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Combined display of all available logs of ProofWiki. You can narrow down the view by selecting a log type, the username (case-sensitive), or the affected page (also case-sensitive).
- 03:34, 11 December 2021 TheoLaLeo talk contribs created page Book:Elliott Mendelson/Introduction to Mathematical Logic/Sixth Edition (Created page with "{{Book|Introduction to Mathematical Logic|2015|Routledge|978-1482237726|Elliott Mendelson|ed = 6th}} === Subject Matter === * Mathematical Logic === Contents === ::Preface ::Introduction :1. The Propositional Calculus ::Propositional Connectives:Truth Tables ::Tautologies ::Adequate Sets of Connectives ::An Axiom System for the Propositional Calculus ::Independence: Many-Valued Logics ::Other Axiomatizations :2. First-Order Logic...")
- 04:50, 5 December 2021 TheoLaLeo talk contribs created page Powerset Not Subset of its Set/Proof 3 (Created page with "== Theorem == {{:Powerset Not Subset of its Set}} == Proof == <onlyinclude> {{AimForCont}} that $\powerset A \subseteq A$. Since $A \in \powerset A$, this implies: :$A \in A$ But this contradicts Set is Not Element of Itself. {{qed}} {{AoF}} </onlyinclude> Category:Power Set")
- 04:39, 5 December 2021 TheoLaLeo talk contribs created page Powerset Not Subset of its Set/Proof 2 (Created page with "== Theorem == {{:Powerset Not Subset of its Set}} == Proof == <onlyinclude> {{AimForCont}} that $\powerset A \subseteq A$. Let $I: \powerset A \to A$ be the identity mapping. $I$ is an injection by Identity Mapping is Injection. But by No Injection from Power Set to Set, this is a contradiction. {{qed}} </onlyinclude> Category:Power Set Category:Injection")
- 04:27, 5 December 2021 TheoLaLeo talk contribs created page Powerset Not Subset of its Set/Proof 1 (Created page with "== Theorem == {{:Powerset Not Subset of its Set}} == Proof == <onlyinclude> {{AimForCont}} that $\powerset A \subseteq A$, and define: :$C = \set {x \in \powerset A : x \notin x}$ We have that $C \subseteq \powerset A$, as it contains only the $x \in \powerset A$ meeting the condition $x \notin x$. Since $\powerset A \subseteq A$, we have: :$C \subseteq A$ and thus :$C \in \powerset A$ We can derive a similar contradiction to Russell's...")
- 17:47, 3 December 2021 TheoLaLeo talk contribs created page Definition talk:Inductive Set (Created page with "What's the difference between this and infinite successor set? They both use the same successor definition --~~~~")
- 07:22, 3 December 2021 TheoLaLeo talk contribs created page Powerset Not Subset of its Set (Created page with "== Theorem == Let $A$ be a set. $A \not \subseteq \powerset A$ == Proof == {{AimForCont}} that $A \subseteq \powerset A$ and define: :$C = \set { x \in \powerset A : x \notin x }$ We have that $C \subseteq \powerset A$. Since $\powerset A \subseteq A$, we have: :$C \subseteq A$ and thus :$C \in \powerset A$ We can derive a similar contradiction to Russell's Paradox. If $C \in C$, then it must meet $C$'s conditi...")
- 23:15, 2 December 2021 TheoLaLeo talk contribs created page Intersection of Class of Sets is Set (Created page with "== Theorem == The intersection of a non-empty class $\Bbb C$ is a set. == Proof == Since $\Bbb C$ is a non-empty class, there is an $S \in \Bbb C$. Since $S$ is an element, it is not a proper class, and is thus a set. By definition of Definition:Intersection of Class of S...")
