is a transitive set. {{qed}} </onlyinclude> Category:Ordinal is Transitive") originally created as "Ordinal is Transitive/Proof 2"
• 07:46, 25 June 2022 ‎Element of Ordinal is Ordinal/Proof 1 (hist | edit) ‎[3,116 bytes] ‎Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Element of Ordinal is Ordinal}} == Proof == <onlyinclude> Let $\On$ denote the class of all ordinals. From Class of All Ordinals is Minimally Superinductive over Successor Mapping, $\On$ is superinductive. By Zero is Smallest Ordinal, $0$ is the smallest element of $\On$. We identify the Definition:Zero (Number)|natural n...")
• 06:57, 25 June 2022 ‎Zero is Smallest Ordinal (hist | edit) ‎[815 bytes] ‎Prime.mover (talk | contribs) (Created page with "== Theorem == The natural number $0$ is the smallest ordinal. == Proof == Let $\On$ denote the class of all ordinals. By the Zero is Ordinal, $0$ is an element of $\On$. We identify the natural number $0$ via the Definition:Von Neumann Construction of Natural Numbers|von Neu...")
• 23:26, 24 June 2022 ‎Element of Ordinal is Ordinal/Proof 2 (hist | edit) ‎[1,247 bytes] ‎Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Element of Ordinal is Ordinal}} == Proof == <onlyinclude> By the definition of ordinal, $n$ is transitive. Thus $m \subseteq n$. By Subset of Strictly Well-Ordered Set is Strictly Well-Ordered, it follows that $m$ is strictly well-ordered by the epsilon restriction $\Epsilon {\restriction_m}$. It is now to be shown that $...")
• 23:15, 24 June 2022 ‎Natural Number is Ordinal/Proof 2 (hist | edit) ‎[898 bytes] ‎Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Natural Number is Ordinal}} == Proof == <onlyinclude> From the von Neumann construction of the natural numbers, $\N$ is identified with the minimally inductive set $\omega$. From Superinductive Class under Successor Mapping contains All Ordinals, it follows by the Principle of Mathematical Induction that every Definition:Natural Number|natural nu...")
• 23:09, 24 June 2022 ‎Natural Number is Ordinal/Proof 1 (hist | edit) ‎[1,084 bytes] ‎Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Natural Number is Ordinal}} == Proof == <onlyinclude> Consider the class of all ordinals $\On$. From Class of All Ordinals is Minimally Superinductive over Successor Mapping, $\On$ is superinductive. Hence {{afortiori}} $\On$ is inductive. From the Definition:Von Neumann Construction of Natural Numbers|von Neumann construction of the natural...")
• 23:08, 24 June 2022 ‎Natural Number is Ordinal (hist | edit) ‎[753 bytes] ‎Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $n \in \N$ be a natural number. Then $n$ is an ordinal. </onlyinclude> == Proof 1 == {{:Natural Number is Ordinal/Proof 1}} == Proof 2 == {{:Natural Number is Ordinal/Proof 2}} == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|...")
• 22:11, 24 June 2022 ‎Union of Set of Ordinals is Ordinal/Proof 1 (hist | edit) ‎[698 bytes] ‎Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Union of Set of Ordinals is Ordinal}} == Proof == <onlyinclude> By Class of All Ordinals is Well-Ordered by Subset Relation, a set of ordinals forms a chain. The result then follows from Union of Chain of Ordinals is Ordinal. </onlyinclude> == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|ed...")
• 22:06, 24 June 2022 ‎Union of Set of Ordinals is Ordinal/Proof 2 (hist | edit) ‎[1,324 bytes] ‎Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Union of Set of Ordinals is Ordinal}} == Proof == <onlyinclude> {{begin-eqn}} {{eqn | l = x \in \bigcup A | o = \leadsto | r = \exists y \in A: x \in y | c = }} {{eqn | o = \leadsto | r = \exists y \subseteq \bigcup A: x \subseteq y | c = }} {{eqn | o = \leadsto | r = x \subseteq \bigcup A | c = }} {{end-eqn}} From this, we conclude that $\ds \bigcup A$ is a transitive class....")
