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Axiom of pairing

In axiomatic set theory and the branches of logic and computer science that use it, the axiom of pairing is one of the axioms of Zermelo–Fraenkel set theory. It was introduced by Zermelo as a special case of his axiom of elementary sets. In the formal language of the Zermelo–Fraenkel axioms, the axiom reads: ∀ A ∀ B ∃ C ∀ D In words: Given any set A and any set B, there is a set C such that, given any set D, D is a member of C if and only if D is equal to A or D is equal to B. Or in simpler words: Given two sets, there is a set whose members are the two given sets; as noted, what the axiom is saying is that, given two sets A and B, we can find a set C whose members are A and B. We can use the axiom of extensionality to show. We call the set C the pair of A and B, denote it, thus the essence of the axiom is: Any two sets have a pair. The set is abbreviated, called the singleton containing A. Note that a singleton is a special case of a pair. Being able to construct a singleton is necessary, for example, to show the non-existence of the an infinitely descending chains x = from the Axiom of regularity.

The axiom of pairing allows for the definition of ordered pairs. For any sets a and b, the ordered pair is defined by the following: =. Note that this definition satisfies the condition = ⟺ a = c ∧ b = d. Ordered n-tuples can be defined recursively as follows: =; the axiom of pairing is considered uncontroversial, it or an equivalent appears in just about any axiomatization of set theory. In the standard formulation of the Zermelo–Fraenkel set theory, the axiom of pairing follows from the axiom schema of replacement applied to any given set with two or more elements, thus it is sometimes omitted; the existence of such a set with two elements, such as, can be deduced either from the axiom of empty set and the axiom of power set or from the axiom of infinity. In the absence of some of the stronger ZFC axioms, the axiom of pairing can still, without loss, be introduced in weaker forms. In the presence of standard forms of the axiom schema of separation we can replace the axiom of pairing by its weaker version: ∀ A ∀ B ∃ C ∀ D.

This weak axiom of pairing implies that any given sets A and B are members of some set C. Using the axiom schema of separation we can construct the set whose members are A and B. Another axiom which implies the axiom of pairing in the presence of the axiom of empty set is ∀ A ∀ B ∃ C ∀ D, it differs from the standard one by use of D ∈ A instead of D = A. Using for A and x for B, we get for C. Use for A and y for B, getting for C. One may continue in this fashion to build up any finite set, and this could be used to generate all hereditarily finite sets without using the axiom of union. Together with the axiom of empty set and the axiom of union, the axiom of pairing can be generalised to the following schema: ∀ A 1 … ∀ A n ∃ C ∀ D that is: Given any finite number of sets A1 through An, there is a set C whose members are A1 through An; this set C is again unique by the axiom of extensionality, is denoted. Of course, we can't refer to a finite number of sets rigorously without having in our hands a set t

Gjesing Church

Gjesing Church Gesing Church, is a modern church in Esbjerg in the southwest of Jutland, Denmark. Designed by architects Niels Munk and Keld Wohlert, the red-brick building with a steeply pitched roof was completed in 1983. In 1979, Gjesing Parish was established in the northern suburbs of Esbjerg; the area had formed part of Bryndum Parish. A temporary building served the new parish until, following a competition between four architectural firms in 1979, a complex designed by Niels Munk and Keld Wohlert of Solrød Strand was inaugurated on 30 January 1983. In 2010, the complex was extended with a handicapped toilet and three offices; the church is adjacent to Esbjerg Storcenter, a shopping centre, in a residential area with apartment buildings and detached houses. Under the same tall roof, the complex consists of the church proper to the east and a parish hall to the west. A corridor connects the two and leads to offices, a classroom for confirmation candidates and atriums; the ground plan, which consists of a porch and waiting room to the west and a sacristy and a chapel to the east, can be described as two large rectangles with two steep half-roofs, set off a little from one another.

