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Von Neumann–Bernays–Gödel set theory

In the foundations of mathematics, von Neumann–Bernays–Gödel set theory (NBG) is an axiomatic set theory that is a conservative extension of Zermelo–Fraenkel–choice set theory (ZFC). NBG introduces the notion of class, which is a collection of sets defined by a formula whose quantifiers range only over sets. NBG can define classes that are larger than sets, such as the class of all sets and the class of all ordinals. Morse–Kelley set theory (MK) allows classes to be defined by formulas whose quantifiers range over classes. NBG is finitely axiomatizable, while ZFC and MK are not.

A key theorem of NBG is the class existence theorem, which states that for every formula whose quantifiers range only over sets, there is a class consisting of the sets satisfying the formula. This class is built by mirroring the step-by-step construction of the formula with classes. Since all set-theoretic formulas are constructed from two kinds of atomic formulas (membership and equality) and finitely many logical symbols, only finitely many axioms are needed to build the classes satisfying them. This is why NBG is finitely axiomatizable. Classes are also used for other constructions, for handling the set-theoretic paradoxes, and for stating the axiom of global choice, which is stronger than ZFC's axiom of choice.

John von Neumann introduced classes into set theory in 1925. The primitive notions of his theory were function and argument. Using these notions, he defined class and set. Paul Bernays reformulated von Neumann's theory by taking class and set as primitive notions. Kurt Gödel simplified Bernays' theory for his relative consistency proof of the axiom of choice and the generalized continuum hypothesis.

The uses of classes
Classes have several uses in NBG:
 * They produce a finite axiomatization of set theory.
 * They are used to state a "very strong form of the axiom of choice" —namely, the axiom of global choice: There exists a global choice function $$G$$ defined on the class of all nonempty sets such that $$G(x) \in x$$ for every nonempty set $$x.$$ This is stronger than ZFC's axiom of choice: For every set $$s$$ of nonempty sets, there exists a choice function $$f$$ defined on $$s$$ such that $$f(x) \in x$$ for all $$x \in s.$$
 * The set-theoretic paradoxes are handled by recognizing that some classes cannot be sets. For example, assume that the class $$Ord$$ of all ordinals is a set. Then $$Ord$$ is a transitive set well-ordered by $$\in$$. So, by definition, $$Ord$$ is an ordinal. Hence, $$Ord \in Ord$$, which contradicts $$\in$$ being a well-ordering of $$Ord.$$ Therefore, $$Ord$$ is not a set. A class that is not a set is called a proper class, $$Ord$$ is a proper class.
 * Proper classes are useful in constructions. In his proof of the relative consistency of the axiom of global choice and the generalized continuum hypothesis, Gödel used proper classes to build the constructible universe. He constructed a function on the class of all ordinals that, for each ordinal, builds a constructible set by applying a set-building operation to previously constructed sets. The constructible universe is the image of this function.

Axiom schema versus class existence theorem
Once classes are added to the language of ZFC, it is easy to transform ZFC into a set theory with classes. First, the axiom schema of class comprehension is added. This axiom schema states: For every formula $$\phi(x_1, \ldots, x_n)$$ that quantifies only over sets, there exists a class $$A$$ consisting of the $n$-tuples satisfying the formula—that is, $$\forall x_1 \cdots \,\forall x_n [(x_1, \ldots, x_n) \in A \iff \phi(x_1, \ldots, x_n)].$$ Then the axiom schema of replacement is replaced by a single axiom that uses a class. Finally, ZFC's axiom of extensionality is modified to handle classes: If two classes have the same elements, then they are identical. The other axioms of ZFC are not modified.

This theory is not finitely axiomatized. ZFC's replacement schema has been replaced by a single axiom, but the axiom schema of class comprehension has been introduced.

To produce a theory with finitely many axioms, the axiom schema of class comprehension is first replaced with finitely many class existence axioms. Then these axioms are used to prove the class existence theorem, which implies every instance of the axiom schema. The proof of this theorem requires only seven class existence axioms, which are used to convert the construction of a formula into the construction of a class satisfying the formula.

Classes and sets
NBG has two types of objects: classes and sets. Intuitively, every set is also a class. There are two ways to axiomatize this. Bernays used many-sorted logic with two sorts: classes and sets. Gödel avoided sorts by introducing primitive predicates: $$\mathfrak{Cls}(A)$$ for "$$A$$ is a class" and $$\mathfrak{M}(A)$$ for "$$A$$ is a set" (in German, "set" is Menge). He also introduced axioms stating that every set is a class and that if class $$A$$ is a member of a class, then $$A$$ is a set. Using predicates is the standard way to eliminate sorts. Elliott Mendelson modified Gödel's approach by having everything be a class and defining the set predicate $$M(A)$$ as $$\exists C(A \in C).$$ This modification eliminates Gödel's class predicate and his two axioms.

Bernays' two-sorted approach may appear more natural at first, but it creates a more complex theory. In Bernays' theory, every set has two representations: one as a set and the other as a class. Also, there are two membership relations: the first, denoted by "∈", is between two sets; the second, denoted by "η", is between a set and a class. This redundancy is required by many-sorted logic because variables of different sorts range over disjoint subdomains of the domain of discourse.

The differences between these two approaches do not affect what can be proved, but they do affect how statements are written. In Gödel's approach, $$A \in C$$ where $$A$$ and $$C$$ are classes is a valid statement. In Bernays' approach this statement has no meaning. However, if $$A$$ is a set, there is an equivalent statement: Define "set $$a$$ represents class $$A$$" if they have the same sets as members—that is, $$\forall x(x \in a \iff x \;\eta \;A).$$ The statement $$a \;\eta \;C$$ where set $$a$$ represents class $$A$$ is equivalent to Gödel's $$A \in C.$$

The approach adopted in this article is that of Gödel with Mendelson's modification. This means that NBG is an axiomatic system in first-order predicate logic with equality, and its only primitive notions are class and the membership relation.

Definitions and axioms of extensionality and pairing
A set is a class that belongs to at least one class: $$A$$ is a set if and only if $$\exist C(A \in C)$$. A class that is not a set is called a proper class: $$A$$ is a proper class if and only if $$\forall C(A \notin C)$$. Therefore, every class is either a set or a proper class, and no class is both.

Gödel introduced the convention that uppercase variables range over classes, while lowercase variables range over sets. Gödel also used names that begin with an uppercase letter to denote particular classes, including functions and relations defined on the class of all sets. Gödel's convention is used in this article. It allows us to write: indent=1
 * $$\exist x\, \phi(x)$$ instead of $$\exist x \bigl(\exist C (x \in C) \land \phi(x)\bigr)$$
 * $$\forall x\, \phi(x)$$ instead of $$\forall x \bigl(\exist C (x \in C) \implies \phi(x)\bigr)$$

The following axioms and definitions are needed for the proof of the class existence theorem.

Axiom of extensionality. If two classes have the same elements, then they are identical.
 * $$\forall A \,\forall B \,[\forall x(x \in A \iff x \in B) \implies A = B]$$

This axiom generalizes ZFC's axiom of extensionality to classes.

Axiom of pairing. If $$x$$ and $$y$$ are sets, then there exists a set $$p$$ whose only members are $$x$$ and $$y$$.
 * $$\forall x \,\forall y \,\exists p \,\forall z \,[z \in p \iff (z = x \,\lor\, z = y)]$$

As in ZFC, the axiom of extensionality implies the uniqueness of the set $$p$$, which allows us to introduce the notation $$\{x, y\}.$$

Ordered pairs are defined by:
 * $$(x, y) = \{\{x\}, \{x, y\}\}$$

Tuples are defined inductively using ordered pairs:
 * $$(x_1) = x_1,$$
 * $$\text{For } n > 1\!: (x_1, \ldots, x_{n-1}, x_n) = ((x_1, \ldots, x_{n-1}), x_n).$$

Class existence axioms and axiom of regularity
Class existence axioms will be used to prove the class existence theorem: For every formula in $$n$$ free set variables that quantifies only over sets, there exists a class of $n$-tuples that satisfy it. The following example starts with two classes that are functions and builds a composite function. This example illustrates the techniques that are needed to prove the class existence theorem, which lead to the class existence axioms that are needed.

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 * Example 1: If the classes $$F$$ and $$G$$ are functions, then the composite function $$G \circ F$$ is defined by the formula: $$\exists t[(x, t) \in F \,\land\, (t, y) \in G].$$ Since this formula has two free set variables, $$x$$ and $$y,$$ the class existence theorem constructs the class of ordered pairs: $$G \circ F\,=\,\{(x, y): \exists t[(x, t) \in F \,\land\, (t, y) \in G]\}.$$

Because this formula is built from simpler formulas using conjunction $$\land$$ and existential quantification $$\exists$$, class operations are needed that take classes representing the simpler formulas and produce classes representing the formulas with $$\land$$ and $$\exists$$. To produce a class representing a formula with $$\land$$, intersection used since $$x \in A \cap B \iff x \in A \land x \in B.$$ To produce a class representing a formula with $$\exists$$, the domain is used since $$x \in Dom(A) \iff \exists t[(x,t) \in A].$$

Before taking the intersection, the tuples in $$F$$ and $$G$$ need an extra component so they have the same variables. The component $$y$$ is added to the tuples of $$F$$ and $$x$$ is added to the tuples of $$G$$: $$F' = \{(x, t, y): (x, t) \in F\}\,$$ and $$\,G' = \{(t, y, x): (t, y) \in G\}$$ In the definition of $$F',$$ the variable $$y$$ is not restricted by the statement $$(x, t) \in F,$$ so $$y$$ ranges over the class $$V$$ of all sets. Similarly, in the definition of $$G',$$ the variable $$x$$ ranges over $$V.$$ So an axiom is needed that adds an extra component (whose values range over $$V$$) to the tuples of a given class.

Next, the variables are put in the same order to prepare for the intersection: $$F = \{(x, y, t): (x, t) \in F\}\,$$ and $$\,G = \{(x, y, t): (t, y) \in G\}$$ To go from $$F'$$ to $$F$$ and from $$G'$$ to $$G$$ requires two different permutations, so axioms that support permutations of tuple components are needed.

The intersection of $$F$$ and $$G$$ handles $$\land$$: $$F \cap G = \{(x, y, t): (x, t) \in F \,\land\, (t, y) \in G\}$$

Since $$(x, y, t)$$ is defined as $$((x, y), t)$$, taking the domain of $$F \cap G$$ handles $$\exists t$$ and produces the composite function: $$G \circ F = Dom(F \cap G) = \{(x, y): \exists t ((x, t) \in F \,\land\, (t, y) \in G)\}$$ So axioms of intersection and domain are needed.
 * }

The class existence axioms are divided into two groups: axioms handling language primitives and axioms handling tuples. There are four axioms in the first group and three axioms in the second group.

