Question: How big is a weasel's paw?

weasel

The marten lives in both deciduous, mixed and coniferous forests and scrubland.


Question: How big is a weasel's paw?

weasel

The species particularly thrives in forests with a low canopy cover and a dense undergrowth.


Question: How big is a weasel's paw?

weasel

[2] Behavior Weasel hunts normally only at night.


Question: How big is a weasel's paw?

weasel

During the day it stays put and chooses as its resting place, for example, a rock burrow, a quarry, a hollow tree, a bird of prey nest and often a squirrel or magpie's nest, normally found in dense spruce.


Question: How big is a weasel's paw?

weasel

Where they stay, the weasel is constantly changing.


Question: How big is a weasel's paw?

weasel

In bad air, for example in a harsh twinkle, it stays still sometimes for days.


Question: How big is a weasel's paw?

weasel

As soon as the storm passes, the weasel sets out on its trek.


Question: How big is a weasel's paw?

weasel

The weasel female gives birth between March and May to chicks, of which there are usually 2 to 5.


Question: How big is a weasel's paw?

weasel

The nest is most often occupied by a hollow tree or stone quarry.


Question: How big is a weasel's paw?

weasel

Nutrition The marten eats mixed food, and prefers meat food when available, especially small mammals.


Question: How big is a weasel's paw?

weasel

[1] At night, it explores all possible socks in its habitat looking for moles and other rodents, birds, frogs and insects.


Question: How big is a weasel's paw?

weasel

Birds' eggs, berries, fruits and honey are also included on its menu.


Question: How big is a weasel's paw?

weasel

Occasionally, you can also see it visiting deer scavengers, among other things.


Question: How big is a weasel's paw?

weasel

In case of emergency, it can also eat mushrooms.


Question: How big is a weasel's paw?

weasel

Use and hunting The marten is bred on fur orchards.


Question: How big is a weasel's paw?

weasel

[1] The use of weasel fur has increased; today it already competes with mink.


Question: How big is a weasel's paw?

weasel

In addition to full fur coats, collars, headdresses and bracelets are made from it, among other things.


Question: How big is a weasel's paw?

weasel

Over the past fifteen years, the annual catch rate in Finland has ranged from around ten thousand to an abundant twenty thousand individuals per year.


Question: How big is a weasel's paw?

weasel

Traps and immediately-killed irons are used as a method of catching.


Question: How big is a weasel's paw?

weasel

There are several trap and gear models: ground-caught and set in wood.


Question: How big is a weasel's paw?

weasel

The marten is also often hunted by tracing a snow trail, as well as a barking dog, such as a swamp using an upright ear.


Question: How big is a weasel's paw?

weasel

The marten hunting season begins on August 1st and ends at the end of March [3].


Question: How big is a weasel's paw?

weasel

Starting in autumn 2018, the start of hunting time was brought forward.


Question: How big is a weasel's paw?

weasel

Sources Elsewhere Transclusion error: {{En}} is only for use in File namespace.


Question: How big is a weasel's paw?

weasel

Use {{lang-en}} or {{en icon}} instead.


Question: How big is a weasel's paw?

weasel

Transclusion error: {{En}} is only for use in File namespace.


Question: How big is a weasel's paw?

weasel

Use {{lang-en}} or {{en icon}} instead.


Question: How big is a weasel's paw?

weasel


Question: How big is a weasel's paw?

weasel

Use {{lang-en}} or {{en icon}} instead.


Question: How big is a weasel's paw?

weasel

Category:Weasel Animals Category:Fur Animals Category:Key Articles of Screening


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Richard D. Winters (January 21, 1918 — January 2, 2011) was a United States Army officer and decorated war veteran.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

He was born in Pennsylvania and graduated from Franklin and Marshall College in 1941.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

During World War II, he commanded Company E of the 506th Parachute Infantry Regiment of the 101st Infantry Division, also known as “Easy Company”.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Winters was demobilized from the war as a major.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

He has been awarded numerous decorations including the Purple Heart (Purple Heart) and Bronze Star (Bronze Star).


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Winters became famous thanks to the television series Battlemates, in which the resourceful and highly respected officer was played by Damian Lewis.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

After World War II, Winters returned to civilian work and to his work in New Jersey.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

During the Korean War in 1951, Winters was recalled to duty.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

He served for a short time at Fort Dix as the regimental training officer.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

After demobilisation, he worked in a few different jobs before starting his own company selling agricultural products.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

He has also frequently been a visiting lecturer at the United States Armed Forces Military Academy at West Point.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

He retired in 1997 and lived on his farm in Pennsylvania until his death.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Winters' autobiography Beyond Band of Brothers: The War Memoirs of Major Dick Winters was published in 2006.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Military Career Enlistment and Paratroops Winters was listed in the Army on 25 August 1941.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

In September, he underwent rookie training at Camp Croft in South Carolina, where he also stayed to train other recruits and volunteers.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

In April 1942, he was selected for the Officer Candidate School, which was located at Fort Benning, Georgia.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

There he became friends with Lewis Nixon, with whom he later served in the 101st Infantry Division during World War II.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

He graduated on 6 June and was appointed as a colonel ().