Light from glazed rifts in the roofing above comes into the nave from the south and north gables, with a similar approach in the church hall. The entire complex is built of red brick, with zig-zag patterns in the gable brickwork, is roofed with copper; the free-standing bell tower stands to the southwest. Inside, the harmoniously designed nave has an open roof trussed ceiling, finished with white sheeting, whitewashed walls. Three old bricks from the mother church in Bryndum are built into the wall behind the pulpit; the floor is laid with red tiles. A white altar table of pine stands beside the east wall; the font takes the form of an intersected cylinder consisting of four blocks of Bornholm granite. Seen from above, the blocks form a cross; the pulpit, which resembles a normal speaking podium, rises only above the floor. Like the altar, the pulpit and the pews are made of pine. Gjesing Kirke website

Rafael Camacho Guzmán

Rafael Camacho Guzmán, was a Mexican trade union and political leader, member of the Institutional Revolutionary Party and governor of Queretaro from 1979 to 1985. Rafael Camacho Guzmán was born in the city of Queretaro in 1916, he studied agronomy in Guanajuato. Speaker worked in the XEBZ in Mexico City, it was the official broadcaster of the Presidency of the Republic during the government of Miguel Aleman. He participated in the founding of the Union of the Industry of the Radio and Television and was its director from 1961 to 1979, he was one of the most prominent of the Confederation of Workers of Mexico and close to Fidel Velazquez leaders. He was elected Governor of Queretaro in the 1979 election. During his administration he held various public works such as the "Corregidora of Queretaro" Stadium, the new auditorium Josefa Ortiz de Domínguez and network highways of the Sierra Gorda, he was a Mexican trade union and political leader, member of the Institutional Revolutionary Party and governor of Queretaro from 1979 to 1985.

Rafael Camacho Guzmán began his political career as leader of the Industrial Union of Workers and Television and Radio Artists, which placed as one of the most prominent of the Confederation of Workers of Mexico and close to Fidel Velazquez leaders, this allowed him to occupy various positions of popular election among which are the Senator in 1976 and Governor of Querétaro in 1979. During his administration he held various public works such as the Estadio La Corregidora and highways of the Sierra Gorda, he studied agricultural engineering. Announcer since 1942. Co-founder and secretary general of the Union of Industry Radio and Allied of Mexico.. He was a worker delegate from Mexico to the International Labour Organization, secretary of the Labour Congress and president of the Executive Council of the Inter-American Regional Labor Organization. Of the board of the National Council of Advertising and governor of Querétaro, he died in Mexico City, Federal District, at the age of 82 years. His remains rest in the Pantheon of the Colonia Cimatario

CSS Neuse

CSS Neuse was a steam-powered ironclad ram of the Confederate States Navy that served in the latter part the American Civil War and was scuttled to avoid capture by advancing Union Army forces. In the early 1960s, she produced 15,000 artifacts from her raised lower hull, the largest number found on a recovered Confederate vessel; the remains of her lower hull and a selection of her artifacts are on exhibit in Kinston, North Carolina at the CSS Neuse Interpretive Center State Historic Site, which belongs to the North Carolina Department of Natural and Cultural Resources. The ironclad is listed on the National Register of Historic Places. A contract for the construction of Neuse was signed on 17 October 1862 between the shipbuilding company of Thomas Howard and Elijah Ellis and the Confederate Navy. Work began in October of that year on the bank across the Neuse River from the small village of Whitehall, North Carolina; the gunboat's design was identical to her sister ironclad CSS Albemarle, but Neuse differed from Albemarle by having four additional gun ports added to her eight-sided armored casemate.

The hull was 158 feet long by 34 feet wide, she was constructed of locally abundant pine, with some 4 inches of oak used as sturdy backing for her 4-inch-thick wrought iron armor. Many delays in construction were incurred by a lack of available materials the iron plate for her armored casemate and deck. Due to continuing iron plate shortages, Neuse became the first of several Southern ironclads built with unarmored decks; this situation was compounded by the Confederate Army exercising priority over the Navy in the use of the South's inadequate railroad system for transporting vital war materiel. Neuse was equipped with two 6.4-inch Brooke rifled cannon. Both cannons were positioned along the ironclad's center-line in the armored casemate, one forward, the other aft; the field of fire for both pivot rifles was 180-degrees, from port to starboard: Each cannon could fire from one of five gun port positions or could deliver a two cannon broadside. Neuse's projectiles consisted of explosive shells, anti-personnel canister shot, grape shot, blunt-nosed, solid wrought iron "bolts" for use against Union armored ships.