- 22:57, 2 December 2021 TheoLaLeo talk contribs created page Definition:Intersection of Class of Sets (Redirected page to Definition:Class Intersection/Class of Sets) Tag: New redirect
- 22:51, 2 December 2021 TheoLaLeo talk contribs created page Intersection of Class of Sets (Redirected page to Definition:Class Intersection/Class of Sets) Tag: New redirect
- 22:47, 2 December 2021 TheoLaLeo talk contribs created page Definition:Class Intersection/Class of Sets (Created page with "== Definition == <onlyinclude> Let $\Bbb C$ be a class of sets. The '''intersection of $\Bbb C$''' is: :$\ds \bigcap \Bbb C := \set {x: \forall S \in \Bbb C: x \in S}$ That is, the class of all objects that are elements of all the elements of $\Bbb C$. Thus: :$\ds \bigcap \set {S, T} := S \cap T$ </onlyinclude>...")
- 22:29, 2 December 2021 TheoLaLeo talk contribs created page If Set Exists then Empty Set Exists (Created page with "== Theorem == If at least one set exists, then there exists an empty set. == Proof == {{NotZFC}} Let $s$ be a set. By the axiom of class comprehension, there is an empty class: :$\O = \set { x : x \ne x }$ Since $x \in \O$ is never true, it follows vacuously that: :$x \in \O \implies x \in S$ By the Definition...")
- 21:53, 2 December 2021 TheoLaLeo talk contribs created page Intersection of Class and Set is Set (Created page with "== Theorem == Let $C$ be the class: :$C = \set { u : \map \phi {u, p_1, \ldots, p_n} }$ Then for all sets $X$, $C \cap X$ is a set. == Proof == {{NotZFC}} By the definition of class intersection: :$a \in C \cap X \implies a \in C \land a \in X$ Thus: :$a \in C \cap X \implies a \in X$ The subclass definition gives: :$C \cap X \subse...")
- 06:52, 30 November 2021 TheoLaLeo talk contribs created page Free Variable/Examples/Series Example (Created page with "<onlyinclude> In the inequality: :$\displaystyle \sum_{n \mathop = 0}^\infty a z^n < z^2$ $a$ and $z$ are both '''free variables''', as the inequality may or may not hold depending on their values. </onlyinclude>")
- 06:34, 30 November 2021 TheoLaLeo talk contribs created page Free Variable/Examples/Cardinality Example (Created page with "<onlyinclude> In set theory: :$\card S = \aleph_0$ $\aleph_0$ is a '''free variable'''. </onlyinclude>")
- 06:23, 30 November 2021 TheoLaLeo talk contribs created page Free Variable/Examples/Derivative Example (Created page with "<onlyinclude> In differential calculus: :$\ds \lim_{h \mathop \to 0} \frac {\map f {x + h} - \map f x} h$ $x$ is a '''free variable'''. </onlyinclude>")
- 06:21, 30 November 2021 TheoLaLeo talk contribs created page Free Variable/Examples (Created page with "== Examples of Free Variables == <onlyinclude> === Calculus Example === {{:Free Variable/Examples/Derivative Example}} </onlyinclude> Category:Bound Variables")
- 06:19, 30 November 2021 TheoLaLeo talk contribs created page Creating Free Variable/Examples/Derivative Example (Created page with "<onlyinclude> In differential calculus: :$\ds \lim_{h \mathop \to 0} \frac {\map f {x + h} - \map f x} h$ $x$ is a '''free variable'''. </onlyinclude>")
- 22:00, 29 November 2021 TheoLaLeo talk contribs created page Definition:Free Variable/Predicate Logic (Created page with "== Definition == <onlyinclude> In predicate logic, a '''free variable''' is a variable which has no '''bound occurrences''' in a WFF. </onlyinclude> == Also known as == A '''free variable''' is often referred to as an '''unknown''', particularly in mathematical contexts. In the field of logic, a '''free variable''' can als...")