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

After officer training, Winters decided to join the Parachute Corps, but just at that time there was no vacant place to serve.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Accordingly, he returned to Camp Croft to train enlisted personnel until, after five weeks, he received orders to join the 506th Parachute Infantry Regiment, which was stationed at Camp Toccoa, Georgia.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

He was initially deployed as one platoon leader of E Company of the 2nd Battalion, with the pronouncement form “Easy Company”.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

In October 1942, he was promoted to lieutenant (), after which he served as a unit officer in the company.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Upon Winters joining E Company, its commander-in-chief was Lieutenant Herbert Sobel, who was later promoted to captain.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

The 506th Parachute Infantry Regiment was the first military unit in the United States to be given anti-landing troop training.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Many of the men in the regiment had very little military experience, so it was necessary for training to be extremely strenuous.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Only 148 of the 500 or so officers made it through training.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

From the crew side, out of 5300 volunteers, only 1800 made it through.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Transfer to England In the autumn of 1943, the regiment was sent by ship across the Atlantic to England to Liverpool, where it arrived on 15 October.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

The regiment moved to Aldbourne in Wiltshire, where it began close training for the Normandy landings.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

The situation between Winters and Sobel became tense during November—December.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Winters secretly doubted Sobel's ability to lead men in battle.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

At the same time, the crew had learned to trust Winters' skills and, on the other hand, doubt Sobel's competence as their leader.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

However, Winters has said he never wanted to compete with Sobel for the leadership of E Company.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Eventually, the situation escalated so bad that Sobel accused Winters of failing to carry out his order.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Winters felt the accusation was unjust and demanded it be brought to a court martial.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

The battalion commander quashed the charges against Winters, which resulted in Sobel bringing up another charge against Winters.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

For the duration of the investigation into the allegations, Winters was transferred to the Staff as the Battalion Finance Officer ().


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

As a result of Sobel's actions, a group of squad leaders () issued the regimental commander, Colonel Robert Sink, a threat to resign from service unless Sobel was transferred to other duties.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Winters tried to talk them into abandoning their intentions without greater impact.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Colonel Sink did not like the threat and many of the group leaders were demoted and transferred out of the company.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

However, he realized that something had to be done about the situation and decided to move Sobel to train new paratroops.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

In February 1944, Lieutenant Thomas Meehan III was named leader of E Company.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Despite his differences with Sobel, Winters has said that he felt part of E Company's success was due to Sobel's rigorous and disciplined training, and how he had built the teams.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

The Normandy landings began in the early morning hours of 6 June 1944 with the landing of paratroops.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Winters jumped with E Company and landed near the village of Sainte-Mère-Église.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

He had lost his gun during the jump, but still managed to survive in enemy territory.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

He was able to locate his location and find several other men of his company and also some men of the 82nd Inland Division.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

During the invasion, Lieutenant Meehan had died after being hit by German anti-aircraft fire (C-47 42-93095, Captain Capelluto, poka 66) [1].


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Thus Winters became Chief of E Company for the duration of the Normandy landings operation.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

The aircraft carrying Winters and his trophy (poka 67) [3] was C-47 42-100646 [4], piloted by Captain Sammons [5], with Winters himself serving as skip master on the plane.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

The plane survived World War II and later became the OH-LCB of Aero Ltd and DO-7 of the Air Force Transport Air Squadron.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

In Finland, the plane was used as a jumping machine for paratroopers.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Later that day, Winters led an attack south of the village of Le Grand-Chem on a German 105 mm howitzer battery near Brécourt Manor.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

The cannons were guarded by a German team of about 50 people (), while Winters had only 13 men at his disposal.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

This attack is still used as a model example of belting at the United States Armed Forces Military Academy at West Point.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

In addition to destroying the battery, in the attack, the Allies got hold of a map detailing the Germans' defenses at Utah Beach.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Winters was promoted to captain on 1 July 1944.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

The next day, he was awarded the Distinguished Service Cross, the second highest decoration in the United States Army.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

A short time later, the 506th Parachute Infantry Regiment was moved back to Aldbourne, England, to regroup for Operation Market Garden.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

The objective of Operation Market Garden Operation Market Garden was to take over the bridges crossing the major rivers of the Netherlands occupied by German forces and thereby facilitate the Allied advance into Germany.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

At the time, the 101st Parachute Division was under the command of Major General Maxwell D. Taylor.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

The area north of the British XXX Army Corps was designated as the drop zone for the Corps.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

The aim was to take over the Son and Veghel bridges northwest of Eindhoven, along with three other smaller bridges.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

The corps jumped on 17 September and succeeded in the first day in capturing four of the five bridges.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

The Son Bridge was left unoccupied when the Germans blew it up.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Fighting continued and on October 5 the Germans began an attack on the back of the 2nd Battalion.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

On the same day, a four-person patrol of E Company encountered the Germans at a crossroads about 1200 yards east of the Company's command post.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

All four men in the patrol were wounded, but managed to return to the command post and report on the situation.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Winters realised the gravity of the situation and included one of the first team in the group ().