Axioms for handling language primitives:

Membership. There exists a class $$E$$ containing all the ordered pairs whose first component is a member of the second component.
 * $$\exists E \,\forall x \,\forall y \,[(x,y) \in E \iff x \in y]\!$$

Intersection (conjunction). For any two classes $$A$$ and $$B$$, there is a class $$C$$ consisting precisely of the sets that belong to both $$A$$ and $$B$$.
 * $$\forall A \,\forall B \,\exists C \,\forall x \,[x \in C \iff (x \in A \,\land\, x \in B)]$$

Complement (negation). For any class $$A$$, there is a class $$B$$ consisting precisely of the sets not belonging to $$A$$.
 * $$\forall A \,\exists B \,\forall x \,[x \in B \iff \neg(x \in A)]$$

Domain (existential quantifier). For any class $$A$$, there is a class $$B$$ consisting precisely of the first components of the ordered pairs of $$A$$.
 * $$\forall A \,\exists B \,\forall x \,[x \in B \iff \exists y((x,y) \in A)]$$

By the axiom of extensionality, class $$C$$ in the intersection axiom and class $$B$$ in the complement and domain axioms are unique. They will be denoted by: $$A \cap B,$$ $$\complement A,$$ and $$Dom(A),$$ respectively.

The first three axioms imply the existence of the empty class and the class of all sets: The membership axiom implies the existence of a class $$E.$$ The intersection and complement axioms imply the existence of $$E \cap \complement E$$, which is empty. By the axiom of extensionality, this class is unique; it is denoted by $$\empty.$$ The complement of $$\empty$$ is the class $$V$$ of all sets, which is also unique by extensionality. The set predicate $$M(A)$$, which was defined as $$\exists C(A \in C)$$, is now redefined as $$A \in V$$ to avoid quantifying over classes.

Axioms for handling tuples:

Product by $$V$$. For any class $$A$$, there is a class $$B$$ consisting of the ordered pairs whose first component belongs to $$A$$.
 * $$\forall A \,\exists B \,\forall u \,[u \in B \iff \exists x \, \exists y \,(u = (x, y) \land x \in A)]$$

Circular permutation. For any class $$A$$, there is a class $$B$$ whose 3tuples are obtained by applying the circular permutation $$(y,z,x) \mapsto (x,y,z)$$ to the 3tuples of $$A$$.
 * $$\forall A \,\exists B \,\forall x \,\forall y \,\forall z \,[(x,y,z) \in B \iff (y,z,x) \in A]$$

Transposition. For any class $$A$$, there is a class $$B$$ whose 3tuples are obtained by transposing the last two components of the 3tuples of $$A$$.
 * $$\forall A \,\exists B \,\forall x \,\forall y \,\forall z \,[(x,y,z) \in B \iff (x,z,y) \in A]$$

By extensionality, the product by $$V$$ axiom implies the existence of a unique class, which is denoted by $$A \times V.$$ This axiom is used to define the class $$V^n$$ of all $n$-tuples: $$V^1 = V$$ and $$V^{n+1} = V^n \times V.\,$$ If $$A$$ is a class, extensionality implies that $$A \cap V^n$$ is the unique class consisting of the $n$-tuples of $$A.$$ For example, the membership axiom produces a class $$E$$ that may contain elements that are not ordered pairs, while the intersection $$E \cap V^2$$ contains only the ordered pairs of $$E$$.

The circular permutation and transposition axioms do not imply the existence of unique classes because they specify only the 3tuples of class $$B.$$ By specifying the 3tuples, these axioms also specify the $n$-tuples for $$n \ge 4$$ since: $$(x_1, \ldots, x_{n-2}, x_{n-1}, x_n) = ((x_1, \ldots, x_{n-2}), x_{n-1}, x_n).$$ The axioms for handling tuples and the domain axiom imply the following lemma, which is used in the proof of the class existence theorem.

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One more axiom is needed to prove the class existence theorem: the axiom of regularity. Since the existence of the empty class has been proved, the usual statement of this axiom is given.

Axiom of regularity. Every nonempty set has at least one element with which it has no element in common. $$\forall a\,[a \neq \empty \implies \exists u(u \in a \land u \cap a = \empty)].$$

This axiom implies that a set cannot belong to itself: Assume that $$x \in x$$ and let $$a = \{x\}.$$ Then $$x \cap a \ne \empty$$ since $$x \in x \cap a.$$ This contradicts the axiom of regularity because $$x$$ is the only element in $$a.$$ Therefore, $$x \notin x.$$ The axiom of regularity also prohibits infinite descending membership sequences of sets: $$\cdots \in x_{n+1} \in x_n \in \cdots \in x_1 \in x_0.$$

Gödel stated regularity for classes rather than for sets in his 1940 monograph, which was based on lectures given in 1938. In 1939, he proved that regularity for sets implies regularity for classes.

Class existence theorem
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The theorem's proof will be done in two steps:
 * 1) Transformation rules are used to transform the given formula $$\phi$$ into an equivalent formula that simplifies the inductive part of the proof. For example, the only logical symbols in the transformed formula are $$\neg$$, $$\land$$, and $$\exists$$, so the induction handles logical symbols with just three cases.
 * 2) The class existence theorem is proved inductively for transformed formulas. Guided by the structure of the transformed formula, the class existence axioms are used to produce the unique class of $n$-tuples satisfying the formula.

Transformation rules. In rules 1 and 2 below, $$\Delta$$ and $$\Gamma$$ denote set or class variables. These two rules eliminate all occurrences of class variables before an $$\in$$ and all occurrences of equality. Each time rule 1 or 2 is applied to a subformula, $$i$$ is chosen so that $$z_i$$ differs from the other variables in the current formula. The three rules are repeated until there are no subformulas to which they can be applied. This produces a formula that is built only with $$\neg$$, $$\land$$, $$\exists$$, $$\in$$, set variables, and class variables $$Y_k$$ where $$Y_k$$ does not appear before an $$\in$$.

 * 1) $$\,Y_k \in \Gamma$$ is transformed into $$\exists z_i(z_i = Y_k \,\land\, z_i \in \Gamma).$$
 * 2) Extensionality is used to transform $$\Delta = \Gamma$$ into $$\forall z_i(z_i \in \Delta \iff z_i \in \Gamma).$$
 * 3) Logical identities are used to transform subformulas containing $$\lor, \implies, \iff,$$ and $$\forall$$ to subformulas that only use $$\neg, \land,$$ and $$\exists.$$

Transformation rules: bound variables. Consider the composite function formula of example 1 with its free set variables replaced by $$x_1$$ and $$x_2$$: $$\exists t[(x_1, t) \in F \,\land\, (t, x_2) \in G].$$ The inductive proof will remove $$\exists t$$, which produces the formula $$(x_1, t) \in F \land (t, x_2) \in G.$$ However, since the class existence theorem is stated for subscripted variables, this formula does not have the form expected by the induction hypothesis. This problem is solved by replacing the variable $$t$$ with $$x_3.$$ Bound variables within nested quantifiers are handled by increasing the subscript by one for each successive quantifier. This leads to rule 4, which must be applied after the other rules since rules 1 and 2 produce quantified variables.

 * 1) If a formula contains no free set variables other than $$x_1, \dots, x_n,$$ then bound variables that are nested within $$q$$ quantifiers are replaced with $x_{n+q}$. These variables have (quantifier) nesting depth $$q$$.

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 * Example 2: Rule 4 is applied to the formula $$\phi(x_1)$$ that defines the class consisting of all sets of the form $$\{\empty, \{\empty, \dots\}, \dots\}.$$ That is, sets that contain at least $$\empty$$ and a set containing $$\empty$$ — for example, $$\{\empty, \{\empty, a, b, c\}, d, e\}$$ where $$a, b, c, d,$$ and $$e$$ are sets.

$$\begin{align} \phi(x_1) \,&=\, \exists u\;\,[\,u \in x_1 \,\land\, \neg\exists v\;\,(\;v\, \in \,u\,)] \,\land\, \,\exists w\;\bigl(w \in x_1 \,\land\, \exists y\;\,[(\;y\, \in w \;\land\; \neg\exists z\;\,(\;z\, \in \,y\,)]\bigr) \\ \phi_r(x_1) \, &=\, \exists x_2[x_2 \!\in\! x_1 \,\land\, \neg\exists x_3(x_3 \!\in\! x_2)] \,\land\, \,\exists x_2\bigl(x_2 \!\in\! x_1 \,\land\, \exists x_3[(x_3 \!\in\! x_2 \,\land\, \neg\exists x_4(x_4 \!\in\! x_3)]\bigr) \end{align}$$

Since $$x_1$$ is the only free variable, $$n = 1.$$ The quantified variable $$x_3$$ appears twice in $$x_3 \in x_2$$ at nesting depth 2. Its subscript is 3 because $$n+q = 1+2 = 3.$$ If two quantifier scopes are at the same nesting depth, they are either identical or disjoint. The two occurrences of $$x_3$$ are in disjoint quantifier scopes, so they do not interact with each other.
 * }

Proof of the class existence theorem. The proof starts by applying the transformation rules to the given formula to produce a transformed formula. Since this formula is equivalent to the given formula, the proof is completed by proving the class existence theorem for transformed formulas.

Gödel pointed out that the class existence theorem "is a metatheorem, that is, a theorem about the system [NBG], not in the system …" It is a theorem about NBG because it is proved in the metatheory by induction on NBG formulas. Also, its proof—instead of invoking finitely many NBG axioms—inductively describes how to use NBG axioms to construct a class satisfying a given formula. For every formula, this description can be turned into a constructive existence proof that is in NBG. Therefore, this metatheorem can generate the NBG proofs that replace uses of NBG's class existence theorem.

A recursive computer program succinctly captures the construction of a class from a given formula. The definition of this program does not depend on the proof of the class existence theorem. However, the proof is needed to prove that the class constructed by the program satisfies the given formula and is built using the axioms. This program is written in pseudocode that uses a Pascal-style case statement. $$\begin{array}{l} \mathbf{function} \;\text{Class}(\phi, \,n) \\ \quad\begin{array}{rl} \mathbf{input}\!: \;\,&\phi \text{ is a transformed formula of the form } \phi(x_1, \ldots, x_n, Y_1, \ldots, Y_m); \\ &n \text{ specifies that a class of } n\text{-tuples is returned.} \\ \;\;\;\;\mathbf{output}\!: \;\,&\text{class } A \text{ of } n\text{-tuples satisfying } \\ &\,\forall x_1 \cdots \,\forall x_n [(x_1, \ldots, x_n) \in A \iff \phi(x_1, \ldots, x_n, Y_1, \ldots, Y_m)]. \end{array} \\ \mathbf{begin} \\ \quad \mathbf{case} \;\phi \;\mathbf{of} \\ \qquad \begin{alignat}{2} x_i \in x_j: \;\;&\mathbf{return} \;\,E_{i,j,n}; &&\text{// } E_{i,j,n} \;\,= \{(x_1, \dots, x_n): x_i \in x_j\} \\ x_i \in Y_k: \;\;&\mathbf{return} \;\,E_{i,Y_k,n}; &&\text{// } E_{i,Y_k,n} = \{(x_1, \dots, x_{n}): x_i \in Y_k\} \\ \neg\psi: \;\;&\mathbf{return} \;\,\complement_{V^n}\text{Class}(\psi, \,n); &&\text{// } \complement_{V^n}\text{Class}(\psi, \,n) = V^n \setminus \text{Class}(\psi, \,n) \\ \psi_1 \land \psi_2: \;\;&\mathbf{return} \;\,\text{Class}(\psi_1, \,n) \cap \text{Class}(\psi_2, \,n);&& \\ \;\;\;\;\,\exists x_{n+1}(\psi): \;\;&\mathbf{return} \;\,Dom(\text{Class}(\psi, \,n+1)); &&\text{// } x_{n+1} \text{ is free in } \psi; \text{ Class}(\psi, \,n+1) \\ &\ &&\text{// returns a class of } (n+1)\text{-tuples} \end{alignat} \\ \quad \mathbf{end} \\ \mathbf{end} \end{array}$$