• 09:27, 24 June 2022 ‎Successor Set of Ordinal is Ordinal/Proof 1 (hist | edit) ‎[971 bytes] ‎Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Successor Set of Ordinal is Ordinal}} == Proof == <onlyinclude> We have the result that Class of All Ordinals is Minimally Superinductive over Successor Mapping. Hence $\On$ is {{afortiori}} a superinductive class {{WRT}} the successor mapping. Hence, by definition of superinductive class: :$\On$ is Definition:Closed Class under Mapping|close...")
• 09:25, 24 June 2022 ‎Successor Set of Ordinal is Ordinal/Proof 2 (hist | edit) ‎[2,483 bytes] ‎Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Successor Set of Ordinal is Ordinal}} == Proof == <onlyinclude> From Ordinal is Transitive, it follows by Successor Set of Transitive Set is Transitive that $S^+$ is transitive. We now have to show that $S^+$ is strictly well-ordered by the epsilon restriction $\Epsilon \! \restriction_{S^+}$. So suppose that a subset $A \...")
• 21:44, 23 June 2022 ‎Levi-Civita Connection in Local Orthonormal Frame (hist | edit) ‎[1,739 bytes] ‎Julius (talk | contribs) (Created page with "== Theorem == Let $\struct {M, g}$ be a Riemannian with or without boundary. Let $U \subseteq M$ be an open subset of $M$. Let $\sqbrk {\cdot, \cdot}$ be the Lie bracket. Let $\tuple {E_i}$ be a smooth local orthonormal frame. Let $c^k_{ij} : U \to \R$ be...")
• 18:53, 23 June 2022 ‎Equivalence of Definitions of Limit Ordinal (hist | edit) ‎[1,140 bytes] ‎Prime.mover (talk | contribs) (Created page with "== Theorem == {{TFAE|def = Limit Ordinal}} === Definition 1 === {{:Definition:Limit Ordinal/Definition 1}} === Definition 2 === {{:Definition:Limit Ordinal/Definition 2}} == Proof == Let $x \in \On$ be an element of the class of all ordinals. From Categories of Elements under Well-Ordering, $x$ falls into one of...")
• 18:00, 23 June 2022 ‎Class of All Ordinals is Well-Ordered by Subset Relation/Proof 2 (hist | edit) ‎[783 bytes] ‎Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Class of All Ordinals is Well-Ordered by Subset Relation}} == Proof == <onlyinclude> From Class of All Ordinals is $g$-Tower, $\On$ is a $g$-tower. The result follows from $g$-Tower is Well-Ordered under Subset Relation. {{qed}} </onlyinclude> == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author...")
• 17:53, 23 June 2022 ‎Class of All Ordinals is G-Tower (hist | edit) ‎[674 bytes] ‎Prime.mover (talk | contribs) (Created page with "== Theorem == Let $\On$ denote the class of all ordinals. Let $g$ be the successor mapping: :$\forall x \in \On: \map g x = x \cup \set x$ Then $\On$ is a $g$-tower. == Proof == From Successor Mapping is Progressing, $g$ is a progressing mapping. From Class of All Ordinals is Minimally Superinductive over Successor Mapping, $\On$...")
• 17:46, 23 June 2022 ‎Class of All Ordinals is Well-Ordered by Subset Relation/Proof 1 (hist | edit) ‎[592 bytes] ‎Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Class of All Ordinals is Well-Ordered by Subset Relation}} == Proof == <onlyinclude> By Subset Relation is Ordering, $\subseteq$ is an ordering of any class. Let $A$ be a subclass of $\On$. By Intersection of Ordinals is Smallest, $A$ has a smallest element under the Definition:Subset Relation|subs...")
• 17:41, 23 June 2022 ‎Properties of Class of All Ordinals/Superinduction Principle (hist | edit) ‎[2,222 bytes] ‎Prime.mover (talk | contribs) (Created page with "== Theorem == Let $\On$ denote the class of all ordinals. <onlyinclude> Let $A$ be a class which satisfies the following $3$ conditions: {{begin-axiom}} {{axiom | n = 1 | lc= $A$ contains the natural number $0$: | q = | m = \quad 0 \in A }} {{axiom | n = 2 | lc= $A$ is closed under Definition...")
• 17:35, 23 June 2022 ‎Properties of Class of All Ordinals/Union of Chain of Ordinals is Ordinal (hist | edit) ‎[1,410 bytes] ‎Prime.mover (talk | contribs) (Created page with "== Theorem == Let $\On$ denote the class of all ordinals. <onlyinclude> Let $C$ be a chain of elements of $\On$. Then its union $\ds \bigcup C$ is also an element of $\On$. </onlyinclude> == Proof == {{ProofWanted}} == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 =...")