Launched in November 1863 while still needing fitting out, Neuse got up steam in April 1864 for duty on the inland waters of North Carolina as part of the force under Commander R. F. Pinkney, CSN. Shortly thereafter, the ironclad grounded off Kinston due to her inexperienced crew, conscripted from the Confederate Army. After that, due to a lack of available Confederate Army shore support, she never left the river area around Kinston, serving instead as a floating ironclad fortification. In March 1865, with Kinston under siege by Union forces, gunpowder trails were laid down which led to a cache of explosives placed in her bow. Neuse burned to just below her waterline and sank into the river mud preventing capture by the advancing Union Army forces, commanded by Major General John M. Schofield. At some point following the war, her sunken hulk, lying in shallow river water and mud, was salvaged of its valuable metals: cannon and their fittings, iron ram, casemate armor, both propellers and their shafts, her steam power plant.

Whatever bits and pieces remained, including her projectiles, lay undisturbed in and around the wreck until Neuse was raised nearly a century later. After nearly a century, the remaining lower hull of the ironclad was discovered and raised in 1963. Neuse's hull was temporarily installed in the Governor Caswell Memorial, beside the river, in Kinston. Since 2013, Neuse and her artifacts have been on display in a new, climate-controlled building in downtown Kinston. There are only four recovered Civil War era ironclad wrecks, CSS Neuse, CSS Muscogee, USS Monitor, USS Cairo. Other Union and Confederate ironclad wreck sites remain untouched; the successful Confederate submarine H. L. Hunley, which sank the Union blockading sloop-of-war USS Housatonic, was recovered and is undergoing extensive restoration and long term conservation at the Warren Lasch Conservation Center in North Charleston, South Carolina. A replica of the CSS Neuse, better known as CSS Neuse II, was the brainchild of Kinston activist and businessman Ted Sampley and built by Alton Stapleford.

Neuse II is on grounds display at a separate site in Kinston and contains a complete fitted-out interior that shows all shipboard details. Neuse is the only Confederate ironclad that has a full-size replica on display. Since April 2002 Neuse's sister ironclad, CSS Albemarle has had a ​3⁄8 scale replica, 63 feet long, at anchor near the Port O' Plymouth Museum in Plymouth, North Carolina; this ironclad replica is capable of sailing on the river. Bisbee, Saxon T.. Engines of Rebellion: Confederate Ir

Regent's Park

Regent's Park is one of the Royal Parks of London. It occupies high ground in inner north-west London, administratively split between the City of Westminster and the London Borough of Camden, it contains London Zoo. The Park was designed by James and Decimus Burton, its construction was financed by James after the Crown Estate rescinded its pledge to do so. The park is Grade I listed on the Register of Historic Parks and Gardens; the park has an outer ring road called the Outer Circle and an inner ring road called the Inner Circle, which surrounds the most tended section of the park, Queen Mary's Gardens. Apart from two link roads between these two, the park is reserved for pedestrians; the south and most of the west side of the park are lined with elegant white stucco terraces of houses designed by John Nash and Decimus Burton. Running through the northern end of the park is Regent's Canal, which connects the Grand Union Canal to London's historic docks; the 166-hectare park is open parkland with a wide range of facilities and amenities, including gardens.