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

They set out to reconnoiter the German positions and noticed machine gun fire south towards the battalion's command post.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

The group, led by Winters, set out to attack towards the machine gun position and overran it.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Shortly thereafter, the group came under enemy fire from German positions.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Winters estimated the Germans to be at least a platoon, so he called the rest of the first platoon as his confirmation and then set out to charge towards the Germans.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

After the battle, they discovered that they had been up against at least 300 German soldiers.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Four days later on 9 October, Winters was appointed deputy commander of the 2nd Battalion of the regiment, which normally came under the duties of major.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Winters, however, served in the position as captain.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Bastogne December 16.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

1944, in Belgium, the Germans began a counter-offensive against the Allies.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Two days later, the 101st Landing Division was moved to the Bastogne area with the intention of defending the line north-east of the town near the village of Foy.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

This stage, known as the Battle of the Ardennes, later became known as the Battle of the Bulge.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

The Germans had at their disposal about 24 divisions, a large number of which were armoured divisions.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

They had surrounded the Bastogne area, but the 101st Infantry Division reinforced with parts of the 10th Panzer Division managed to hold off the Germans for nearly a week before the US 3rd Army managed to break the Germans siege.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

After the siege, the 2nd Battalion attacked the village of Foy on 9 January 1945.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

About two months later, the 2nd Battalion moved to Haguenau and Winters was promoted to major and soon appointed commander of the 2nd Battalion.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

The 2nd Battalion thereafter did little to experience fighting.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Final stages of the war In April 1945, the battalion operated on defensive duties along the Rhine before being transferred to Bavaria towards the end of the month.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

In early May, the 101st Infantry Division received orders to take the village of Berchtesgaden in the Alps.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

The 2nd Battalion set out to advance across Germany amid the surrendering German soldiers.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

On the afternoon of 5 May, troops took over the village.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

The battalion was stationed there when the war in Europe ended with the German surrender three days later.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Winters remained in Europe as the United States began to withdraw its troops.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

He had enough points to return to his home, but the military said he was still needed.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Later, he was offered a permanent post in the army, but declined.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

He eventually boarded a ship at Marseille on 4 November 1945 and returned to the United States.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

He left army duties on 29 November, but was not formally demobilized until 22 January 1946.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

The Medal of Honor, the highest decoration, has been proposed for Winters since the attention the Band of Brothers received, but the proposals have not gone through.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

[6] [7] After the war, Winters worked in New Jersey for a company called the Nixon Nitration Works of Edison, which was the family business of his friend Lewis Nixon that he met at officer school.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

He became general manager of the company in 1950.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Two years earlier he had married Ethel Estoppey.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Along with the job, he continued studies at Rutgers University, where he studied business studies and human resources management.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

In June 1951, he was recalled to Army service due to the Korean War.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

He was assigned to the 11th Landing Division at Fort Campbell, Kentucky, where he was to report within six months.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

After receiving the invitation, he traveled to Washington, D.C. to meet with General Tony McAuliffe.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

He had seen enough war in his mind, and he didn't want to go to Korea.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

McAuliffe understood the war veteran's wish but said Winters was needed because of his leadership experience.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Thus, Winters enrolled at Fort Dix, New Jersey, where he served as the regimental planning and training officer.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

However, Winters soon realised he did not like his post, as many officer cookies lacked self-discipline and did not attend organised lessons.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

As a result, he volunteered for Ranger training.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

He then received orders to leave for Korea, for which he travelled to Seattle.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

There he received an offer to resign at will from the service he accepted.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

After separating from the Army in 1951, Winters and his wife Ethel bought a small farm on which they built a house and raised their two children.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Winters started his own company in 1972 that sold animal feed to Pennsylvanian farmers.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Shortly after the company was founded, the family moved to Hershey, Pennsylvania.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Winters retired in 1997.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

1990s and the beginning of the new millennium At the end of the millennium, Winters appeared in several series and books, recounting his own and E Company's experiences in World War II.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Stephen Ambrose wrote the 1992 book Band of Brothers: Easy Company, 506th Regiment, 101st Airborne from Normandy to Hitler's Eagle's Nest, which was later made into a drama series The Fighters.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Also published about Winters was a 2005 book Biggest Brother: The Life of Major Dick Winters, The Man Who Led the Band of Brothers, written by Larry Alexander.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Winters has also written his own memoirs Beyond Band of Brothers: The War Memoirs of Major Dick Winters along with historian and retired Colonel Cole C. Kingseed.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Despite his many heroic achievements, Winters has remained humble.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

The Combatsmates series ends with a documentary interview episode in which he quotes a passage from a letter he received from Sergeant Mike Ranny: Sources On the topic elsewhere Transclusion error: {{En}} is only for use in File namespace.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Use {{lang-en}} or {{en icon}} instead.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Transclusion error: {{En}} is only for use in File namespace.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Use {{lang-en}} or {{en icon}} instead.


Question: What's Richard D. Winters after he became a veteran?

Richard Winters

Category:United States soldiers of World War II Class:Those born in 1918 Class:Those who died in 2011


Question: What is an infinite ordinal?

Ordinal

Ordinal, or ordinal number, is a generalization of the concept of natural number used in set doctrine to describe ways to place objects belonging to a collection in sequential order, one after another.


Question: What is an infinite ordinal?

Ordinal

Every finite number of items can be placed in order simply by counting them: in this case, each object receives an ordinal number of some integer.


Question: What is an infinite ordinal?

Ordinal

In practice, ordinal numbers can be thought of as, say, number bundles that allow items to be set in a prescribed order.


Question: What is an infinite ordinal?

Ordinal

In the case of finite collections, the ordinal numbers correspond to completely natural numbers, that is, positive integers.