- 21:28, 29 November 2021 TheoLaLeo talk contribs created page Definition:Bound Variable (Predicate Logic) (Created page with "== Definition == <onlyinclude> In predicate logic, a '''bound variable''' is a variable which exists in a WFF only as '''bound occurrences'''. </onlyinclude> == Examples == {{:Bound Variable/Examples}} == Also known as == A '''bound variable''' is also popularly seen with the name '''dumm...")
- 19:50, 29 November 2021 TheoLaLeo talk contribs created page Equivalence of Definitions of Unique Existential Quantifier/Definition 1 iff Definition 3 (Created page with "== Theorem == {{TFAE|def = Uniqueness Quantifier}} === Definition 1 === {{:Definition:Existential Quantifier/Unique/Definition 1}} === Definition 3 === {{:Definition:Existential Quantifier/Unique/Definition 3}} == Proof == <onlyinclude> Suppose Definition 1, that for some $x$: :(1) : $\map P x$ an...")
- 19:36, 29 November 2021 TheoLaLeo talk contribs created page Equivalence of Definitions of Unique Existential Quantifier/Definition 2 iff Definition 3 (Created page with "== Theorem == {{TFAE|def = Uniqueness Quantifier}} === Definition 2 === {{:Definition:Existential Quantifier/Unique/Definition 2}} === Definition 3 === {{:Definition:Existential Quantifier/Unique/Definition 3}} == Proof == <onlyinclude> Suppose Definition 2, that for some $x$: :(1) : $\forall y :...")
- 19:19, 29 November 2021 TheoLaLeo talk contribs created page Equivalence of Definitions of Unique Existential Quantifier/Definition 1 iff Definition 2 (Created page with "== Theorem == {{TFAE|def = Uniqueness Quantifier}} === Definition 1 === {{:Definition:Existential Quantifier/Unique/Definition 1}} === Definition 2 === {{:Definition:Existential Quantifier/Unique/Definition 2}} == Proof == <onlyinclude> Suppose Definition 1, that for some $x$, both: :(1) : $\map P...")
- 19:03, 29 November 2021 TheoLaLeo talk contribs created page Equivalence of Definitions of Unique Existential Quantifier (Created page with "== Theorem == {{TFAE|def = Uniqueness Quantifier}} === Definition 1 === {{:Definition:Existential Quantifier/Unique/Definition 1}} === Definition 2 === {{:Definition:Existential Quantifier/Unique/Definition 2}} {{TFAE|def = Existential Quantifier/Unique}} === Definition 3 === {{:Definition:Existent...")
- 19:02, 29 November 2021 TheoLaLeo talk contribs created page Definition:Uniqueness Quantifier (Redirected page to Definition:Existential Quantifier/Unique) Tag: New redirect
- 18:50, 29 November 2021 TheoLaLeo talk contribs created page Definition:Existential Quantifier/Unique/Definition 1 (Created page with "== Definition == <onlyinclude> There exists a unique object $x$ such that $\map P x$, denoted $\exists ! x: \map P x$, {{iff}}: :$\exists x : \paren { \map P x \land \forall y : \paren { \map P y \implies x = y } }$ </onlyinclude> == Also denoted as == {{:Definition:Existential Quantifier/Unique/Also denoted as}}")
- 18:49, 29 November 2021 TheoLaLeo talk contribs created page Definition:Existential Quantifier/Unique/Definition 2 (Created page with "== Definition == <onlyinclude> There exists a unique object $x$ such that $\map P x$, denoted $\exists ! x: \map P x$, {{iff}}: :$\exists x : \forall y : \paren { \map P y \iff x = y }$ </onlyinclude> == Also denoted as == {{:Definition:Existential Quantifier/Unique/Also denoted as}}")
- 18:49, 29 November 2021 TheoLaLeo talk contribs created page Definition:Existential Quantifier/Unique/Definition 3 (Created page with "== Definition == <onlyinclude> There exists a unique object $x$ such that $\map P x$, denoted $\exists ! x: \map P x$, {{iff}} both: :$\exists x : \map P x$ and: :$\forall y : \forall z : \paren { \paren {\map P y \land \map P z } \implies y = z }$ </onlyinclude> == Also denoted as == {{:Definition:Existential Quantifier/Unique/Also denoted as}}")