Let $$\phi$$ be the formula of example 2. The function call $$A = Class(\phi, 1)$$ generates the class $$A,$$ which is compared below with $$\phi.$$ This shows that the construction of the class $$A$$ mirrors the construction of its defining formula $$\phi.$$

$$\begin{alignat}{2} &\phi \;&&= \;\;\exists x_2\,(x_2 \!\in\! x_1 \land \;\;\neg\;\;\;\;\exists x_3\;(x_3 \!\in\! x_2)) \,\land \;\;\,\exists x_2\,(x_2 \!\in\! x_1 \land \;\;\,\exists x_3\,(x_3 \!\in\! x_2 \,\land\;\;\neg\;\;\;\;\exists x_4\;(x_4 \!\in\! x_3))) \\ &A \;&&= Dom\,(\;E_{2,1,2}\; \cap \;\complement_{V^2}\,Dom\,(\;E_{3,2,3}\;)) \,\cap\, Dom\,(\;E_{2,1,2}\;\cap \, Dom\,(\;\,E_{3,2,3}\; \cap \;\complement_{V^3}\,Dom\,(\;E_{4,3,4}\;))) \end{alignat}$$

Extending the class existence theorem
Gödel extended the class existence theorem to formulas $$\phi$$ containing relations over classes (such as $$Y_1 \subseteq Y_2$$ and the unary relation $$M(Y_1)$$), special classes (such as $$Ord$$), and operations (such as $$(x_1, x_2)$$ and $$x_1 \cap Y_1$$). To extend the class existence theorem, the formulas defining relations, special classes, and operations must quantify only over sets. Then $$\phi$$ can be transformed into an equivalent formula satisfying the hypothesis of the class existence theorem.

The following definitions specify how formulas define relations, special classes, and operations:
 * 1) A relation $$R$$ is defined by: $$R(Z_1, \dots, Z_k) \iff \psi_R(Z_1, \dots, Z_k).$$
 * 2) A special class $$C$$ is defined by: $$u \in C \iff \psi_C(u).$$
 * 3) An operation $$P$$ is defined by: $$u \in P(Z_1, \dots, Z_k) \iff \psi_P(u, Z_1, \dots, Z_k).$$

A is defined by:
 * 1) Variables and special classes are terms.
 * 2) If $$P$$ is an operation with $$k$$ arguments and $$\Gamma_1, \dots, \Gamma_k$$ are terms, then $$P(\Gamma_1, \dots, \Gamma_k)$$ is a term.

The following transformation rules eliminate relations, special classes, and operations. Each time rule 2b, 3b, or 4 is applied to a subformula, $$i$$ is chosen so that $$z_i$$ differs from the other variables in the current formula. The rules are repeated until there are no subformulas to which they can be applied. $$\, \Gamma_1, \dots, \Gamma_k, \Gamma,$$ and $$\Delta$$ denote terms.

 * 1) A relation $$R(Z_1, \dots, Z_k)$$ is replaced by its defining formula $$\psi_R(Z_1, \dots, Z_k).$$
 * 2) Let $$\psi_C(u)$$ be the defining formula for the special class $$C.$$ 1. $\Delta \in C$ is replaced by $\psi_C(\Delta).$

2. $C \in \Delta$ is replaced by $\exists z_i[z_i = C \land z_i \in \Delta].$
 * 1) Let $$\psi_P(u, Z_1, \dots, Z_k)$$ be the defining formula for the operation $$P(Z_1, \dots, Z_k).$$ 1. $\Delta \in P(\Gamma_1, \dots, \Gamma_k)$ is replaced by $\psi_P(\Delta, \Gamma_1, \dots, \Gamma_k).$

2. $P(\Gamma_1, \dots, \Gamma_k) \in \Delta$ is replaced by $\exists z_i[z_i = P(\Gamma_1, \dots, \Gamma_k) \land z_i \in \Delta].$
 * 1) Extensionality is used to transform $$\Delta = \Gamma$$ into $$\forall z_i(z_i \in \Delta \iff z_i \in \Gamma).$$

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$$Y_1 \subseteq Y_2 \iff \forall z_1(z_1 \in Y_1 \implies z_1 \in Y_2) \quad\text{(rule 1)}$$
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 * Example 3: Transforming $$Y_1 \subseteq Y_2.$$
 * }

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$$\begin{alignat}{2} x_1 \cap Y_1 \in x_2 &\iff \exists z_1[z_1 = x_1 \cap Y_1 \,\land\, z_1 \in x_2] &&\text{(rule 3b)} \\ &\iff \exists z_1[\forall z_2(z_2 \in z_1 \iff z_2 \in x_1 \cap Y_1) \,\land\, z_1 \in x_2] &&\text{(rule 4)} \\ &\iff \exists z_1[\forall z_2(z_2 \in z_1 \iff z_2 \in x_1 \land z_2 \in Y_1) \,\land\, z_1 \in x_2] \quad&&\text{(rule 3a)} \\ \end{alignat}$$
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 * Example 4: Transforming $$x_1 \cap Y_1 \in x_2.$$

This example illustrates how the transformation rules work together to eliminate an operation.
 * }

$$

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Set axioms
The axioms of pairing and regularity, which were needed for the proof of the class existence theorem, have been given above. NBG contains four other set axioms. Three of these axioms deal with class operations being applied to sets.

Definition. $$F$$ is a function if $$F \subseteq V^2 \land \forall x\, \forall y\, \forall z\, [(x,y) \in F \,\land\, (x,z) \in F \implies y = z].$$

In set theory, the definition of a function does not require specifying the domain or codomain of the function (see Function (set theory)). NBG's definition of function generalizes ZFC's definition from a set of ordered pairs to a class of ordered pairs.

ZFC's definitions of the set operations of image, union, and power set are also generalized to class operations. The image of class $$A$$ under the function $$F$$ is $$F[A] = \{y: \exists x(x \in A \,\land\, (x, y) \in F)\}.$$ This definition does not require that $$A \subseteq Dom(F).$$ The union of class $$A$$ is $$\cup A = \{x: \exists y(x \in y\, \,\land\, y \in A)\}.$$ The power class of $$A$$ is $$\mathcal{P}(A) = \{x: x \subseteq A\}.$$ The extended version of the class existence theorem implies the existence of these classes. The axioms of replacement, union, and power set imply that when these operations are applied to sets, they produce sets.

Axiom of replacement. If $$F$$ is a function and $$a$$ is a set, then $$F[a]$$, the image of $$a$$ under $$F$$, is a set. $$\forall F \,\forall a \,[F \text{ is a function}\implies \exists b \,\forall y\,(y \in b \iff \exists x(x \in a \,\land\, (x, y) \in F))].$$

Not having the requirement $$A \subseteq Dom(F)$$ in the definition of $$F[A]$$ produces a stronger axiom of replacement, which is used in the following proof.

$$ $$

Axiom of union. If $$a$$ is a set, then there is a set containing $$\cup a.$$ $$\forall a\, \exists b\, \forall x\,[\,\exists y(x \in y\, \,\land\, y \in a) \implies x \in b\,].$$

Axiom of power set. If $$a$$ is a set, then there is a set containing $$\mathcal{P}(a).$$
 * $$\forall a\, \exists b\, \forall x\, (x \subseteq a \implies x \in b).$$

$$ $$

Axiom of infinity. There exists a nonempty set $$a$$ such that for all $$x$$ in $$a$$, there exists a $$y$$ in $$a$$ such that $$x$$ is a proper subset of $$y$$. $$\exists a\, [\exists u(u \in a) \,\land\, \forall x(x \in a \implies \exists y(y \in a \,\land\, x \subset y))].$$

The axioms of infinity and replacement prove the existence of the empty set. In the discussion of the class existence axioms, the existence of the empty class $$\empty$$ was proved. We now prove that $$\empty$$ is a set. Let function $$F = \empty$$ and let $$a$$ be the set given by the axiom of infinity. By replacement, the image of $$a$$ under $$F$$, which equals $$\empty$$, is a set.

NBG's axiom of infinity is implied by ZFC's axiom of infinity: $$\,\exists a\, [\empty \in a \,\land\, \forall x(x \in a \implies x \cup \{x\} \in a)].\,$$ The first conjunct of ZFC's axiom, $$\empty \in a$$, implies the first conjunct of NBG's axiom. The second conjunct of ZFC's axiom, $$\forall x(x \in a \implies x \cup \{x\} \in a)$$, implies the second conjunct of NBG's axiom since $$x \subset x \cup \{x\}.$$ To prove ZFC's axiom of infinity from NBG's axiom of infinity requires some of the other NBG axioms (see Weak axiom of infinity).

Axiom of global choice
The class concept allows NBG to have a stronger axiom of choice than ZFC. A choice function is a function $$f$$ defined on a set $$s$$ of nonempty sets such that $$f(x) \in x$$ for all $$x \in s.$$ ZFC's axiom of choice states that there exists a choice function for every set of nonempty sets. A global choice function is a function $$G$$ defined on the class of all nonempty sets such that $$G(x) \in x$$ for every nonempty set $$x.$$ The axiom of global choice states that there exists a global choice function. This axiom implies ZFC's axiom of choice since for every set $$s$$ of nonempty sets, $$G\vert_s$$ (the restriction of $$G$$ to $$s$$) is a choice function for $$s.$$ In 1964, William B. Easton proved that global choice is stronger than the axiom of choice by using forcing to construct a model that satisfies the axiom of choice and all the axioms of NBG except the axiom of global choice. The axiom of global choice is equivalent to every class having a well-ordering, while ZFC's axiom of choice is equivalent to every set having a well-ordering.

Axiom of global choice. There exists a function that chooses an element from every nonempty set.
 * $$\exists G\,[G \text{ is a function}\, \land \forall x(x \ne \empty \implies \exists y(y \in x \land (x,y) \in G))].$$

Von Neumann's 1925 axiom system
Von Neumann published an introductory article on his axiom system in 1925. In 1928, he provided a detailed treatment of his system. Von Neumann based his axiom system on two domains of primitive objects: functions and arguments. These domains overlap—objects that are in both domains are called argument-functions. Functions correspond to classes in NBG, and argument-functions correspond to sets. Von Neumann's primitive operation is function application, denoted by [a, x] rather than a(x) where a is a function and x is an argument. This operation produces an argument. Von Neumann defined classes and sets using functions and argument-functions that take only two values, A and B. He defined x ∈ a if [a, x] ≠ A.