Question: What is an infinite ordinal?

Ordinal

Instead, there are more concepts of “order” for infinite collections.


Question: What is an infinite ordinal?

Ordinal

That of them, which essentially can be viewed as a generalization of the order of a finite set “one after another”, is called well-order.


Question: What is an infinite ordinal?

Ordinal

well-order can be thought of as meaning that items are marked with name or number tags “one after the other”, but allow that some name-tags come only after infinitely many predecessors.


Question: What is an infinite ordinal?

Ordinal

For example, after the set of natural numbers {0,1,2,...}, one can add the symbol ω, which is the smallest infinite ordinal, to the end of the queue.


Question: What is an infinite ordinal?

Ordinal

Thus the set {0,1,2,... , ω) continues up to infinity.


Question: What is an infinite ordinal?

Ordinal

Some ordinals are the immediate successor of one of its predecessors, the successor ordinal', as is the case with natural numbers {1,2,...}, but other ordinals such as ω are not immediate successors of any other ordinal, and such are called boundary ordinals.


Question: What is an infinite ordinal?

Ordinal

Ordinals must be separated from cardinal numbers, which can be used to indicate how many items there are in any collection.


Question: What is an infinite ordinal?

Ordinal

In the case of finite numbers, the cardinal and ordinal numbers correspond inversely uniquely to each other; for example, one can go from an ordinal number to the corresponding cardinal number by counting the name tags.


Question: What is an infinite ordinal?

Ordinal

In contrast, two or more different infinite ordinals may correspond to the same cardinal (see Hilbert's hotel).


Question: What is an infinite ordinal?

Ordinal

Similar to other numbers, ordinals can also be added together, multiplied, or increased to potency, but in the case of infinite ordinals, the combination and multiplication are not commutative.


Question: What is an infinite ordinal?

Ordinal

[1] Ordinals were introduced by Georg Cantor in 1883 [2] [3] to deal with infinite queues and to classify derived sets, which he had introduced in 1872 when studying the singleness of trigonometric sets.


Question: What is an infinite ordinal?

Ordinal

[4] Ordinals as an extension of natural numbers Natural numbers, which in this context are also read zero, can be used for two purposes: they may express the magnitude of a set, or they can express the position of the object when placed in sequential order.


Question: What is an infinite ordinal?

Ordinal

In conventional language, natural numbers correspond to cardinal, i.e., basic number words (one, two, three, etc.), in the latter case ordinal or ordinal (first, second, third, etc.) In mathematics, however, these do not have an essential difference in the case of finite numbers, as every finite cardinal number is matched by one finite ordinal number and vice versa.


Question: What is an infinite ordinal?

Ordinal

Thus, all the ways in which a given finite number of items can be placed in sequential order are mutually isomorphic.


Question: What is an infinite ordinal?

Ordinal

In the case of infinite sets, however, the thing is different; the “magnitude” of the set and the place of the embryo in the queue must be described by different concepts, of which the former leads to infinite cardinal numbers, the latter to ordinals.


Question: What is an infinite ordinal?

Ordinal

This must be done because even the “magnitude” of an infinite set, the awesomeness, that is, cardinality, is uniquely determinable, but there are several mutually non-isomorphic ways veryarrange its embryos, as further described in more detail.


Question: What is an infinite ordinal?

Ordinal

The concept of a cardinal number is related to a set that need not be assumed to have any particular mathematical structure.


Question: What is an infinite ordinal?

Ordinal

Instead, ordinals are so closely related to so-called well-ordered sets that some mathematicians believe that the concepts of ordinal and well-order need not make any distinction.


Question: What is an infinite ordinal?

Ordinal

a well-ordered set is a perfectly ordered set, that is, a set of which one of two different elements is always smaller than the other, and in which, moreover, there is no infinite descending set (although there may be infinite ascending sets); the subset has the smallest embryo.


Question: What is an infinite ordinal?

Ordinal

Ordinals can be used to denote the embryos of any well-ordered set (the smallest of them gets the symbol 0, the second smallest number 1, the next number 2, and so on), and can measure the “length” of the set to give the smallest ordinal not labeled any element of the set.


Question: What is an infinite ordinal?

Ordinal

This “length” is called the order type of a set.


Question: What is an infinite ordinal?

Ordinal

Each ordinal is defined by the set of ordinals preceding it; in fact, ordinals are most often defined today by “identifying” each ordinal with the set of ordinals preceding it.


Question: What is an infinite ordinal?

Ordinal

For example, ordinal 42 is the ordinal type of the set of ordinals smaller than that; this set includes ordinals from 0, i.e., the smallest ordinal up to ordinal 41, which is the immediate predecessor of 42, and can be identified with the set {0,1,2,... ,41}.


Question: What is an infinite ordinal?

Ordinal

Conversely, every set of ordinals (S) that is down-closed (that is, for which it applies that if any ordinal α belongs to the set S and if another ordinal β < α, then β also belongs to the set S) is itself an ordinal or can be identified with some ordinal.


Question: What is an infinite ordinal?

Ordinal

There are also infinite ordinals.


Question: What is an infinite ordinal?

Ordinal

The smallest infinite ordinal is ω, which is the order type of natural numbers (finite ordinals).


Question: What is an infinite ordinal?