The northern side of the park is the home of London Zoo and the headquarters of the Zoological Society of London. There are several public gardens with flowers and specimen plants, including Queen Mary's Gardens in the Inner Circle, in which the Open Air Theatre stands. Winfield House, the official residence of the U. S. Ambassador to the United Kingdom, stands in private grounds in the western section of the park. South of the Inner Circle is dominated by Regent's University London, home of the European Business School London, Regent's American College London and Webster Graduate School among others. Abutting the northern side of Regent's Park is Primrose Hill, another open space which, with a height of 256 feet, has a clear view of central London to the south-east, as well as Belsize Park and Hampstead to the north. Primrose Hill is the name given to the surrounding district; the public areas of Regent's Park are managed by a government agency. The Crown Estate Paving Commission is responsible for managing certain aspects of the built environment of Regent's Park.

The park lies within the boundaries of the City of Westminster and the London Borough of Camden, but those authorities have only peripheral input to the management of the park. The Crown Estate owns the freehold of Regent's Park. In the Middle Ages the land was part of the manor of Tyburn, acquired by Barking Abbey; the 1530s Dissolution of the Monasteries meant Henry VIII appropriated it, under that statutory forfeiture with minor compensation scheme. It has been state property since, it was set aside as a hunting and forestry park, Marylebone Park, from that Dissolution until 1649 after which it was let as small-holdings for hay and dairy produce. Although the park was the idea of the Prince Regent George IV, was named for him, James Burton, the pre-eminent London property developer, was responsible for the social and financial patronage of the majority of John Nash's London designs, for their construction. Architectural scholar Guy Williams has written, "John Nash relied on James Burton for moral and financial support in his great enterprises.

Decimus had showed precocious talent as a draughtsman and as an exponent of the classical style... John Nash needed the son's aid, as well as the father's". Subsequent to the Crown Estate's refusal to finance them, James Burton agreed to finance the construction projects of John Nash at Regent's Park, which he had been commissioned to construct: in 1816, Burton purchased many of the leases of the proposed terraces around, proposed villas within Regent's Park, and, in 1817, Burton purchased the leases of five of the largest blocks on Regent Street; the first property to be constructed in or around Regent's Park by Burton was his own mansion: The Holme, designed by his son, Decimus Burton, completed in 1818. Burton's extensive financial involvement "effectively guaranteed the success of the project". In return, Nash agreed to promote the career of Decimus Burton; such were James Burton’s contributions to the project that the Commissioners of Woods described James, not Nash, as "the architect of Regent's Park".

Contrary to popular belief, the dominant architectural influence in many of the Regent's Park projects – including Cornwall Terrace, York Terrace, Chester Terrace, Clarence Terrace, the villas of the Inner Circle, all of which were constructed by James Burton's company – was Decimus Burton, not John Nash, appointed architectural "overseer" for Decimus's projects. To the chagrin of Nash, Decimus disregarded his advice and developed the Terraces according to his own style, to the extent that Nash sought the demolition and complete rebuilding of Chester Terrace, but in vain. Decimus's terraces were built by his father James; the Regent Park scheme was integrated with other schemes built for the Prince Regent by the triplet of Nash, James Burton, Decimus Burton: these included Regent Street and Carlton House Terrace in a grand sweep of town planning stretching from St. James's Park to Parliament Hill; the scheme is considered one of the first examples of a garden suburb and continues to influence the design of suburbs.

The park was first opened to

The Whole Story

The Whole Story is the first compilation album by English singer Kate Bush. Released in November 1986, it was Bush's third UK number one album; the compilation went on to become her best-selling release to date, being certified four times platinum in the United Kingdom. The album includes eleven of Bush's previous singles, with a unreleased track entitled "Experiment IV", released as a single and reached the UK Top 30. A newly recorded version of Bush's debut single; the album mix of "The Man with the Child in His Eyes" features on this album instead of the single version. A home video compilation of the same name was released which includes the promotional videos for each song on the album. In 2014, during Kate Bush's Before the Dawn residency at the Hammersmith Apollo, The Whole Story charted at number 8 in the UK. All tracks are written by Kate Bush. All tracks are written by Kate Bush. Kate Bush – keyboards, producer Ian Cooper – cutting engineer Jon Kelly – producer Andrew Powell – producer