Von Neumann's work in set theory was influenced by Georg Cantor's articles, Ernst Zermelo's 1908 axioms for set theory, and the 1922 critiques of Zermelo's set theory that were given independently by Abraham Fraenkel and Thoralf Skolem. Both Fraenkel and Skolem pointed out that Zermelo's axioms cannot prove the existence of the set {Z0, Z1, Z2, ...} where Z0 is the set of natural numbers and Zn+1 is the power set of Zn. They then introduced the axiom of replacement, which would guarantee the existence of such sets. However, they were reluctant to adopt this axiom: Fraenkel stated "that Replacement was too strong an axiom for 'general set theory'", while "Skolem only wrote that 'we could introduce' Replacement".

Von Neumann worked on the problems of Zermelo set theory and provided solutions for some of them:
 * A theory of ordinals
 * Problem: Cantor's theory of ordinal numbers cannot be developed in Zermelo set theory because it lacks the axiom of replacement.
 * Solution: Von Neumann recovered Cantor's theory by defining the ordinals using sets that are well-ordered by the ∈-relation, and by using the axiom of replacement to prove key theorems about the ordinals, such as every well-ordered set is order-isomorphic with an ordinal. In contrast to Fraenkel and Skolem, von Neumann emphasized how important the replacement axiom is for set theory: "In fact, I believe that no theory of ordinals is possible at all without this axiom."
 * A criterion identifying classes that are too large to be sets
 * Problem: Zermelo did not provide such a criterion. His set theory avoids the large classes that lead to the paradoxes, but it leaves out many sets, such as the one mentioned by Fraenkel and Skolem.
 * Solution: Von Neumann introduced the criterion: A class is too large to be a set if and only if it can be mapped onto the class V of all sets. Von Neumann realized that the set-theoretic paradoxes could be avoided by not allowing such large classes to be members of any class. Combining this restriction with his criterion, he obtained his axiom of limitation of size: A class C is not a member of any class if and only if C can be mapped onto V.
 * Finite axiomatization
 * Problem: Zermelo had used the imprecise concept of "definite propositional function" in his axiom of separation.
 * Solutions: Skolem introduced the axiom schema of separation that was later used in ZFC, and Fraenkel introduced an equivalent solution. However, Zermelo rejected both approaches "particularly because they implicitly involve the concept of natural number which, in Zermelo's view, should be based upon set theory." Von Neumann avoided axiom schemas by formalizing the concept of "definite propositional function" with his functions, whose construction requires only finitely many axioms. This led to his set theory having finitely many axioms. In 1961, Richard Montague proved that ZFC cannot be finitely axiomatized.
 * The axiom of regularity
 * Problem: Zermelo set theory starts with the empty set and an infinite set, and iterates the axioms of pairing, union, power set, separation, and choice to generate new sets. However, it does not restrict sets to these. For example, it allows sets that are not well-founded, such as a set x satisfying x ∈ x.
 * Solutions: Fraenkel introduced an axiom to exclude these sets. Von Neumann analyzed Fraenkel's axiom and stated that it was not "precisely formulated", but it would approximately say: "Besides the sets ... whose existence is absolutely required by the axioms, there are no further sets." Von Neumann proposed the axiom of regularity as a way to exclude non-well-founded sets, but did not include it in his axiom system. In 1930, Zermelo became the first to publish an axiom system that included regularity.

Von Neumann's 1929 axiom system

In 1929, von Neumann published an article containing the axioms that would lead to NBG. This article was motivated by his concern about the consistency of the axiom of limitation of size. He stated that this axiom "does a lot, actually too much." Besides implying the axioms of separation and replacement, and the well-ordering theorem, it also implies that any class whose cardinality is less than that of V is a set. Von Neumann thought that this last implication went beyond Cantorian set theory and concluded: "We must therefore discuss whether its [the axiom's] consistency is not even more problematic than an axiomatization of set theory that does not go beyond the necessary Cantorian framework."

Von Neumann started his consistency investigation by introducing his 1929 axiom system, which contains all the axioms of his 1925 axiom system except the axiom of limitation of size. He replaced this axiom with two of its consequences, the axiom of replacement and a choice axiom. Von Neumann's choice axiom states: "Every relation R has a subclass that is a function with the same domain as R."

Let S be von Neumann's 1929 axiom system. Von Neumann introduced the axiom system S + Regularity (which consists of S and the axiom of regularity) to demonstrate that his 1925 system is consistent relative to S. He proved: These results imply: If S is consistent, then von Neumann's 1925 axiom system is consistent. Proof: If S is consistent, then S + Regularity is consistent (result 1). Using proof by contradiction, assume that the 1925 axiom system is inconsistent, or equivalently: the 1925 axiom system implies a contradiction. Since S + Regularity implies the axioms of the 1925 system (result 2), S + Regularity also implies a contradiction. However, this contradicts the consistency of S + Regularity. Therefore, if S is consistent, then von Neumann's 1925 axiom system is consistent.
 * 1) If S is consistent, then S + Regularity is consistent.
 * 2) S + Regularity implies the axiom of limitation of size. Since this is the only axiom of his 1925 axiom system that S + Regularity does not have, S + Regularity implies all the axioms of his 1925 system.

Since S is his 1929 axiom system, von Neumann's 1925 axiom system is consistent relative to his 1929 axiom system, which is closer to Cantorian set theory. The major differences between Cantorian set theory and the 1929 axiom system are classes and von Neumann's choice axiom. The axiom system S + Regularity was modified by Bernays and Gödel to produce the equivalent NBG axiom system.

Bernays' axiom system
In 1929, Paul Bernays started modifying von Neumann's new axiom system by taking classes and sets as primitives. He published his work in a series of articles appearing from 1937 to 1954. Bernays stated that: "The purpose of modifying the von Neumann system is to remain nearer to the structure of the original Zermelo system and to utilize at the same time some of the set-theoretic concepts of the Schröder logic and of Principia Mathematica which have become familiar to logicians. As will be seen, a considerable simplification results from this arrangement."

Bernays handled sets and classes in a two-sorted logic and introduced two membership primitives: one for membership in sets and one for membership in classes. With these primitives, he rewrote and simplified von Neumann's 1929 axioms. Bernays also included the axiom of regularity in his axiom system.

Gödel's axiom system (NBG)
In 1931, Bernays sent a letter containing his set theory to Kurt Gödel. Gödel simplified Bernays' theory by making every set a class, which allowed him to use just one sort and one membership primitive. He also weakened some of Bernays' axioms and replaced von Neumann's choice axiom with the equivalent axiom of global choice. Gödel used his axioms in his 1940 monograph on the relative consistency of global choice and the generalized continuum hypothesis.

Several reasons have been given for Gödel choosing NBG for his monograph:
 * Gödel gave a mathematical reason—NBG's global choice produces a stronger consistency theorem: "This stronger form of the axiom [of choice], if consistent with the other axioms, implies, of course, that a weaker form is also consistent."
 * Robert Solovay conjectured: "My guess is that he [Gödel] wished to avoid a discussion of the technicalities involved in developing the rudiments of model theory within axiomatic set theory."
 * Kenneth Kunen gave a reason for Gödel avoiding this discussion: "There is also a much more combinatorial approach to L [the constructible universe], developed by ... [Gödel in his 1940 monograph] in an attempt to explain his work to non-logicians. ... This approach has the merit of removing all vestiges of logic from the treatment of L."
 * Charles Parsons provided a philosophical reason for Gödel's choice: "This view [that 'property of set' is a primitive of set theory] may be reflected in Gödel's choice of a theory with class variables as the framework for ... [his monograph]."

Gödel's achievement together with the details of his presentation led to the prominence that NBG would enjoy for the next two decades. In 1963, Paul Cohen proved his independence proofs for ZF with the help of some tools that Gödel had developed for his relative consistency proofs for NBG. Later, ZFC became more popular than NBG. This was caused by several factors, including the extra work required to handle forcing in NBG, Cohen's 1966 presentation of forcing, which used ZF, and the proof that NBG is a conservative extension of ZFC.

NBG, ZFC, and MK
NBG is not logically equivalent to ZFC because its language is more expressive: it can make statements about classes, which cannot be made in ZFC. However, NBG and ZFC imply the same statements about sets. Therefore, NBG is a conservative extension of ZFC. NBG implies theorems that ZFC does not imply, but since NBG is a conservative extension, these theorems must involve proper classes. For example, it is a theorem of NBG that the global axiom of choice implies that the proper class V can be well-ordered and that every proper class can be put into one-to-one correspondence with V.

One consequence of conservative extension is that ZFC and NBG are equiconsistent. Proving this uses the principle of explosion: from a contradiction, everything is provable. Assume that either ZFC or NBG is inconsistent. Then the inconsistent theory implies the contradictory statements ∅ = ∅ and ∅ ≠ ∅, which are statements about sets. By the conservative extension property, the other theory also implies these statements. Therefore, it is also inconsistent. So although NBG is more expressive, it is equiconsistent with ZFC. This result together with von Neumann's 1929 relative consistency proof implies that his 1925 axiom system with the axiom of limitation of size is equiconsistent with ZFC. This completely resolves von Neumann's concern about the relative consistency of this powerful axiom since ZFC is within the Cantorian framework.

Even though NBG is a conservative extension of ZFC, a theorem may have a shorter and more elegant proof in NBG than in ZFC (or vice versa). For a survey of known results of this nature, see.

Morse–Kelley set theory has an axiom schema of class comprehension that includes formulas whose quantifiers range over classes. MK is a stronger theory than NBG because MK proves the consistency of NBG, while Gödel's second incompleteness theorem implies that NBG cannot prove the consistency of NBG.

For a discussion of some ontological and other philosophical issues posed by NBG, especially when contrasted with ZFC and MK, see Appendix C of.