Ordinal

It can be identified with the set of natural numbers, as the set of naturals is well-ordered, like every set of ordinals, and since it is down-closed, it can be identified with the associated ordinal, which is exactly the way in which ω is defined.


Question: What is an infinite ordinal?

Ordinal

Perhaps a clearer intuitive understanding of ordinals is obtained by studying some of the first of them.


Question: What is an infinite ordinal?

Ordinal

As noted above, they begin with the natural numbers 0, 1, 2, 3, 4, 5,... After all natural numbers comes the first infinite ordinal ω, followed by ω+1, ω+2, ω+3, and so on.


Question: What is an infinite ordinal?

Ordinal

(How the summation of infinite ordinals is defined is told below in the section the calculations of ordinals; at this point it is worth thinking of ω+1, ω+2, ω+3, and so on as just the names of the ordinals.)


Question: What is an infinite ordinal?

Ordinal

After all such comes ω·2 (i.e., ω+ω), ω·2+1, ω·2+2 and so on, after all such ω·3 and similarly later ω·4.


Question: What is an infinite ordinal?

Ordinal

The set of all ordinals formed in this way (ω·m</i>+<i data-parsoid='{"dsr":[8698,8703,2,2]}'>n, where m and n are natural numbers) must itself have an associated ordinal, and is ω2.


Question: What is an infinite ordinal?

Ordinal

Further we obtain equivalently ω3, ω4, and so on, as well as finally ωω, ω² and much later finally ε0, which can be represented as a limit in the form 0 = ω ω ω. {\ displaystyle\ epsilon _ {0} =\ omega ^ {\ omega ^ {\ omega ^ {\ omega ^ { ... }}}} [5] All of these are still relatively small, ordinal numbers of numberable sets.


Question: What is an infinite ordinal?

Ordinal

This can be continued infinitely far; in fact, “infinitely far” is exactly what ordinals are suitable for: every time one says “and so on” when enumerating ordinals, this saying defines a larger ordinal.


Question: What is an infinite ordinal?

Ordinal

The smallest overnumbering ordinal is the set of all numbering ordinals for which the notation ω1 is used.


Question: What is an infinite ordinal?

Ordinal

Definitions of Well-ordered sets In a well-ordered set, every non-empty subset has the smallest embryo.


Question: What is an infinite ordinal?

Ordinal

According to the axiom of choice, this is consistent with the fact that the set is perfectly arranged and that there is no infinite descending queue, which is perhaps more easily figurable.


Question: What is an infinite ordinal?

Ordinal

in the practice of well-order, the most important meaning lies in the fact that it allows transfinite induction: if from the fact that some property is on all predecessors of some embryo, it follows that it is also present on that embryo, then this property is on all embryos of the whole set, provided that the set is veryorganized.


Question: What is an infinite ordinal?

Ordinal

This also matters in computer science: if any method of calculation can veryarranged so that each step is followed by a “lower” step, then the calculation will not continue indefinitely.


Question: What is an infinite ordinal?

Ordinal

Two well-ordered sets A and B are said to be order-somorphic if one element of set A can be set to correspond to every element of set A, such that if the element x of set A is smaller than the germ y of set A', then there is also an embryo corresponding to the embryo x in set B smaller than the corresponding embryo y embryo among B and vice versa.


Question: What is an infinite ordinal?

Ordinal

Such an equivalence of embryos of different sets is called an order somorphism.


Question: What is an infinite ordinal?

Ordinal

Order-somorphic sets are similar in terms of order, and of them it can be said that their embryos are merely “named in different ways.”


Question: What is an infinite ordinal?

Ordinal

The order somorphism of sets is obviously an equivalence relation.


Question: What is an infinite ordinal?

Ordinal

If there is an order somorphism between two well-ordered sets, there is only one: this fact contributes to the justification of considering these sets essentially the same and looking for a “canonical” representative for the isomorphism type (class).


Question: What is an infinite ordinal?

Ordinal

Ordinals are precisely such “canonical” representatives of isomorphism types and can be used as means of marking every well-ordered set.


Question: What is an infinite ordinal?

Ordinal

Formally if partial order < is defined in set S and partial order <' among S', partially ordered sets (S, <) and (S, <') are order somorphisms if there exists a bijection f between them that preserves order, i.e., f (a) < f (b) if and only if a < b.


Question: What is an infinite ordinal?

Ordinal

Every well-ordered set (S, <) is ordinal somorphic to the set of ordinals smaller than the ordinal number expressing the order type of the set (S, <).


Question: What is an infinite ordinal?

Ordinal

Initially, ordinals were conceived as isomorph classes of well-ordered sets, that is, equivalence classes related to an equivalence relation denoting order somorphism.


Question: What is an infinite ordinal?

Ordinal

However, this involves the technical problem that this equivalence class is too broad to be treated as a set in the standard Zermelon—Frankel axioms formulation of mass doctrine.


Question: What is an infinite ordinal?

Ordinal

However, this difficulty is overcome.


Question: What is an infinite ordinal?

Ordinal

Ordinals can be said to be the order types of troops.


Question: What is an infinite ordinal?

Ordinal

[1] The definition of an ordinal as an equivalence class In the original definition of an ordinal number, found for example in Principia Mathematica, the order type of a well-ordered set is defined as the set of all the well-ordered orders that are with this ordinal somorphic; that is, ordinal numbers are equivalence classes formed by well-ordered sets.


