Set theory

Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most to objects that are relevant to mathematics; the language of set theory can be used to define nearly all mathematical objects. The modern study of set theory was initiated by Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, such as Russell's paradox, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with or without the axiom of choice, are the best-known. Set theory is employed as a foundational system for mathematics in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.

Mathematical topics emerge and evolve through interactions among many researchers. Set theory, was founded by a single paper in 1874 by Georg Cantor: "On a Property of the Collection of All Real Algebraic Numbers". Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, mathematicians had struggled with the concept of infinity. Notable is the work of Bernard Bolzano in the first half of the 19th century. Modern understanding of infinity began in 1870–1874 and was motivated by Cantor's work in real analysis. An 1872 meeting between Cantor and Richard Dedekind influenced Cantor's thinking and culminated in Cantor's 1874 paper. Cantor's work polarized the mathematicians of his day. While Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not. Cantorian set theory became widespread, due to the utility of Cantorian concepts, such as one-to-one correspondence among sets, his proof that there are more real numbers than integers, the "infinity of infinities" resulting from the power set operation.

This utility of set theory led to the article "Mengenlehre" contributed in 1898 by Arthur Schoenflies to Klein's encyclopedia. The next wave of excitement in set theory came around 1900, when it was discovered that some interpretations of Cantorian set theory gave rise to several contradictions, called antinomies or paradoxes. Bertrand Russell and Ernst Zermelo independently found the simplest and best known paradox, now called Russell's paradox: consider "the set of all sets that are not members of themselves", which leads to a contradiction since it must be a member of itself and not a member of itself. In 1899 Cantor had himself posed the question "What is the cardinal number of the set of all sets?", obtained a related paradox. Russell used his paradox as a theme in his 1903 review of continental mathematics in his The Principles of Mathematics. In 1906 English readers gained the book Theory of Sets of Points by husband and wife William Henry Young and Grace Chisholm Young, published by Cambridge University Press.

The momentum of set theory was such. The work of Zermelo in 1908 and the work of Abraham Fraenkel and Thoralf Skolem in 1922 resulted in the set of axioms ZFC, which became the most used set of axioms for set theory; the work of analysts such as Henri Lebesgue demonstrated the great mathematical utility of set theory, which has since become woven into the fabric of modern mathematics. Set theory is used as a foundational system, although in some areas—such as algebraic geometry and algebraic topology—category theory is thought to be a preferred foundation. Set theory begins with a fundamental binary relation between an object o and a set A. If o is a member of A, the notation o. A set is described by listing elements separated by commas, or by a characterizing property of its elements, within braces. Since sets are objects, the membership relation can relate sets as well. A derived binary relation between two sets is the subset relation called set inclusion. If all the members of set A are members of set B A is a subset of B, denoted A ⊆ B.

For example, is a subset of, so is but is not. As insinuated from this definition, a set is a subset of itself. For cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined. A is called a proper subset of B if and only if A is a subset of B, but A is not equal to B. 1, 2, 3 are members of the set but are not subsets of it. Just as arithmetic features binary operations on numbers, set theory features binary operations on sets. The: Union of the sets A and B, denoted A ∪ B, is the set of all objects that are a member of A, or B, or both; the union of and is the set. Intersection of the sets A and B, denoted A ∩ B, is the set of all objects that are members of both A and B; the intersection of and is the set. Set difference of U and A, denoted U \ A, is the set of all members of U that are not members of A; the set difference \ is, conversely, the set difference \ is. When A is a subset of U, the set difference U \ A is called the complement of A in U.

In this case, if the choice of U is clear from the context, the notation Ac is sometimes used instead of U \ A if U

West Canada Lake Wilderness Area

The West Canada Lake Wilderness Area is an Adirondack Park Forest Preserve management unit in the State of New York, USA. The area contains 168 bodies of water covering 2,400 acres, 78.3 miles of foot trails, 11 lean-tos. The 138-mile Northville-Placid Trail traverses it; the area is located in the town of Ohio in Herkimer County and the towns of Morehouse, Lake Pleasant and Indian Lake in Hamilton County. It is bounded on the north by the Moose River Plains area and private lands in the vicinity of Little Moose Lake, Squaw Brook, Snowy Mountain and Squaw Mountain; the terrain ranges from rolling hills to steep mountains such as Snowy Mountain. Water drains from the area into three basins: the Mohawk and the Black; the forest cover consists chiefly of mixed hardwood-softwood types with large diameter trees of both types on the more fertile soils. There is considerable area in spruce-balsam swamp and beaver meadows. Among the spots that attract hikers and campers in addition to the mentioned lakes are T Lake Falls and T Lake Mountain, West Canada Creek, Panther Mountain, Snowy Mountain and Cedar River.