Models
ZFC, NBG, and MK have models describable in terms of the cumulative hierarchy Vα and the constructible hierarchy Lα. Let V include an inaccessible cardinal κ, let X ⊆ Vκ, and let Def(X) denote the class of first-order definable subsets of X with parameters. In symbols where "$$(X,\in)$$" denotes the model with domain $$X$$ and relation $$\in$$, and "$$\models$$" denotes the satisfaction relation: $$\operatorname{Def}(X) := \Bigl\{ \{ x \mid x \in X \text{ and } (X,\in) \models \phi(x,y_1,\ldots,y_n) \}: \phi \text{ is a first-order formula and } y_{1},\ldots,y_{n} \in X \Bigr\}.$$

Then:
 * $$(V_\kappa,\in)$$ and $$(L_\kappa,\in)$$ are models of ZFC.
 * (Vκ, Vκ+1, ∈) is a model of MK where Vκ consists of the sets of the model and Vκ+1 consists of the classes of the model. Since a model of MK is a model of NBG, this model is also a model of NBG.
 * (Vκ, Def(Vκ), ∈) is a model of Mendelson's version of NBG, which replaces NBG's axiom of global choice with ZFC's axiom of choice. The axioms of ZFC are true in this model because (Vκ, ∈) is a model of ZFC. In particular, ZFC's axiom of choice holds, but NBG's global choice may fail. NBG's class existence axioms are true in this model because the classes whose existence they assert can be defined by first-order definitions. For example, the membership axiom holds since the class $$E$$ is defined by: $$E = \{x \in V_\kappa: (V_\kappa,\in) \models \exists u \ \exists v [x = (u, v) \land u \in v]\}.$$
 * (Lκ, Lκ+, ∈), where κ+ is the successor cardinal of κ, is a model of NBG. NBG's class existence axioms are true in (Lκ, Lκ+, ∈). For example, the membership axiom holds since the class $$E$$ is defined by: $$E = \{x \in L_\kappa: (L_\kappa,\in) \models \exists u \ \exists v [x = (u, v) \land u \in v]\}.$$ So E ∈ 𝒫(Lκ). In his proof that GCH is true in L, Gödel proved that 𝒫(Lκ) ⊆ Lκ+. Therefore, E ∈ Lκ+, so the membership axiom is true in (Lκ, Lκ+, ∈). Likewise, the other class existence axioms are true. The axiom of global choice is true because Lκ is well-ordered by the restriction of Gödel's function (which maps the class of ordinals to the constructible sets) to the ordinals less than κ. Therefore, (Lκ, Lκ+, ∈) is a model of NBG.
 * If $$M$$ is a nonstandard model of $$\mathrm{ZFC}$$, then $$(M,\mathrm{Def}(M))\vDash \mathrm{GB}+\Delta^1_1\text{-CA}$$ is equivalent to "there exists an $$X$$ such that $$(M,X)\vDash\mathrm{GB}+\Delta^1_1\text{-CA}$$", where $$\mathrm{Def}(M)$$ is the set of subsets of $$M$$ that are definable over $$M$$. This provides a second-order part for extending a given first-order nonstandard model of $$\mathrm{ZFC}$$ to a nonstandard model of $$\mathrm{GB}$$, if there is such an extension at all.

Category theory
The ontology of NBG provides scaffolding for speaking about "large objects" without risking paradox. For instance, in some developments of category theory, a "large category" is defined as one whose objects and morphisms make up a proper class. On the other hand, a "small category" is one whose objects and morphisms are members of a set. Thus, we can speak of the "category of all sets" or "category of all small categories" without risking paradox since NBG supports large categories.

However, NBG does not support a "category of all categories" since large categories would be members of it and NBG does not allow proper classes to be members of anything. An ontological extension that enables us to talk formally about such a "category" is the conglomerate, which is a collection of classes. Then the "category of all categories" is defined by its objects: the conglomerate of all categories; and its morphisms: the conglomerate of all morphisms from A to B where A and B are objects. On whether an ontology including classes as well as sets is adequate for category theory, see.

title: Von Neumann–Bernays–Gödel set theory


text:
Mandala of the Two Realms

The Mandala of the Two Realms (Traditional Chinese: 両界曼荼羅; Pinyin: Liǎngjiè màntúluó; Rōmaji: Ryōkai mandara), also known as the Mandala of the Two Divisions (Traditional Chinese: 両部曼荼羅; Pinyin: Liǎngbù màntúluó; Rōmaji: Ryōbu mandara), is a set of two mandalas depicting both the Five Wisdom Buddhas of the Diamond Realm as well as the Five Wisdom Kings of the Womb Realm. The number of deities arranged around the cores varies, but may range as high as 414.

The Diamond Realm represents the unchanging cosmic principle of the Buddha, while the Womb Realm depicts the active, physical manifestation of Buddha in the natural world. The mandalas are thus considered a compact expression of the entirety of the Dharma in Mahayana Buddhism, and form the root of the Vajrayana teachings. Chinese Tangmi and Japanese Shingon and Tendai temples, in particular, often prominently display the Mandalas of the Two Realms mounted at right angles to the image platform on the central altar.

The two mandalas are believed to have evolved separately in India, and were joined for the first time in China, perhaps by Kūkai's teacher Hui-kuo.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Mandala of the Two Realms


text:
Rambling Rose

Rambling Rose may refer to:

Roses

 * Garden roses, rambling types of the rose flowering plant
 * Rosa multiflora, sometimes known as rambler rose

Music

 * "Rambling Rose" (1948 song), by Joe Burke, lyrics by Joseph McCarthy, recorded by Perry Como and others
 * "Ramblin' Rose", a 1962 song written by Noel and Joe Sherman, popularized by Nat King Cole
 * Ramblin' Rose (album) by Nat King Cole, 1962
 * "Ramblin' Rose", a song by Marijohn Wilkin, Fred Burch, and Obey Wilson, recorded by Jerry Lee Lewis and others

Other uses

 * Ramblin' Rose, a fishing vessel featured on Deadliest Catch documentary
 * Rambling Rose (novel), a 1972 novel by Calder Willingham
 * Rambling Rose (film), a 1991 drama film based on the novel

title: Rambling Rose


text:
François-Auguste Gevaert

François-Auguste Gevaert (31 July 1828 – 24 December 1908) was a Belgian musicologist and composer.

Life
Gevaert was born in Huise, near Oudenaarde. His father was a baker, and he was intended for the same profession, but better counsels prevailed and he was permitted to study music. He was sent in 1841 to the Ghent Conservatory, where he studied under Édouard de Sommere and Martin-Joseph Mengal. Then he was appointed organist of the Jesuit church in that city.

Soon Gevaert's compositions attracted attention, and he won the Belgian Prix de Rome which entitled him to two years' travel. The journey was postponed during the production of his first opera and other works. He finally embarked on it in 1849. After a short stay in Paris he went to Spain, and subsequently to Italy.

In 1867 Gevaert, having returned to Paris, became Chef de Chant at the Academie de Musique there, in succession to the popular operatic composer Fromental Halévy. Four years later, he was appointed head of the Brussels Conservatoire. In that role, he "exerted a far-reaching influence through his historical concerts, producing works of all nations and periods." Gevaert died in Brussels.

Though during his lifetime Gevaert's own music enjoyed considerable success in Belgium (it included no fewer than a dozen operas, including Quentin Durward and Le Capitaine Henriot), it is now forgotten, save for some of his choral pieces, which have recently been issued on CD by the Fuga Libera label. Nowadays he is mostly remembered, even in his native land, less as a composer than as a teacher, historian, and lecturer. His many prose writings include a Treatise on Instrumentation (still sometimes used today), a book on harmony, and a Vade Mecum for organists. Notable students of Gevaert included Alfred Wotquenne, who is best known for having provided the first thorough listing of C. P. E. Bach's compositions, whilst Gevaert's daughter Jacqueline Marthe married the singer and art historian Hippolyte Fierens-Gevaert.

Honours

 * 1881: Grand Officier in the Order of Leopold.
 * 1896: Grand Cordon in the Order of Leopold.

Selected works

 * Te Deum (1843)
 * Ouverture Flandre au lion (1848)
 * Fantasia sobre motivos españoles (1850)
 * Requiem (1853)
 * "Vers l'avenir" (1905)
 * Grand' Messe de Noël Puer Natus est Nobis (1907)
 * Quartet for clarinet, horn, bassoon and piano

Operas
 * Georgette, ou le moulin de Fontenoy (1853)
 * Le billet de Marguerite (1854)
 * Les lavandières de Santarem (1855)
 * Quentin Durward (1858, libretto by Michel Carré and Eugène Cormon after Walter Scott
 * Le diable au moulin (1859)
 * La Château Trompette (1860)
 * Le Capitaine Henriot (1864)

Secular cantatas
 * België (1847)
 * Le roi Léar (1847)
 * Évocation patriotique (1856)
 * De nationale verjaerdag (1857)
 * Le retour de l'armée (1859)
 * Jacob van Artevelde (1864) (about Jacob van Artevelde)

title: François-Auguste Gevaert


text:
Mag

Mag, MAG, Mags or mags may refer to:

Arts, entertainment and media

 * MAG (video game), released in 2010
 * Mág (film), a 1988 Czech film
 * Mag (Slovenian magazine), published from 1995 to 2010
 * The Mag, a British music magazine

Businesses and organisations

 * MacKenzie Art Gallery, Regina, Saskatchewan, Canada
 * Manchester Airport Holdings, trading as MAG, a British holding company
 * Maricopa Association of Governments, the regional agency for the greater Maricopa region in Arizona, U.S.
 * Mines Advisory Group, a non-governmental organization
 * MAG motorcycle engines, manufactured by Motosacoche
 * Hungarian General Machine Factory (Magyar Általános Gépgyár, MÁG), a former automobile and aircraft manufacturer

Military

 * a United States Marine Corps aviation group
 * FN MAG, a machine gun
 * MAG-7, a shotgun

People

 * Maumoon Abdul Gayoom (born 1937) President of the Maldives from 1978 to 2008
 * Mag Bodard (1916–2019), Italian-born French film producer
 * Mags Darcy (born 1987), Irish camogie player
 * Mags McCarthy (born 1989 or 1990), Irish country music singer
 * Mags Murray (1961–2020), Irish politician
 * Mags Portman (1974–2019), British medical doctor
 * Gallus Mag, nickname of a notorious 19th century female bouncer in New York City
 * Mags, stage name of Magne Furuholmen (born 1962), Norwegian musician and visual artist

Places

 * Mag, a village near Săliște, Sibiu County, Romania
 * Mag (river)
 * Madagascar, UNDP country code MAG

Science and technology

 * Magnesium alloy wheels, commonly called "mags"
 * Magnetometer, sometimes abbreviated "mag"
 * Magnitude (astronomy), abbreviated "mag" as a measure of brightness of an object in the sky
 * Male accessory gland, in human biology
 * Maximum Absorbency Garment, a diaper-like garment worn by astronauts
 * Metal active gas welding
 * Myelin-associated glycoprotein

Other uses

 * Mount Albert Grammar School (MAGS), Auckland, New Zealand
 * Magnology, or The MAG, a community convention in Germany
 * Hag and Mag, demons in Mandaeism
 * Men's artistic gymnastics
 * Madang Airport, Papua New Guinea, IATA airport code MAG
 * Mag., an abbreviation of Magister
 * MAG mini truck, a captive import utility truck better known as the Hafei Ruiyi

title: Mag


text:
Theatre of ancient Greece

A theatrical culture flourished in ancient Greece from 700 BC. At its centre was the city-state of Athens, which became a significant cultural, political, and religious place during this period, and the theatre was institutionalised there as part of a festival called the Dionysia, which honoured the god Dionysus. Tragedy (late 500 BC), comedy (490 BC), and the satyr play were the three dramatic genres emerged there. Athens exported the festival to its numerous colonies. Modern Western theatre comes, in large measure, from the theatre of ancient Greece, from which it borrows technical terminology, classification into genres, and many of its themes, stock characters, and plot elements.

Etymology
The word τραγῳδία, from which the word "tragedy" is derived, is a compound of two Greek words: τράγος or "goat" and ᾠδή meaning "song", from ἀείδειν.

This etymology indicates a link with the practices of the ancient Dionysian cults. It is impossible, however, to know with certainty how these fertility rituals became the basis for tragedy and comedy.

Origins

The Ancient Greeks valued the power of the spoken word, and it was their main method of communication and storytelling. Bahn and Bahn write, "To Greeks, the spoken word was a living thing and infinitely preferable to the dead symbols of a written language." Socrates himself believed that once something has been written down, it lost its ability for change and growth. For these reasons, among many others, oral storytelling flourished in Greece.