• 17:30, 23 June 2022 ‎Properties of Class of All Ordinals/Zero is Ordinal (hist | edit) ‎[1,199 bytes] ‎Prime.mover (talk | contribs) (Created page with "== Theorem == Let $\On$ denote the class of all ordinals. <onlyinclude> The natural number zero is an element of $\On$. </onlyinclude> == Proof == {{ProofWanted}} == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Properties of Class of All Ordinals|next = Properties...")
• 17:26, 23 June 2022 ‎Properties of Class of All Ordinals (hist | edit) ‎[1,282 bytes] ‎Prime.mover (talk | contribs) (Created page with "== Theorem == Let $\On$ denote the class of all ordinals. $\On$ has the following properties: === Zero is an Ordinal === {{:Properties of Class of All Ordinals/Zero is Ordinal}} === Successor of Ordinal is an Ordinal === {{:Properties of Class of All Ordinals/Successor of Ordinal is Ordinal}} === Prop...")
• 16:53, 23 June 2022 ‎Class of All Ordinals is Minimally Superinductive over Successor Mapping (hist | edit) ‎[5,421 bytes] ‎Prime.mover (talk | contribs) (Created page with "== Theorem == The class of all ordinals $\On$ is the class which is minimally superinductive under the successor mapping. == Proof == {{ProofWanted}} == Sources == * {{BookReference|Set Theory and the Continuum Problem|2010|Raymond M. Smullyan|author2 = Melvin Fitting|ed = revised|edpage = Revised Edition|prev = Def...")
• 22:57, 22 June 2022 ‎Conditional Entropy of Join as Sum/Corollary 5 (hist | edit) ‎[1,488 bytes] ‎Usagiop (talk | contribs) (Created page with "== Corollary 5 to Conditional Entropy of Join as Sum == Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $\AA, \DD \subseteq \Sigma$ be finite sub-$\sigma$-algebras. Then: <onlyinclude> :$\ds \map H \AA \le \map H {\AA \mid \DD} $ </onlyinclude> where: :$\map H \cdot$ denotes the entropy :$\map H {\cdot \mid \cdot}$ denotes the...")
• 22:55, 22 June 2022 ‎Conditional Entropy of Join as Sum/Corollary 4 (hist | edit) ‎[1,178 bytes] ‎Usagiop (talk | contribs) (Created page with "== Corollary 4 to Conditional Entropy of Join as Sum == Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $\AA, \CC, \DD \subseteq \Sigma$ be finite sub-$\sigma$-algebras. Then: <onlyinclude> :$\ds \CC \subseteq \DD \implies \map H {\AA \mid \CC} \ge \map H {\AA \mid \DD} $ </onlyinclude> where: :$\map H {\cdot \mid \cdot}$ denotes the Definition:Conditional Entropy of Finite...")
• 22:54, 22 June 2022 ‎Conditional Entropy of Join as Sum/Corollary 3 (hist | edit) ‎[1,376 bytes] ‎Usagiop (talk | contribs) (Created page with "== Corollary 3 to Conditional Entropy of Join as Sum == Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $\AA, \CC \subseteq \Sigma$ be finite sub-$\sigma$-algebras. Then: <onlyinclude> :$\ds \AA \subseteq \CC \implies \map H \AA \le \map H \CC $ </onlyinclude> where: :$\map H \cdot$ denotes the entropy == Proof == {{proofWan...")
• 22:51, 22 June 2022 ‎Conditional Entropy of Join as Sum/Corollary 2 (hist | edit) ‎[1,335 bytes] ‎Usagiop (talk | contribs) (Created page with "== Corollary 2 to Conditional Entropy of Join as Sum == Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space. Let $\AA, \CC, \DD \subseteq \Sigma$ be finite sub-$\sigma$-algebras. Then: <onlyinclude> :$\ds \AA \subseteq \CC \implies \map H {\AA \mid \DD} \le \map H {\CC \mid \DD} $ </onlyinclude> where: :$\map H {\cdot \mid \cdot}$ denotes the Definition:Conditional Entropy of Finite...")

(newest | oldest) View (newer 50 | older 50) (20 | 50 | 100 | 250 | 500)