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- 05:11, 27 January 2020 Union of Subclass is Subset of Union of Class (hist) [1,333 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $A$ and $B$ be classes. Let $\displaystyle \bigcup A$ and $\displaystyle \bigcup B$ denote the Definition:Union of Cl...")
- 04:08, 27 January 2020 Intersection of Empty Set/Class Theory (hist) [1,541 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $V$ be a basic universe. Let $\O$ denote the empty class. Then the Definition:Int...")
- 03:53, 27 January 2020 Intersection of Non-Empty Class is Set (hist) [1,658 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $V$ be a basic universe. Let $A \subseteq V$ be a non-empty Definition:Class (Class Theory)|...")
- 03:37, 27 January 2020 Intersection of Class Exists and is Unique (hist) [764 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $V$ be a basic universe. Let $A \subseteq V$ be a class. Let $\displaystyle \bigcap A$ d...")
- 05:59, 26 January 2020 Equality of Ordered Pairs/Necessary Condition/Proof from Wiener Formalization (hist) [2,234 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Equality of Ordered Pairs/Necessary Condition}} == Proof == <onlyinclude> First a lemma: {{:Equality of Ordered Pairs/Lem...")
- 05:22, 26 January 2020 Equality of Ordered Pairs/Necessary Condition/Proof from Empty Set Formalization (hist) [2,860 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Equality of Ordered Pairs/Necessary Condition}} == Proof == <onlyinclude> First a lemma: {{:Equality of Ordered Pairs/Lem...")
- 04:07, 26 January 2020 Equality of Ordered Pairs/Necessary Condition/Proof from Kuratowski Formalization (hist) [2,670 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Equality of Ordered Pairs/Necessary Condition}} == Proof == <onlyinclude> {{:Equality of Ordered Pairs/Necessary Condition/Proof from Wiener Formalization}}...")
- 02:57, 24 January 2020 Equality of Ordered Pairs/Sufficient Condition (hist) [1,101 bytes] Prime.mover (talk | contribs) (Created page with "{{refactor|Separating necessary and sufficient conditions into separate pages -- work in progress}} == Theorem == Let $\tuple {a, b}$ and $\tuple {c, d}$ be Definition:Orde...")
- 02:57, 24 January 2020 Equality of Ordered Pairs/Necessary Condition (hist) [2,187 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == <onlyinclude> Let $\tuple {a, b}$ and $\tuple {c, d}$ be ordered pairs such that $\tuple {a, b} = \tuple {c, d}$. Then $a = c$ and $...")
- 02:45, 24 January 2020 Equality of Ordered Pairs/Lemma (hist) [1,497 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $\set {a, b}$ and $\set {a, d}$ be doubletons such that $\set {a, b} = \set {a, d}$. Then: :$b = d$ == Proof == We have that: :...")
- 18:17, 23 January 2020 Equivalence of Definitions of Axiom of Pairing for Classes (hist) [1,974 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == The following formulations of the '''axiom of pairing''' in the context of '''class theory''...")
- 17:29, 23 January 2020 Basic Universe has Infinite Number of Elements (hist) [3,589 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $V$ be a basic universe. Then $V$ has an infinite number of Definition:Element of Class|element...")
- 03:13, 23 January 2020 Equivalence of Definitions of Axiom of Pairing (hist) [1,647 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == The following formulations of the '''axiom of pairing''' in the context of '''Definition:Axiomatic Set Theory|axiomatic set th...")
- 03:32, 22 January 2020 Doubleton Class of Equal Sets is Singleton Class (hist) [1,337 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $V$ be a basic universe. Let $a, b \in V$ be sets. Consider the Definition:Doubleton Class|doubleton cl...")
- 03:22, 22 January 2020 Doubleton Class can be Formed from Two Sets (hist) [1,101 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $V$ be a basic universe. Let $a, b \in V$ be sets. Then the Definition:Doubleton Class|doubleton class...")
- 01:20, 22 January 2020 Singleton Classes are Equal iff Sets are Equal (hist) [1,022 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $a$ and $b$ be sets. Let $\set a$ and $\set b$ denote the singleton classes of $a$ and $b$. Then: :$\se...")