Greek tragedy, as it is presently known, was created in Athens around 532 BC, when Thespis was the earliest recorded actor. Being a winner of the first theatrical contest held in Athens, he was the, or leader, of the dithyrambs performed in and around Attica, especially at the Rural Dionysia. By Thespis' time, the dithyramb had evolved far away from its cult roots. Under the influence of heroic epic, Doric choral lyric and the innovations of the poet Arion, it had become a narrative, ballad-like genre. Because of these, Thespis is often called the "Inventor of Tragedy"; however, his importance is disputed, and Thespis is sometimes listed as late as 16th in the chronological order of Greek tragedians; the statesman Solon, for example, is credited with creating poems in which characters speak with their own voice, and spoken performances of Homer's epics by rhapsodes were popular in festivals prior to 534 BC. Thus, Thespis's true contribution to drama is unclear at best, but his name has been given a longer life in English as a common term for performer—i.e., a "thespian."

The dramatic performances were important to the Athenians – this is made clear by the creation of a tragedy competition and festival in the City Dionysia (or Great Dionysia). This was organized possibly to foster loyalty among the tribes of Attica (recently created by Cleisthenes). The festival was created roughly around 508 BC. While no drama texts exist from the sixth century BC, the names of three competitors besides Thespis are known: Choerilus, Pratinas, and Phrynichus. Each is credited with different innovations in the field.

Some information is known about Phrynichus. He won his first competition between 511 BC and 508 BC. He produced tragedies on themes and subjects later exploited in the Golden Age such as the Danaids, Phoenician Women and Alcestis. He was the first poet we know of to use a historical subject – his Fall of Miletus, produced in 493–2, chronicled the fate of the town of Miletus after it was conquered by the Persians. Herodotus reports that "the Athenians made clear their deep grief for the taking of Miletus in many ways, but especially in this: when Phrynichus wrote a play entitled The Fall of Miletus and produced it, the whole theatre fell to weeping; they fined Phrynichus a thousand drachmas for bringing to mind a calamity that affected them so personally and forbade the performance of that play forever." He is also thought to be the first to use female characters (though not female performers).

Until the Hellenistic period, all tragedies were unique pieces written in honour of Dionysus and played only once; what is primarily extant today are the pieces that were still remembered well enough to have been repeated when the repetition of old tragedies became fashionable (the accidents of survival, as well as the subjective tastes of the Hellenistic librarians later in Greek history, also played a role in what survived from this period).

New inventions during the classical period
After the Achaemenid destruction of Athens in 480 BC, the town and acropolis were rebuilt, and theatre became formalized and an even greater part of Athenian culture and civic pride. This century is normally regarded as the Golden Age of Greek drama. The center-piece of the annual Dionysia, which took place once in winter and once in spring, was a competition between three tragic playwrights at the Theatre of Dionysus. Each submitted three tragedies, plus a satyr play (a comic, burlesque version of a mythological subject). Beginning in a first competition in 486 BC each playwright submitted a comedy. Aristotle claimed that Aeschylus added the second actor (deuteragonist), and that Sophocles introduced the third (tritagonist). Apparently, the Greek playwrights never used more than three actors based on what is known about Greek theatre.

Tragedy and comedy were viewed as completely separate genres, and no plays ever merged aspects of the two. Satyr plays dealt with the mythological subject matter of the tragedies, but in a purely comedic manner.

Hellenistic period

The power of Athens declined following its defeat in the Peloponnesian War against Sparta. From that time on, the theatre started performing old tragedies again. Although its theatrical traditions seem to have lost their vitality, Greek theatre continued into the Hellenistic period (the period following Alexander the Great's conquests in the fourth century BC).

The primary Hellenistic theatrical form was not tragedy but New Comedy, comic episodes about the lives of ordinary citizens. The only extant playwright from the period is Menander. One of New Comedy's most important contributions was its influence on Roman comedy, an influence that can be seen in the surviving works of Plautus and Terence.

Architecture
Most ancient Greek cities lay on or near hills, so seating was generally built into the slope of a hill, producing a natural viewing area known as the (literally "seeing place"). In cities without suitable hills, banks of earth were piled up. At the foot of the hill was a flattened, generally circular performance space with an average diameter of 78 ft, known as the (literally "dancing place"), where a chorus of typically 12 to 15 people performed plays in verse accompanied by music. There were often tall, arched entrances called or, through which actors and chorus members entered and exited the orchestra. In some theatres, behind the orchestra, was a backdrop or scenic wall known as the.

The term eventually came to mean the whole area of, , and.

Theatron
The was the seating area, built into a hill to create a natural viewing space. The first seats in Greek theatres (other than just sitting on the ground) were wooden, but around 499 BC, the practice of inlaying stone blocks into the side of the hill to create permanent, stable seating became more common. They were called the and reserved for priests and a few of the most respected citizens. The separated the upper and lower seating areas.

After 465 BC, playwrights began using a backdrop or scenic wall, called the (from which the word scene derives), that hung or stood behind the orchestra and also served as an area where actors could change their costumes. After 425 BC a stone scene wall, called a, became a common supplement to. The was a long wall with projecting sides, which may have had doorways for entrances and exits. Just behind the was the  ("in front of the scene"), which is similar to the modern day proscenium. The upper story was called the. Some theatres also had a raised speaking place on the orchestra called the. By the end of the 5th century BC, around the time of the Peloponnesian War, the was two stories high.

The death of a character was always heard behind the, for it was considered inappropriate to show a killing in view of the audience. Conversely, there are scholarly arguments that death in Greek tragedy was portrayed off stage primarily because of dramatic considerations, and not prudishness or sensitivity of the audience.

A temple nearby, especially on the right side of the scene, is almost always part of the Greek theatre complex. This could justify, as a transposition, the recurrence of the pediment with the later solidified stone scene.

Orchestra
The orchestra was a circular piece of ground at the bottom of the theatron where the chorus and actors performed; the word means "dancing space", as the chorus also danced in early periods. Originally unraised, Greek theatre would later incorporate a raised stage for easier viewing. This practice would become common after the advent of "New Comedy," which incorporated dramatic portrayal of individual characters. The coryphaeus was the head chorus member, who could enter the story as a character able to interact with the characters of a play. Plays often began in the morning and lasted into the evening.

Acoustics
The theatres were built on a large scale to accommodate a large number of performers on stage and in the audience—up to fourteen thousand. Physics and mathematics played a significant role in the construction of these theatres, as their designers had to be able to create acoustics in them such that the actors' voices could be heard throughout the theatre, including the very top row of seats. The Greek's understanding of acoustics compares very favorably with the current state of the art.

Scenic elements
There were several scenic elements commonly used in Greek theatre:

 * mechane, a crane that gave the impression of a flying actor (thus, deus ex machina)
 * ekkyklêma, a wheeled platform often used to bring dead characters into view for the audience
 * pinakes, pictures hung to create scenery
 * thyromata, more complex pictures built into the second-level scene (3rd level from the ground)

Masks

The Ancient Greek term for a mask is prosopon (lit., "face"), and was a significant element in the worship of Dionysus at Athens likely used in ceremonial rites and celebrations. Many masks worshipped the higher power, the gods, making masks also very important for religion. Most of the evidence comes from only a few vase paintings of the 5th century BC, such as one showing a mask of the god suspended from a tree with decorated robe hanging below it and dancing and the Pronomos vase, which depicts actors preparing for a satyr play. No physical evidence remains available to us, as the masks were made of organic materials and not considered permanent objects, ultimately being dedicated at the altar of Dionysus after performances. Nevertheless, the mask is known to have been used since the time of Aeschylus and considered to be one of the iconic conventions of classical Greek theatre.

Masks were also made for members of the chorus, who play some part in the action and provide a commentary on the events in which they are caught up. Although there are twelve or fifteen members of the tragic chorus, they all wear the same mask because they are considered to be representing one character.

Stylized comedy and tragedy masks said to originate in ancient Greek theatre have come to widely symbolize the performing arts generally.

Mask details

Illustrations of theatrical masks from 5th century display helmet-like masks, covering the entire face and head, with holes for the eyes and a small aperture for the mouth and an integrated wig. These paintings never show actual masks on the actors in performance. They are most often shown being handled by the actors before or after a performance. This demonstrates the way in which the mask was to 'melt' into the face and allow the actor to vanish into the role. Effectively, the mask transformed the actor as much as memorization of the text. Therefore, performance in ancient Greece did not distinguish the masked actor from the theatrical character.

The mask-makers were called skeuopoios or "maker of the props", thus suggesting that their role encompassed multiple duties and tasks. The masks were most likely made out of light weight, organic materials like stiffened linen, leather, wood, or cork, with the wig consisting of human or animal hair. Due to the visual restrictions imposed by these masks, it was imperative that the actors hear in order to orient and balance themselves. Thus, it is believed that the ears were covered by substantial amounts of hair and not the helmet-mask itself. The mouth opening was relatively small, preventing the mouth being seen during performances. Vervain and Wiles posit that this small size discourages the idea that the mask functioned as a megaphone, as originally presented in the 1960s. Greek mask-maker Thanos Vovolis suggests that the mask serves as a resonator for the head, thus enhancing vocal acoustics and altering its quality. This leads to increased energy and presence, allowing for the more complete metamorphosis of the actor into his character.

Mask functions
In a large open-air theatre, like the Theatre of Dionysus in Athens, the classical masks were able to create a sense of dread in the audience creating large scale panic, especially since they had intensely exaggerated facial features and expressions. They enabled an actor to appear and reappear in several different roles, thus preventing the audience from identifying the actor to one specific character. Their variations help the audience to distinguish sex, age, and social status, in addition to revealing a change in a particular character's appearance, e.g., Oedipus, after blinding himself. Unique masks were also created for specific characters and events in a play, such as the Furies in Aeschylus' Eumenides and Pentheus and Cadmus in Euripides' The Bacchae. Worn by the chorus, the masks created a sense of unity and uniformity, while representing a multi-voiced persona or single organism and simultaneously encouraged interdependency and a heightened sensitivity between each individual of the group. Only 2 to 3 actors were allowed on the stage at one time, and masks permitted quick transitions from one character to another. There were only male actors, but masks allowed them to play female characters.

The modern method to interpret a role by switching between a few simple characters goes back to changing masks in the theatre of ancient Greece.

Other costume details
The actors in these plays that had tragic roles wore boots called cothurnus (buskin), that elevated them above the other actors. The actors with comedic roles only wore a thin-soled shoe called a soccus or sock. For this reason, dramatic art is sometimes called "sock and buskin."

Male actors playing female roles would wear a wooden structure on their chests (posterneda) to imitate the look of breasts and another structure on their stomachs (progastreda) to make them appear softer and more lady like. They would also wear white body stockings under their costumes to make their skin appear fairer.

Most costuming detail comes from pottery paintings from that time as costumes and masks were fabricated out of disposable material, so there are little to no remains of any costume from that time. The biggest source of information is the Pronomos Vase where actors are painted at a show's after party.