Among the area's chief attributes are its numerous ponds and streams, most of which support a brook trout population. List of Wilderness Areas in the Adirondack Park Adirondack Park Agency. "Adirondack State Land Master Plan", Updated 2014

Steven Van McHone

Steven Van McHone was a murderer executed by the U. S. state of North Carolina. He was convicted of killing his mother, Mildred Johnson Adams, stepfather Wesley Dalton Adams Sr. on June 3, 1990 in Surry County, North Carolina. Steven McHone's stepbrother Wesley Adams Jr. and his wife Wendy were visiting Wesley Adams Sr. and his wife, Mildred. Just after midnight on June 3, McHone was heard arguing with the couple by Wesley Jr. and Wendy, who were in bed. Mildred Adams entered their room and asked if they had moved a handgun, in the house, they replied that they hadn't and she left and closed the door. Before Wesley Jr. could get dressed, three shots were heard. Wendy said she heard the elder Wesley tell her husband to call 9-1-1. Wesley Jr. saw his father and stepbrother wrestling and McHone was holding a handgun. Wesley Jr. managed to disarm McHone but when he returned to the phone, the wrestling resumed and the two headed out of sight of him, up a hallway. His father returned about a minute saying that his stepmother was "facedown out back."

McHone reappeared raising it into a firing position. His stepfather moved towards him and attempted to grab the gun, but McHone fired, throwing Wesley Sr. with such force that he hit and knocked down Wesley Jr. McHone and Wesley Jr. struggled again, with McHone cursing Wesley Jr. and crying, asking Wesley Jr. to shoot McHone for what he had done. The first response team arrived at the scene and found Wesley Sr. dead with a large chest wound, Mildred Adams alive, but with a gunshot wound to the head. About 2 a.m. a deputy sheriff with the Surry County Sheriff's Department arrived and McHone was placed into a patrol car. The deputy sheriff would testify that although he smelled alcohol on McHone's breath, he did not believe that he was drunk to the point of not being aware of his actions and the consequences. McHone was sentenced to death on March 7, 1991, his appeals centered on the fact he was under the influence of alcohol and other drugs at the time of the murders. Voluntary intoxication is a defense to a capital indictment under North Carolina state law.

The trial counsel had failed to object to inappropriate statements of the prosecutor and unsatisfactory jury instruction. On November 9, 2005, the execution had been stayed after Surry County Superior Court Judge Anderson Cromer had agreed to hear testimony from paramedic Teresa Durham, who said in an affidavit that Mildred Adams had said McHone had not shot her; this ruling was overturned by the North Carolina Supreme Court the next day, without reason, in their one page decision. Clemency was denied by Governor Mike Easley on November 10 and McHone's appeal to the Supreme Court of the United States was denied without comment. Easley had met with the family of McHone. For his last meal he ordered medium rare porterhouse steak, steak fries, chocolate cheese cake, a 20 oz. Mountain Dew. After entering the execution chamber strapped to a gurney at 1:50 a.m. EST, he did not make an official final statement but did appear to say "I'm so sorry" to his stepbrother. McHone exchanged smiles and laughter with his attorney and with two friends, who were among nine people who served as official witnesses.

At about 2 a.m. executioners added a sedative to McHone's intravenous lines. He appeared to be asleep. Two lethal chemicals were added, at 2:05 a.m. McHone's torso shook and his face became pale, he appeared not to move afterward. He was pronounced dead at 2:10 a.m.. Wesley Adams Jr. released a statement after the execution: "We have sympathy and pray for comfort for those who will grieve Steve's passing. We do, feel that justice was upheld and that this fate was sealed many years ago. We feel that the enforcement of duly deliberated and prescribed sentences send a stronger message, as to the sanctity of human life, than does the sparing of those who have taken life willfully and brutally." Capital punishment in North Carolina Capital punishment in the United States List of people executed in North Carolina "McHone dies by lethal injection at 2:10 a.m. executed for murder of mother, stepfather". Winston-Salem Journal. November 11, 2005. Archived from the original on May 5, 2007. "N. C. death row inmate executed for murder of mother, stepfather".

Associated Press. November 11, 2005. Report from the National Coalition to Abolish the Death Penalty November 2005 Executions from Offender Data Screen from North Carolina Department of Correction