- 18:39, 21 January 2020 Singleton Class of Empty Set is Supercomplete (hist) [1,251 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $\O$ denote the empty set. Then the singleton $\set \O$ is Definition:Supercomplete Class|superco...")
- 17:56, 21 January 2020 Singleton Class/Examples/Empty Set (hist) [793 bytes] Prime.mover (talk | contribs) (Created page with "== Example of Singleton Class == <onlyinclude> The singleton class $\O$ is the Definition:Class (Class Theory)|...")
- 17:42, 21 January 2020 Singleton Class/Examples (hist) [254 bytes] Prime.mover (talk | contribs) (Created page with "== Examples of Singleton Classes == <onlyinclude> === Singleton Class of Empty Set === {{:Singleton Class...")
- 16:25, 21 January 2020 Singleton Class can be Formed from Set (hist) [1,065 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $V$ be a basic universe. Let $a \in V$ be a set. Then the singleton class...")
- 12:56, 21 January 2020 Existence of Set is Equivalent to Existence of Empty Set (hist) [2,042 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $V$ be a basic universe. Let $P$ be the axiom: :$V$ has at least one Definition:Element of Class|ele...")
- 11:41, 21 January 2020 Basic Universe is not Empty (hist) [784 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $V$ be a basic universe Then $V$ is not the empty class. == Proof == The Axiom:Axiom of Empty...")
- 10:52, 21 January 2020 Empty Class is Supercomplete (hist) [1,073 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == The empty class is supercomplete. == Proof == Definition:Vacuous Truth|Vacuous...")
- 07:31, 21 January 2020 Empty Class is Subclass of All Classes (hist) [1,182 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == The empty class is a subclass of all classes. == Proof =...")
- 07:23, 21 January 2020 Empty Class is Unique (hist) [1,345 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == There is exactly one empty class. == Proof == Let $P$ be a property s...")
- 03:05, 21 January 2020 Basic Universe is not Set (hist) [1,236 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $V$ be a basic universe. Then $V$ is not a set. == Proof == {{AimForCont}} $V$ were a Definition:Set|...")
- 02:42, 21 January 2020 Basic Universe is Supercomplete (hist) [969 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $V$ be a basic universe. Then $V$ is supercomplete. == Proof == By definition, a Defi...")
- 18:33, 20 January 2020 Class has Subclass which is not Element (hist) [1,827 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $A$ be a class. Then $A$ has at least one subclass $B$ which is not an Definition:Element of...")
- 18:32, 20 January 2020 Not Every Class is a Set/Proof 2 (hist) [908 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Not Every Class is a Set}} == Proof == <onlyinclude> Consider the universal class $V$. From Class has Subclass which is not...")
- 18:28, 20 January 2020 Not Every Class is a Set/Proof 1 (hist) [1,743 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Not Every Class is a Set}} == Proof == <onlyinclude> Let a set $x$ be defined as '''ordinary''' {{iff}} $x \notin x$. Let $\map \phi x$ b...")
- 18:17, 20 January 2020 Not Every Class is a Set (hist) [423 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $A$ be a class. Then it is not necessarily the case that $A$ is also a set. == Proof == $A$ be a...")
- 08:44, 18 January 2020 Class is Subclass of Universal Class (hist) [1,068 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $V$ denote the universal class. Let $A$ be a class. Then $A$ is a Definition:Subclas...")
- 05:53, 16 January 2020 Exists Subset which is not Element/Proof 2 (hist) [870 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Exists Subset which is not Element}} == Proof == <onlyinclude> Consider the power set $\powerset S$ of $S$. From Cantor's Theorem...")
- 05:49, 16 January 2020 Exists Subset which is not Element/Proof 1 (hist) [845 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == {{:Exists Subset which is not Element}} == Proof == <onlyinclude> Let $S$ be a set. Let $T$ be the set of all Definitio...")
- 21:36, 15 January 2020 Exists Subset which is not Element (hist) [492 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $S$ be a set. Then there exists at least one subset of $S$ which is not an element of $S$....")