- 22:35, 28 November 2021 TheoLaLeo talk contribs created page Axiom talk:Axiom of Specification/Class Theory (Created page with "Is the second $x \in B$ in :$\forall A_1, A_2, \ldots, A_n: \exists B: \forall x: \paren {x \in B \iff \paren {x \in B \land \phi {A_1, A_2, \ldots, A_n, x} } }$ purposeful? It's redundant and isn't in any formulations I've seen --~~~~")
- 07:15, 28 November 2021 TheoLaLeo talk contribs created page There Exists No Universal Set/Proof 4 (Created page with "== Theorem == {{:There Exists No Universal Set}} == Proof == <onlyinclude> {{AimForCont}} such a $\UU$ exists. Using the Axiom of Specification, we can create the set of all ordinals: :$\set {x \in \UU: x \text{ is an ordinal } }$ But from Burali-Forti Paradox, this set cannot exist, which is a contradiction. {{qed}} </onlyinclude> == Sources...")
- 05:14, 28 November 2021 TheoLaLeo talk contribs created page Goodman's Paradox (Created page with "== Paradox == All emeralds thus far observed have been green. By philosophical induction, we conclude that all emeralds are green. Let $t$ be some arbitrary time in the future. Define the predicate: :'$x$ is grue' to be true {{iff}} :'$x$ is green and was first observed before $t$' or :'$x$ is blue' All emeralds thus far observed have been grue, as they have been green and have been observed before $t$. By...")
- 05:13, 28 November 2021 TheoLaLeo talk contribs created page Book:Nelson Goodman/Fact, Fiction, and Forecast (Created page with "{{Book|Fact, Fiction, and Forecast|1983|Nelson Goodman|ed = Fourth}} === Subject Matter === * Philosophy === Contents === {{contents wanted}} Category:Books/Philosophy")
- 21:53, 27 November 2021 TheoLaLeo talk contribs created page Talk:Achilles Paradox (Created page with "equations (1) and (2) are a bit misaligned as the first is in a regular math environment and the second is in an align environment, not really sure how to fix that --~~~~")
- 09:07, 27 November 2021 TheoLaLeo talk contribs created page Negated Restricted Universal Quantifier (Created page with "== Theorem == Let $x$ and $A$ be sets. Let $\map P x$ be a propositional function. :$\neg \forall x \in A : \map P x \iff \exists x \in A : \neg \map P x $ == Proof == From left to right: {{begin-eqn}} {{eqn | q = \neg \forall x \in A | o = \map P x }} {{eqn | ll= \leadsto | q = \neg \forall x | o = x \in A \implies \map P x | c = {{defof|Restricted Universal Quantifier}} }} {{eqn | l...")
- 08:58, 27 November 2021 TheoLaLeo talk contribs created page Negated Restricted Existential Quantifier (Created page with "== Theorem == Let $x$ and $A$ be sets. Let $\map P x$ be a propositional function. :$\neg \exists x \in A : \map P x \iff \forall x \in A : \neg \map P x $ == Proof == From left to right: {{begin-eqn}} {{eqn | q = \neg \exists x \in A | o = \map P x }} {{eqn | ll= \leadsto | q = \neg \exists x | o = x \in A \land \map P x | c = {{defof|Restricted Existential Quantifier}} }} {{eqn | ll...")
- 07:31, 27 November 2021 TheoLaLeo talk contribs created page User:TheoLaLeo/Sandbox (Created page with " == Class Theory == === Bigger Axiom Page === Set axioms, quantifiers restricted to $V$ Axiom:Axiom of Extension/Set Theory Axiom:Axiom of Pairing/Set Theory Axiom:Axiom of Unions/Set Theory Axiom:Axiom of Powers/Set Theory Axiom:Axiom of Infinity/Set Theory Class axioms, quantifiers range over classes Axiom:Axiom of Extension/Class Theory Axiom:Axiom of Foundation (Classes) Axiom:Class Compr...")