Question: What is an infinite ordinal?

Ordinal

[6] However, this definition must be rejected in the axiomatization of mass-doctrine presented by Zermelon—Fraenkel, as these equivalence classes are too broad to be sets.


Question: What is an infinite ordinal?

Ordinal

However, the definition can still be used in type theory, as well as in Quine's axiomatic set doctrine (New Foundation) and formulations similar to it (where it also provides a rather surprising alternative solution to the Buralin—Fort paradox of greatest ordinal).


Question: What is an infinite ordinal?

Ordinal

Von Neumann's definition of ordinals Rather than defining ordinals as equivalence classes of well-ordered sets, they are most commonly defined today as certain well-ordered sets that (canonically) represent each class.


Question: What is an infinite ordinal?

Ordinal

Thus, every ordinal number is a well-ordered set, and every well-ordered set is ordinal somorphic to one and only one ordinal number.


Question: What is an infinite ordinal?

Ordinal

The standard definition presented by John von Neumann reads: every ordinal is a well-ordered set of all ordinals smaller than that.


Question: What is an infinite ordinal?

Ordinal

By mathematical notation, λ = [0, λ).


Question: What is an infinite ordinal?

Ordinal

[7] [8] Formally: A set'S is ordinal if and only if S is genuinely well-ordered with respect to the membership of the set, and every germ of S is also a subset of S.


Question: What is an infinite ordinal?

Ordinal

Also, natural numbers are ordinals of the mold of this definition.


Question: What is an infinite ordinal?

Ordinal

For example, the number 2 belongs to the set 4 = {0,1,2,3) and 2 is the set {0,1), so it is also a subset of the set {0,1,2,3}.


Question: What is an infinite ordinal?

Ordinal

By transfinite induction, it can be shown that every well-ordered set is ordinal somorphic to exactly one such ordinal, that is, between them there is an order-preserving bijective description.


Question: What is an infinite ordinal?

Ordinal

All embryos of each ordinal are ordinals too.


Question: What is an infinite ordinal?

Ordinal

If S and T are two ordinals, S is a germ of T if and only if S is a genuine subset of T.


Question: What is an infinite ordinal?

Ordinal

Furthermore, if they are two different ordinals, always either S is an embryo of T or an embryo of T S.


Question: What is an infinite ordinal?

Ordinal

In addition, each set of ordinals is veryarranged.


Question: What is an infinite ordinal?

Ordinal

This generalizes the fact that every subset of the set of natural numbers is veryarranged.


Question: What is an infinite ordinal?

Ordinal

It follows that every ordinal S is a set whose germ are all ordinals smaller than S. and only those.


Question: What is an infinite ordinal?

Ordinal

For example, each set of ordinals has the smallest upper bound, namely the supremum, which is the ordinal obtained by forming the union of all ordinals in the set.


Question: What is an infinite ordinal?

Ordinal

Under the Union axioma, this Union exists no matter how large a number is involved.


Question: What is an infinite ordinal?

Ordinal

The class of all ordinals is not a set.


Question: What is an infinite ordinal?

Ordinal

If it were a set, it could be shown that it is an ordinal and therefore a member of itself, which nonetheless contradicts the fact that membership is genuinely organized.


Question: What is an infinite ordinal?

Ordinal

This is the Buralin—Fort paradox.


Question: What is an infinite ordinal?

Ordinal

For the class of all ordinals, the notations “Ord”, “ON” or “∞” are used variously.


Question: What is an infinite ordinal?

Ordinal

An ordinal is finite if and only if its inverse order is also well-ordered, and this is the case if and only if every subset of it has a largest germ.


Question: What is an infinite ordinal?

Ordinal

Alternative definitions are currently presented as a definition of ordinal.


Question: What is an infinite ordinal?

Ordinal

For example, if one assumes a regularity axioma, the following conditions for a set x are equal: x is an ordinal x is a transitive set and belonging to a set is trichotomous with x, x is the transitive set perfectly arranged by subset relation x is the transitive set of transitive sets transitive set.


Question: What is an infinite ordinal?

Ordinal

These definitions cannot be used in non-well-founded mass doctrines.


Question: What is an infinite ordinal?

Ordinal

In set theories with groin elements, it must additionally be ensured that the definition does not allow grooves to occur in ordinals.


Question: What is an infinite ordinal?

Ordinal

Transphitic queue If α is a boundary ordinal and X is a set, the description from α to X can also be said to be the queue of X's embryos indexed by α.


Question: What is an infinite ordinal?

Ordinal

This concept, a transfinite queue, or ordinal-indexed queue, is a generalization of the notion of queue.


Question: What is an infinite ordinal?

Ordinal

An ordinary queue corresponds to the case where α = ω. Transfinite induction The principle of transfinite induction applies in every well-ordered set, but is primarily related to ordinals.


Question: What is an infinite ordinal?

Ordinal

According to it, If from the fact that any property holds among all those ordinals smaller than the given ordinal α, it follows that this property also holds on the ordinal α, this property is on all ordinals.


Question: What is an infinite ordinal?

Ordinal

Admittedly, if a theorem P (α) is true whenever P (β) is true for all β<α, then P (α) is true with all ordinals all α.


Question: What is an infinite ordinal?