- 20:07, 15 January 2020 Russell's Paradox/Corollary (hist) [1,038 bytes] Prime.mover (talk | contribs) (Created page with "== Corollary to Russell's Paradox == <onlyinclude> :$\not \exists x: \forall y: \paren {\map \RR {x, y} \iff \neg \map \RR {y, y} }$ </onlyinclude> Given a Definition:R...")
- 07:50, 15 January 2020 Frege Set Theory is Logically Inconsistent (hist) [1,100 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == The system of axiomatic set theory that is Frege set theory is Definition:Inconsistent|inc...")
- 07:36, 15 January 2020 Set of Natural Numbers can be Derived using Comprehension Principle (hist) [765 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $\N$ denote the set of natural numbers. By application of the comprehension principle...")
- 07:33, 15 January 2020 Set Union can be Derived using Comprehension Principle (hist) [1,318 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $a$ be a set of sets. By application of the comprehension principle, the Definition:Set...")
- 07:28, 15 January 2020 Power Set can be Derived using Comprehension Principle (hist) [1,014 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $a$ be a set. By application of the comprehension principle, the Definition:Power Set|power set...")
- 07:23, 15 January 2020 Doubleton of Sets can be Derived using Comprehension Principle (hist) [1,015 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $a$ and $b$ be sets. By application of the comprehension principle, the set $\...")
- 07:07, 15 January 2020 Empty Set can be Derived from Comprehension Principle (hist) [1,115 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == The '''empty set''' can be formed by application of the comprehension principle. Hence the '''[...")
- 21:22, 14 January 2020 Axiom of Choice is Independent of ZF/Historical Note (hist) [741 bytes] Prime.mover (talk | contribs) (Created page with "== Historical Note on Axiom of Choice is Independent of ZF == <onlyinclude> {{AuthorRef|Kurt Friedrich Gödel|Kurt Gödel}} showed that the '''Axiom:Axiom of Choice|Axio...")
- 21:19, 14 January 2020 Axiom of Choice is Independent of ZF (hist) [687 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == The Axiom of Choice can be neither proved nor disproved from the axioms of ZF. == Proof == {...")
- 21:11, 14 January 2020 Continuum Hypothesis is Independent of ZFC/Historical Note (hist) [1,252 bytes] Prime.mover (talk | contribs) (Created page with "== Historical Note on Continuum Hypothesis is Independent of ZFC == <onlyinclude> Between $1938$ and $1940$ (accounts differ), {{AuthorRef|Kurt Friedrich Gödel|Kurt Göde...")
- 21:03, 14 January 2020 Continuum Hypothesis is Independent of ZFC (hist) [737 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == The Continuum Hypothesis can be neither proved nor disproved from the axioms of either ZF or ZFC....")
- 20:58, 14 January 2020 Continuum Hypothesis/Historical Note (hist) [1,992 bytes] Prime.mover (talk | contribs) (Created page with "== Historical Note on Continuum Hypothesis == <onlyinclude> The '''Continuum Hypothesis''' was originally conjectured by {{AuthorRef|Georg Cantor}}. In $1940$, {{Auth...")
- 20:58, 14 January 2020 Generalized Continuum Hypothesis/Historical Note (hist) [610 bytes] Prime.mover (talk | contribs) (Created page with "== Historical Note on Continuum Hypothesis == <onlyinclude> The '''Generalized Continuum Hypothesis''' was originally conjectured by {{AuthorRef|Georg Cantor}}. It is...")
- 20:56, 14 January 2020 Generalized Continuum Hypothesis (hist) [779 bytes] Prime.mover (talk | contribs) (Created page with "== Hypothesis == <onlyinclude> Let $S$ be an infinite set. There is no set whose cardinality is stri...")
- 20:40, 14 January 2020 Power Set of Natural Numbers is Cardinality of Continuum (hist) [2,217 bytes] Prime.mover (talk | contribs) (Created page with "== Theorem == Let $\N$ denote the set of natural numbers. Let $\powerset \N$ denote the power set o...")