- 12:59, 26 November 2021 TheoLaLeo talk contribs created page Grelling-Nelson Paradox (Created page with "== Paradox == Define an adjective to be ''autological'' if it is true when applied to itself. For instance, the word "English" is autological, as it is a word in English. The word "multisyllabic" is also autological, as it contains multiple syllables. Define an adjective to be ''heterological'' if it is not true when applied to itself. For instance, the word "long" is heterological, as it is not a long word. The word "monosy...")
- 23:11, 24 November 2021 TheoLaLeo talk contribs created page Axiom of Specification from Replacement and Empty Set (Created page with "== Theorem == The Axiom of Specification is a consequence of: :the Axiom of Replacement and :the Axiom of the Empty Set. == Proof == {{Proofread|proof is tedious}} Let $A$ be an arbitrary set. Let $\map P x$ be an arbitrary propositional function. It is to be shown that there exists a set $B$ consi...")
- 08:28, 24 November 2021 TheoLaLeo talk contribs created page Talk:Axiom of Pairing from Powers and Replacement (Created page with "I don't want to refactor to include another proof, but Power Set of Empty Set and Power Set of Singleton can show that $\{ \O \{ \O \} \}$ is a set much more quickly. --~~~~")
- 06:48, 24 November 2021 TheoLaLeo talk contribs created page Talk:Cardinality of Image of Injection (Created page with "This is a trivial corollary of Injection to Image is Bijection, it could be added as a corollary to that page. The induction proof for the finite case seems like overkill considering this. --~~~~")
- 01:15, 24 November 2021 TheoLaLeo talk contribs created page Image of Empty Set is Empty Set/Corollary 2 (Created page with "== Corollary of Image of Empty Set is Empty Set == <onlyinclude> Let $S = \O$ and $T \ne \O$. There is no surjection $f: S \to T$. </onlyinclude> == Proof == Let $f: S \to T$. By Image of Empty Set is Empty Set/Corollary 1: :$f \sqbrk S = \O$ Thus: :$f \sqbrk S \ne T$ and $f$ is not a surjection. {{qed}} Category:Mapping Theory Category:Empty Set")
- 01:12, 24 November 2021 TheoLaLeo talk contribs created page Empty Set is Empty Set/Corollary 2 (Created page with "== Corollary of Image of Empty Set is Empty Set == <onlyinclude> Let $S = \O$ and $T \ne \O$. There is no surjection $f: S \to T$. </onlyinclude> == Proof == Let $f: S \to T$. By Image of Empty Set is Empty Set/Corollary 1: :$f \sqbrk S = \O$ Thus: :$f \sqbrk S \ne T$ and $f$ is not a surjection. {{qed}} Category:Mapping Theory Category:Empty Set")
- 00:53, 24 November 2021 TheoLaLeo talk contribs created page Image of Empty Set is Empty Set/Corollary 1 (Created page with "== Corollary of Image of Empty Set is Empty Set == <onlyinclude> Let $f: S \to T$ be a mapping. The image of the empty set is the empty set: :$f \sqbrk \O = \O$ </onlyinclude> == Proof == By definition, a mapping is a relation. Thus Image of Empty Set is Empty Set applies. {{qed}} == Sources ==...")