Ordinal

Or more pragmatically: to prove that characteristic P is on all ordinals α, it suffices that it is already known to exist on all β<α.


Question: What is an infinite ordinal?

Ordinal

Transfinite recursion By transfinite induction can, not only prove theorems, but also define concepts.


Question: What is an infinite ordinal?

Ordinal

Such a definition is normally referred to as transinitic recursion.


Question: What is an infinite ordinal?

Ordinal

The proof that the concept thus obtained is well-defined is based on transfinite induction.


Question: What is an infinite ordinal?

Ordinal

Let any of the F ordinals be a definable class function.


Question: What is an infinite ordinal?

Ordinal

The idea is that when defining F (α) for any ordinal α, one can already assume that F (β) is already defined for all ordinals β < α, and thus an expression can be represented for F (α) using the values F (β) of the function.


Question: What is an infinite ordinal?

Ordinal

It follows by transfinite induction that there is one and only one function that implements the recursion formula with all ordinals up to and including α.


Question: What is an infinite ordinal?

Ordinal

As an example, consider the function F, which is defined so that F (α) is the smallest ordinal that does not belong to the set {F (β)}, that is, a set that includes all values F (β) when β < α. The definition assumes that F (β) is made known by this very process for defining F in stages.


Question: What is an infinite ordinal?

Ordinal

This might seem like a perimeter definition, but this is exactly what transfinite recursion allows.


Question: What is an infinite ordinal?

Ordinal

In fact, F (0) is also well defined, since there is no such ordinal that β < 0, and the set {F (β)} is empty.


Question: What is an infinite ordinal?

Ordinal

Thus, F (0) is 0, the most smallest ordinal.


Question: What is an infinite ordinal?

Ordinal

Now that F (0) is known, it can be stated that F (1) is the smallest ordinal not belonging to the set {F (0)} = {0}, and so on.


Question: What is an infinite ordinal?

Ordinal

Admittedly, this example is rather trivial, as it is precisely through transfinite induction that it turns out that F (α) = α with all ordinals α.


Question: What is an infinite ordinal?

Ordinal

Follower and boundary ordinal Every ordinal except zero has the smallest embryo, zero.


Question: What is an infinite ordinal?

Ordinal

It may also have the largest embryo, but may not be.


Question: What is an infinite ordinal?

Ordinal

For example, for ordinal 42 the largest embryo 41 and ordinal ω+6 the largest embryo ω+5.


Question: What is an infinite ordinal?

Ordinal

On the other hand, ωs do not have the largest embryo, it does not have the largest natural number.


Question: What is an infinite ordinal?

Ordinal

If the largest embryo of an ordinal is α, it is an embryo following α and is referred to as successor ordinal and notation α+1.


Question: What is an infinite ordinal?

Ordinal

When the ordinals are defined according to von Neumann, the successor of α is α {α} {\ displaystyle\ alpha\ cup\ {\ alpha\}}, as its embryos are embryos of α as well as α itself.


Question: What is an infinite ordinal?

Ordinal

[7] An ordinal that is non-zero and which is not a successor ordinal is called a boundary ordinal.


Question: What is an infinite ordinal?

Ordinal

The designation is based on the fact that the boundary ordinal really is, in a topological sense, the limit value of all ordinals smaller than that.


Question: What is an infinite ordinal?

Ordinal

When α ι | ι < γ {\ displaystyle\ langle\ alpha _ {\ iota} |\ iota <\ gamma\ rangle} is an ordinal-indexed queue indexed by the limit γ and the queue is ascending, that is α ι < α ρ {\ displaystyle\ alpha _ {\ iota} <\ alpha _ {\ rho}\!}


Question: What is an infinite ordinal?

Ordinal

whenever ι < ρ, {\ displaystyle\ iota <\ rho,\!}


Question: What is an infinite ordinal?

Ordinal

, its limit value is defined by the set {α ι | ι < γ}, {\ displaystyle\ {\ alpha _ {\ iota} |\ iota <\ gamma\},\!}


Question: What is an infinite ordinal?

Ordinal

as the smallest upper limit, that is, the smallest ordinal greater than any of its queues.


Question: What is an infinite ordinal?

Ordinal

There's always such a thing.


Question: What is an infinite ordinal?

Ordinal

In this sense, the boundary ordinal is the limit value of all ordinals indexed by itself.


Question: What is an infinite ordinal?

Ordinal

In other words, it is the supremum of the set of ordinals smaller than that.


Question: What is an infinite ordinal?

Ordinal

Another way to define a boundary ordinal is to say that α is a boundary ordinal if and only if there exists an ordinal smaller than α, and whenever there is an ordinal smaller than α, there is such an ordinal that <σ<α.


Question: What is an infinite ordinal?

Ordinal

So, for example, in the queue 0, 1, 2,..., ω, ω+1 ω there is a boundary ordinal, because for every ordinal smaller than that, in this case, therefore, for every natural number there exists another ordinal greater than that but still smaller than ω.


Question: What is an infinite ordinal?

Ordinal

Thus, the boundary ordinal is either zero, the successor of some (only one) other ordinal, or a boundary ordinal.


Question: What is an infinite ordinal?

Ordinal

This distinction is important because many definitions formulated by transfinite induction are based on it.


Question: What is an infinite ordinal?