Costuming would give off a sense of character, as in gender, age, social status, and class. For example, characters of higher class would be dressed in nicer clothing, although everyone was dressed fairly nicely. Contrary to popular belief, they did not dress in only rags and sandals, as they wanted to impress. Some examples of Greek theatre costuming include long robes called chiton that reached the floor for actors playing gods, heroes, and old men. Actors playing goddesses and women characters that held a lot of power wore purple and gold. Actors playing queens and princesses wore long cloaks that dragged on the ground and were decorated with gold stars and other jewels, and warriors were dressed in a variety of armor and wore helmets adorned with plumes. Costumes were supposed to be colourful and obvious to be easily seen by every seat in the audience.

title: Theatre of ancient Greece


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User talk:Djinn112/Archive2

Salve. Si primum archivum meum legere vis, hanc hypercatenam tangeas.

Superpenguinnus Ad Salventem Nuntium Jinnis Respondet
Thanks for the welcome, Djinn. Actually my forte is Electrical engineering with a focus on microelectronics; computers (specifically operating systems and older hardware) are just a hobby. I'm mostly here to correct minor issues as I run across them, but it's likely I'll write an article if I ever see a blaring omission that needs to be remedied... --uberpenguin 22:39, 2004 Oct 22 (UTC)

Fortunatus Undeseptuagintidecimi Jinni De Pagina Electronici Organi Dicet
Hi! Glad you're here. I found the printout of an e-mail that I thought I'd lost. Back in April, Pea Hix of optigan.com suggested some changes to the article for accuracy's sake and gave permission to use photos from the site. I'm claiming fair use.

If there's anything you care to add, please do as I'd like to nominate this as a featured article. Thanks!! - Lucky 6.9 04:35, 25 Oct 2004 (UTC)

Jinni Quituusjudus Dicet: Felix Sit Natalis Dies!
Happy birthday, Djinn! Best wishes. --Whosyourjudas (talk) 01:31, 29 Oct 2004 (UTC)

Indiceret Nihil De Pagina Eius Nova Locuti
Hi, I noticed that Locutus only redirects to the Picard article and I figured it wouldn’t hurt to have a separate Locutus article so I pieced together one from using available articles that refer to Locutus. I thought you might want to have a look and correct any mistakes I made, and add to it. Greetings, Nought 02:01, 30 Oct 2004 (UTC)

Kyberia Jinni Letterarum Italicarum Nomeniumque Navium Dicet
I changed the link because the article on the Romulan Bird of Prey happens to name a specific episode from a "specific series" (ie, Star Trek: The Original Series) and Balance of Terror which is an episode from that series, and not of the overall Star Trek universe in general. I think it should be changed IMO. Cyberia23 23:59, 12 Nov 2004 (UTC)
 * Oh and go to StarTrek.com where they always use "U.S.S." in front of ship names. I go by what they do, plus I spent 4 years in the U.S. Navy, and we always spelled it the same way. Wikipedia's standard is wrong. Cyberia23 00:04, 13 Nov 2004 (UTC)

Interrogatios Jinni Bebbopus Consultat Sed Mox Amovet
I had previously asked a question here but the question actually belonged in the Help desk, so I asked there and got the answer. I now see why you removed extra spacing from an article and how to adjust my stylesheet so that I see the spacing as wider. I'm going to go find every page I edited with extra spacing and take it out. Sorry for the clutter on your page; I've just removed the questions. Bebop 21:38, 16 Nov 2004 (UTC)

title: User talk:Djinn112/Archive2


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Kickback (video game platform)

Kickback.com was an esports platform that allowed users play competitive video games. Players of any skill level could enter ranked matches and compete for a chance to win tournaments using their skills in-game. Kickback integrated on top of popular existing games, where the service adds matchmaking, anti-cheat and support. These features were available to all users, but were made optional for users playing for fun. The site was backed by Y Combinator in 2015.

History
The website kickback.com launched on February 2, 2015, in San Francisco, California by hosting paid tournaments in Minecraft. Players competed against one another in player-versus-player matches for a chance to win prizes. On March 2, 2015, Kickback announced its Y Combinator backing and integration with PayPal. On December 1, 2015, Kickback began allowing users to play Counter-Strike: Global Offensive.

Games offered
Kickback offered a variety of Counter-Strike: Global Offensive minigames. Users are matched player versus player with the objective of defeating an opponent through in-game kills. The site runs daily events that are free to enter and award real prizes.

title: Kickback (video game platform)


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Ballymahon

Ballymahon on the River Inny is a town in the southern part of County Longford, Ireland. It is 19 km north-east of Athlone, at the junction of the N55 and R392 roads.

History
Ballymahon derives its name from the Irish language term Baile Uí Mhatháin  or Baile Mathuna, meaning "Town of Mahon". This may refer to Mahon (Mathgamain mac Cennétig), a southern chieftain and the elder brother of Brian Boru, who is believed by some to have fought a battle in 960 in the vicinity of Ballymahon at Shrule (in Irish Sruaith Fhuil, River of Blood) where he defeated O'Rourke of Cavan and laid claim to lands in the area. This is disputed by others who claim Mahon may relate to a sub chieftain of the O'Farrells who ruled over this part of County Longford in the 14th century.

The earliest documentary evidence of Ballymahon is from the year 1578, when lands in the area were granted to the Dillon family, later Earls of Roscommon. By 1654, the maps of William Petty's Down Survey shows Ballymahon as a group of houses situated at the southern end of the present town, on the right bank of the River Inny.

Two main families, the Shuldham family of Moigh House and the King-Harman family of Newcastle House, developed the town in the mid-nineteenth century. The buildings in the town are of late Georgian architecture, with two and three-storey gabled houses, colour-washed and in rows of three and four. From 1788 to 1853, the Roman Catholic bishops of Ardagh and Clonmacnoise resided in the town, and the parish church served as a Pro-Cathedral until St Mel's Cathedral was built in Longford.

Geography
The River Inny, a tributary of the River Shannon, flows westwards through Ballymahon in the direction of Lough Ree three miles from the town. A stretch of the Inny from Newcastle Bridge to Ballymahon town is used for beginner to intermediate kayaking, canoeing and time trials.

The Royal Canal also passes westwards through Ballymahon from Dublin to Clondra, County Longford. Following extensive works the canal has been restored and is now fully navigable. Brannigan Harbour, c.1 km from Ballymahon town, is on the Royal Canal and is a common point for boats and barges to stop and pass.

Built heritage
Churches in the town include St Matthew's Roman Catholic church (built 1906) which is constructed in a Gothic-style and located on the northwest side of Ballymahon. St Catherine's Church of Ireland church (built 1800, rebuilt 1824) has a narrow spire which is a prominent feature in the town.

Ballymahon Courthouse and Market House, originally built in the early 19th century, is a community library. It is situated in the centre of the town (built 1819).

Opposite St Catherine's Church is the old RIC police barracks built on the northern edge of what local tradition holds was a castle or fortress of the O'Farrell Clan, earthworks of which can still be seen off the Fairgreen/Thomand Lodge Road. It became a Garda station after the establishment of the Irish Free State and remained in service until 1996 when a new, smaller, station was built on the southern bank of the river opposite the old mill. The barracks was attacked during the War of Independence in August 1920. The attack was led by General Seán MacEoin, Seán Connolly (who fired the first shot in the Easter Rising and was later killed himself in the Selton Hill ambush) and Frank Davis. A small party of local volunteers also took part in the raid which mainly aimed to seize firearms. MacEoin later visited the barracks to campaign during an election campaign in the 1950s. Ruairí Ó Brádaigh was held here in the 1950s during the IRA Border War which ended in 1962.

Transport and communications
Ballymahon is on the N55 road, a National Secondary route leading from Athlone, 22 kilometres by road to the south, towards Belfast and Northern Ireland. The county town of Longford lies 22 kilometres by road to the north.

The R392 road links Ballymahon to Mullingar in the east and to Roscommon in the west, via Lanesborough–Ballyleague. This route was a portion of an ancient ceremonial way from Rathcroghan to the Hill of Tara. Intact portions of the ancient roadway can be seen at the nearby Corlea Trackway and the ceremonial route attests to the straightness of the R392.

The Royal Canal links Ballymahon to Dublin, via several towns such as Mullingar and Maynooth, and to the River Shannon at Clondra.

Bus services include Bus Éireann route 466 which links Ballymahon with Athlone railway station in County Westmeath, and Local Link route 24 to Longford town and Longford railway station.

Education
Ballymahon has a primary school, St. Matthew's National School, which had an enrollment of over 360 pupils as of 2019. There are also two secondary schools in the area: Mercy Secondary School and Ballymahon Vocational School. The nearest third-level college is Athlone Institute of Technology in Athlone, approximately 22 km away.

Culture and sport
Ballymahon is home to the Bog Lane Theatre. It also hosts the Oliver Goldsmith Summer School, a festival of literature which is held on the June Bank Holiday Weekend, with poetry readings held at Goldsmith's birthplace in Pallas. Ballymahon also plays host to the annual Still Voices International Film Festival.

The town's Gaelic football (GAA) club is Ballymahon GAA, and its association football (soccer) club is Ballymahon AFC.

Twin towns
Ballymahon is twinned with the two adjacent towns of Landévant and Landaul in the Morbihan region of Brittany in north-west France. Twinning events and cross-cultural tours are held annually.

People

 * Sir Thomas Molyneux (1661–1733) and William Molyneux (1656–1698), natural philosopher and writer, were descendants of the Dowdall-Molyneux family who had large estates in Ballymulvey, near Ballymahon.
 * Thomas Marlay (c.1680–1756), Lord Chief Justice of Ireland, was born at Creevagh Beg, Ballymahon.
 * Oliver Goldsmith (1728–1774), author, was born in the nearby townland of Pallas and raised in nearby Lissoy (The Pigeons). His mother was resident in the town in her latter years in a building later demolished during the widening of the N55 junction in the 1990s. Ballymahon has a monument on the main street commemorating the poet as well as a commemorative monument and park at his birthplace in Pallas.
 * John Keegan Casey (1846–1870), known as "the poet of the Fenians" was born in 1846 in County Westmeath before his family moved to Gurteen outside Ballymahon where his father was schoolmaster. Casey later taught in the area and became involved in the Fenian movement locally.
 * John Henry Patterson (1867–1947), soldier and author, was born in Forgney near Ballymahon. His book The Man-Eaters of Tsavo (1907) details his experiences while building a railway bridge in Kenya in the late 1890s.
 * Dr Kathleen Lynn, political and women's rights activist and medical doctor, lived in Ballymahon for four years in her youth from 1882 to 1886, while her father served as a Church of Ireland clergyman to the Ballymahon Parish.
 * James Dooley (1877-1950), Australian politician and Premier of New South Wales, was born at Currycreaghan near Ballymahon.

title: Ballymahon


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Talk:Removable singularity

Incorrect statement?
I believe that your statement of Reimann's theorem is incorrect. While I can't seem to find a good description of the theorem online anywhere I am pretty sure that:

a) It is defined as the limit at a of the function existing (i.e. the limit is the same as you approach from all sides, but the function is defined as something else when it is actually at a. (sorry I don't feel like inserting the latex right now).

b) In the example of the Heaviside functin where f(x)=0 x<0 and f(x)=1 x>=1 we have a function that is obviously bounded at 0, but this is not a removable singularity. No redefinition of the Heaviside function will make this continuous.