- 21:37, 23 November 2021 TheoLaLeo talk contribs created page Definition talk:Gödel-Bernays Axioms (Created page with "Is there a source giving this presentation of NBG? Nothing is cited. --~~~~")
- 20:31, 23 November 2021 TheoLaLeo talk contribs created page Russell's Paradox/Proof 1 (Created page with "== Theorem == {{:Russell's Paradox}} == Proof == <onlyinclude> Sets have elements. Some of those elements may themselves be sets. So, given two sets $S$ and $T$, we can ask the question: Is $S$ an element of $T$? The answer will either be ''yes'' or ''no''. In particular, given any set $S$, we can ask the question: Is $S$ an element of $S$? Again, the answer will either be ''yes'' or ''no''. Thus, $\map P S = S \in S$ is a [...")
- 20:26, 23 November 2021 TheoLaLeo talk contribs created page Russell's Paradox/Proof 2 (Created page with "== Theorem == {{:Russell's Paradox}} == Proof == <onlyinclude> {{AimForCont}} the comprehension principle, that for all predicates $P$ where $S$ is not free: :$\exists S : \forall x : \paren {x \in S \iff P(x)}$ Since $x \notin x$ is a predicate $S$ is not free, it follows that: :$\exists S : \forall x : \paren {x \in S \iff x \n...")
- 09:56, 23 November 2021 TheoLaLeo talk contribs created page Russell's Paradox/Corollary/Proof 2 (Created page with "==Theorem== {{:Russell's Paradox/Corollary}} ==Proof== <onlyinclude> {{AimForCont}}: :$\exists x: \forall y: \paren {\map \RR {x, y} \iff \neg \map \RR {y, y} }$ By Existential Instantiation: :$\forall y: \paren {\map \RR {x, y} \iff \neg \map \RR {y, y} }$ By Universal Instantiation: :$\map \RR {x, x} \iff \neg \map \RR {x, x} $ But this contradicts Biconditional of Negated Propositions. We thus conclude: :$\not \exists x: \forall y: \paren {\map \RR {x,...")
- 09:49, 23 November 2021 TheoLaLeo talk contribs created page Russell's Paradox/Corollary/Proof 1 (Created page with "==Theorem== {{:Russell's Paradox/Corollary}} ==Proof== <onlyinclude> {{AimForCont}} there does exist such an $x$. Let $\RR$ be such that $\map \RR {x, x}$. Then $\neg \map \RR {x, x}$. Hence it cannot be the case that $\map \RR {x, x}$. Now suppose that $\neg \map \RR {x, x}$. Then by definition of $x$ it follows that $\map \RR {x, x}$. In both cases a contradiction results. Hence there can be no such $x$. {{qed}} </onlyinclude> Cate...")
- 09:45, 23 November 2021 TheoLaLeo talk contribs created page Biconditional of Negated Propositions (Created page with "== Theorem == A biconditional of a proposition and its negation: <onlyinclude> :$\vdash \neg (p \iff \neg p)$ </onlyinclude> == Proof by Truth Table == {{:Biconditional of Negated Propositions/Proof 1}} == Sources == Category:Biconditional")
- 09:44, 23 November 2021 TheoLaLeo talk contribs created page Biconditional of Negated Propositions/Proof 1 (Created page with "==Theorem== {{:Biconditional of Negated Propositions}} ==Proof== <onlyinclude> We apply the Method of Truth Tables to the proposition. As can be seen by inspection, in each case, the truth values in the appropriate columns match for all boolean interpretations. $\begin{array}{c|cccc|} \hline \neg & (p & \iff & \neg & p)\\ \hline \T & \F & \F & \T & \F \\ \T & \T & \F & \F & \T \\ \hline \end{array}$ {...") Tag: Visual edit: Switched
- 05:57, 23 November 2021 TheoLaLeo talk contribs created page Curry's Paradox (Created page with " ==Paradox== Let $P$ be an arbitrary proposition. Consider the following argument: <blockquote>This argument is valid. Therefore, $P$.</blockquote> Assume for contradiction that the argument is invalid. Then by the definition of argument validity, its premise must be true and its conclusion false. But if its premise is true then the argument is valid, contradicting our assumption. Thus the argument must be valid. This validates the arg...") Tag: Visual edit: Switched