Perhaps this definition of Reimann's theorem makes more sense in the complex analysis sense, but if so you should be more specific. --anon

 * I believe Riemann's theorem applies only to holomorphic functions. Perhaps a different wording could make this clearer. —Caesura(t) 22:07, 8 December 2005 (UTC)
 * Yes, this phenomenon is unique to complex analysis. If the function is bounded at the singularity, that's enough to guarantee not only its continuity, but even infinite differentiability. I clarified this in the article. Oleg Alexandrov (talk) 01:02, 9 December 2005 (UTC)

Method of proof?
The proof given here strikes me as rather strange. In particular, it's necessary to show that the Taylor series won't have a first order term. And what is the motivation for multiplying by (z-a)^2 rather than by (z-a)? Ahlfors, Complex Analysis (on page 124 in the 3rd edition, 1979) gives a neat proof using the Cauchy integral formula instead. Jowa fan (talk) 05:49, 25 January 2010 (UTC)
 * I agree that the proof is not very clear. I guess that the motivation for multiplying by (z − a)2 is to ensure that h(a) exists (namely, it is 0), but this should be stated explicitly in the proof. The first two terms of the Taylor series of h then vanish since a0 = h(a) = 0 and a1 = h(a) = 0. — Emil J. 11:40, 25 January 2010 (UTC)

title: Talk:Removable singularity


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User:Secretlondon/Photos to take/Fruit and Veg

 * Arugula - Rocket
 * Kidney bean
 * Lentil
 * Lima bean - butter bean
 * Runner bean
 * Pea
 * Brussels sprout
 * Celery
 * Cymbopogon - Lemon grass
 * Nettle
 * Green pepper
 * Leek (vegetable)
 * Beetroot
 * Parsnip
 * Radish
 * Rutabaga - Swede
 * Turnip
 * Spinach
 * Courgette
 * Lime


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Therapeutic touch

Therapeutic touch (TT), or non-contact therapeutic touch (NCTT), is a pseudoscientific energy therapy which practitioners claim promotes healing and reduces pain and anxiety. "Therapeutic Touch" is a registered trademark in Canada for the "[s]tructured and standardized healing practice performed by practitioners trained to be sensitive to the receiver's energy field that surrounds the body;...no touching is required."

Practitioners of therapeutic touch state that by placing their hands on, or near, a patient, they are able to detect and manipulate what they say is the patient's energy field. One highly cited study, designed by the then-nine-year-old Emily Rosa and published in the Journal of the American Medical Association in 1998, found that practitioners of therapeutic touch could not detect the presence or absence of a hand placed a few inches above theirs when their vision was obstructed. Simon Singh and Edzard Ernst concluded in their 2008 book Trick or Treatment that "the energy field was probably nothing more than a figment in the imaginations of the healers". The American Cancer Society noted, "Available scientific evidence does not support any claims that TT can cure cancer or other diseases." A 2004 Cochrane review found no good evidence that it helped with wound healing, but the authors withdrew it in 2016 "due to serious concerns over the validity of included studies".

Origin and foundations
Dora Kunz, a theosophy promoter and one-time president (1975–1987) of the Theosophical Society in America, and Dolores Krieger, now Professor Emerita of Nursing Science, New York University, developed therapeutic touch in the 1970s. According to Krieger, therapeutic touch has roots in ancient healing practices, such as the laying on of hands, although it has no connection with religion or with faith healing. Krieger states that, "in the final analysis, it is the healee (client) who heals himself. The healer or therapist, in this view, acts as a human energy support system until the healee's own immunological system is robust enough to take over".

Justification for TT has been sought in two fields: Martha E. Rogers' contemporal "Science of Unitary Human Beings", and quantum mechanics, in particular Fritjof Capra's mystical interpretation of the latter. A 2002 review found that neither justification was tenable: Rogers' theories were found to be inconsistent with the tenets of TT, while the overlap in terminology between the two could be ascribed to a lack of precision in Rogers' works, making them multi-interpretable. The quantum physics justification holds that the possibility to heal at a distance is possible due to a "global interconnectivity" of the universe, which is connected by TT adherents to an interpretation of Bell's theorem and the possibility of quantum nonlocality; this interpretation is not supported by experimental evidence. The 2002 study concluded that "the theory TT possesses is deprived of explanatory power" and "evidence that supports the current picture of physical energy should be regarded as evidence against the theory of TT".

The supposed healing in TT takes place via a discredited physical process called "electron transfer resonance", which physicist Alan Sokal describes as "nonsense".

Scientific investigations
Over the decades, many clinical studies have been performed to investigate TT's efficacy, as well as various meta-analyses and at least one systematic review, yielding varying results and conclusions. O'Mathúna et al., in discussing these studies, note several problems, such as failure to exclude methodologically flawed studies and a susceptibility to the publication bias of complementary medicine journals, which carry a "preponderance of studies with positive results"; they argue that

"in light of background scientific knowledge, the antecedent plausibility of TT is sufficiently low that any methodological flaw in a study will always provide a more plausible explanation for any positive findings."

- O'Mathúna et al.

Emily Rosa, at nine years of age, conceived and executed a study on therapeutic touch. With the help of Stephen Barrett from Quackwatch and the assistance of her mother, Linda Rosa and her step-father Larry Sarner, Emily became the youngest researcher to have a paper accepted by the Journal of the American Medical Association (JAMA), which debunked the claim of therapeutic touch practitioners can reliably sense a "Human Energy Field." Twenty-one practitioners of therapeutic touch participated in her study. The practitioners sat on one side of a cardboard screen, while Emily sat on the other. The practitioners then placed their hands through holes in the screen. Emily flipped a coin to determine which of the practitioner's hands she would place hers over (approximately 4-5 inches above the subject's hand). The practitioners then were to say where her hand was by sensing her biofield. Although all of the participants had asserted that they would be able to do this, the actual results did not support therapeutic touch's fundamental claim. The practitioners had succeeded in locating Emily's hand 44% of the time, a rate within the range of chance. JAMA editor George D. Lundberg, M.D, recommended that third-party payers and the public should question paying for this procedure "until or unless additional honest experimentation demonstrates an actual effect."

There is no good medical evidence for the effectiveness of therapeutic touch. A Cochrane systematic review, first published in 2004, found "[t]here is no robust evidence that TT promotes healing of acute wounds", but in 2016 the authors retracted it after the validity of the reviewed studies were questioned. The American Cancer Society noted, "Available scientific evidence does not support any claims that TT can cure cancer or other diseases."

Skeptical view of therapeutic touch
When examining the existing literature on therapeutic touch, it has been observed that these studies tend to only cite research that favours the desired findings. There have been studies focused on therapeutic touch that have failed to include any research that has contradictory findings. However, it is very important for studies such as these to report all results found from other studies even if they may contradict the present study's hypothesis.

It is also important to use the appropriate controls in such studies. There have been studies such as that by Grad, Cadoret, and Paul that have appeared at first glance to show the effectiveness of therapeutic touch, yet once replicated using the appropriate controls, they were shown to have nonsignificant results, therefore rendering the original results inconclusive.

Researcher bias has been noted in studies examining therapeutic touch, such as that by Turner, in which he included such statements as, "If we can successfully complete this study, this will be the first real scientific evidence there is for therapeutic touch". It is important for researchers not to bias the results in order to achieve their desired outcome, as bias can lead to a misrepresentation of the true effectiveness of therapy such as that of therapeutic touch.

Replication is another important factor. A study on therapeutic touch by Wirth appeared to have successful results in which more than half of the subjects being treated by this therapy had healed by day 16, with no healing shown in the control group. However, closer examination of this study reveals that there were several trials to test therapy, that only two of the five trials were successful, and that the control group actually healed as well or better than the treatment group in the other three trials. This makes the results of such a study inconclusive in showing any effectiveness of therapeutic touch.

Therapeutic touch and nursing education
Sokal, in 2006, reported generally accepted estimates of over 80 colleges and universities spread over 70 countries where therapeutic touch is taught as well as some 80 hospitals in North America where it is practiced. He added that "these figures should be taken with a grain of salt, inasmuch as both advocates and detractors [...] have an interest in exaggerating its incidence".

Owen Hammer and James Underdown from the Independent Investigations Group examined nursing standards in California, where the California Board of Registered Nursing (CBRN) can award registered nurses taking classes in therapeutic touch with continuing education units (CEUs) required for licensure renewal. In 2006, Hammer and Underdown presented the board with the scientific evidence refuting the validity of therapeutic touch as a legitimate treatment, but the board did not change its policy.

title: Therapeutic touch


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Christof Perick

Christof Prick (born 1946) is a German orchestra conductor. He uses the name Christof Perick in English-speaking countries. His father was the concertmaster of the Hamburg Philharmonic.

Biography
Born in Hamburg, Prick studied at the University of Music and Theater in his hometown Hamburg. He was appointed to the Theater Saarbrücken as Germany's youngest general music director in 1974. From 1977 to 1986, he was responsible for the Staatstheater Karlsruhe and the Badische Staatskapelle in the same position. He was Generalmusikdirektor of the Niedersächsisches Staatsorchester and of the Staatsoper Hannover from 1993 to 1996.

His work in contemporary music has included conducting the premiere of Wolfgang Rihm's opera Oedipus in October 1987.

Perick became Music Director of the Charlotte Symphony Orchestra in 2001. He stepped down from this post in 2010 and served as the orchestra's conductor laureate for the 2010–2011 season.

From 2006 to 2011, Perick was Music Director of the Staatstheater Nürnberg, including the post of principal conductor of the Staatsphilharmonie Nürnberg. In addition to his conducting posts, since 1999, he has been a professor of conducting at the Staatliche Hochschule für Musik in Hamburg. In July 2015, the Beethoven Orchester Bonn announced the appointment of Perick as its interim Generalmusikdirektor for the 2016–2017 season.

Recordings
Perick's recordings include a 1988 recording of Franz Schmidt's rarely heard opera Notre Dame.

Personal life
Perick and his wife Ulrike, an actress, make their home in Germany.

title: Christof Perick


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Zmacs

Zmacs is one of the many variants of the Emacs text editor. Zmacs was written for the MIT Lisp machine and runs on its descendants (Symbolics Genera, LMI Lambda, TI Explorer). Zmacs is written in Lisp Machine Lisp (called ZetaLisp on Symbolics Lisp Machines). It is based on the ZWEI programming substrate, which stands for "Zwei Was EINE Initially"; Zwei was a collection of routines which could be used to easily implement other programs, like the Symbolics mail program, Zmail.

A distinctive feature of Zmacs, which can also be found in Hemlock and LispWorks, is that commands look like M-x Compile Buffer instead of M-x compile-buffer as modern Emacsen, like GNU Emacs, generally format commands.

Zmacs also supports buffers and modes. Zmacs also uses the window system of the Lisp Machine with support for mouse and windows. Zmacs supports unlimited backup of files, since the file system of the Lisp Machine supports file versions. It is not compatible with GNU Emacs and its Emacs Lisp.

title: Zmacs


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FXT

FXT may refer to:
 * FreeX FXT, a German paraglider
 * Functional XML Transformation Tool

title: FXT