Ordinal

Often when a function F is defined by transfinite induction for all ordinals, one first defines F (0) and F (α+1) assuming that F (α) is known, then defining F (δ) for boundary ordinals δ so that it is the limit of F (β) for all β<δ (either the limit of ordinals shown above in the sense or by some other concept of limit value if the values of F are not ordinals).


Question: What is an infinite ordinal?

Ordinal

In such definitions, the most interesting step is defining a function for a successor, not boundary ordinals.


Question: What is an infinite ordinal?

Ordinal

Such functions, especially if F is not decreasing and its values are ordinals, are said to be continuous.


Question: What is an infinite ordinal?

Ordinal

The aggregation of ordinals, multiplication, and potency increase are continuous as functions of their second argument.


Question: What is an infinite ordinal?

Ordinal

Indexing Ordinal Classes Every well-ordered set is order-somorphic with one and only one ordinal α {\ displaystyle\ alpha}.


Question: What is an infinite ordinal?

Ordinal

That is, its embryos can be indexed by those ordinals smaller than α {\ displaystyle\ alpha}.


Question: What is an infinite ordinal?

Ordinal

In particular, this also applies to every set of ordinals: any set of ordinals can be indexed by ordinals smaller than some α {\ displaystyle\ alpha}.


Question: What is an infinite ordinal?

Ordinal

In a slightly modified form, the same is true for classes of ordinals in other words, for collections of ordinals that are, in a sense, too broad to be sets: every class of ordinals can be indexed by ordinals (and if the class is unbounded by all ordinals in class, in this way a class bijection is formed with the class of all ordinals.)


Question: What is an infinite ordinal?

Ordinal

Thus, one can talk about the th element of the class γ {\ displaystyle\ gamma} (when it is agreed that the smallest of them is 0:s, the next 1:nen, etc.).


Question: What is an infinite ordinal?

Ordinal

Formally, it is defined by transfinite induction: an embryo of class γ {\ displaystyle\ gamma} :s is defined as the smallest embryo larger than β {\ displaystyle\ beta} :s embryo for all β < γ {\ displaystyle\ beta <\ gamma} (this naturally requires that the thing is already defined by all values β < γ {\ displaystyle\ beta <\ gamma}).


Question: What is an infinite ordinal?

Ordinal

This could be applied, for example, to the category of boundary ordinals: γ {\ displaystyle\ gamma} :s ordinal, which is either boundary orbital or zero, is ω ⋅ γ {\ displaystyle\ omega\ cdot\ gamma}.


Question: What is an infinite ordinal?

Ordinal

In the same way, additively indivisible ordinals, i.e. nonzero ordinals that are not sums of two ordinals genuinely smaller than themselves: γ {\ displaystyle\ gamma} the index is ω γ {\ displaystyle\ omega ^ {\ gamma}}.


Question: What is an infinite ordinal?

Ordinal

The indexing technique of ordinal classes is often useful when dealing with fixed points: for example γ {\ displaystyle\ gamma} :nnelle ordinal α {\ displaystyle\ alpha}, for which ω α = α {\ displaystyle\ omega ^ {\ alpha} =\ alpha}, the notation ε γ {\ displaystyle\ varepsilon _ {\ gamma}} is used.


Question: What is an infinite ordinal?

Ordinal

These are called epsilon figures.


Question: What is an infinite ordinal?

Ordinal

Closed unbounded sets and classes Ordinals of class C {\ displaystyle C} are said to be unbounded, i.e. cofinal, if for every ordinal α {\ displaystyle\ alpha} there exists such as C ordinal β {\ displaystyle\ beta}, that α < β {\ displaystyle\ alpha <\ beta}.


Question: What is an infinite ordinal?

Ordinal

In this case, the class must be a genuine class, for it cannot be a set.


Question: What is an infinite ordinal?

Ordinal

A class is said to be closed if the limit of the queue of ordinals belonging to the class is the class itself, or, conversely, if the indexing class function F {\ displaystyle F} is continuous in the sense that when δ {\ displaystyle\ delta} is the boundary ordinal, F (δ) {\ displaystyle F (\ delta)}, that is, the ordinal of class δ {\ displaystyle\ delta} :s is the limit value of the values F (γ) ({\ displaystyle F (\ gamma) (} when γ < δ {\ displaystyle\ gamma <\ delta}.


Question: What is an infinite ordinal?

Ordinal

This also implies the same as that the category is closed in the topological sense when order-topology is used as the topology.


Question: What is an infinite ordinal?

Ordinal

To avoid talking about topology in the case of genuine classes, the matter can be similarly expressed also so that the intersection of the class and any ordinal is closed in the ordinal topology of this ordinal.


Question: What is an infinite ordinal?

Ordinal

Of particular importance are those classes of ordinals that are both closed and unlimited and are sometimes said to your club.


Question: What is an infinite ordinal?

Ordinal

For example, the category of all boundary ordinals is closed and unbounded, since for every ordinal there is always the greater the boundary ordinal, and the limit ordinal of boundary ordinals is always the boundary ordinal.


Question: What is an infinite ordinal?

Ordinal

Also, the category of addiitively undistributed ordinals is closed and unbounded, as is ε c d o t {\ displaystyle\ varepsilon _ {c} dot}, and the category of all cardinals, in contrast, the category of cofinal cardinals the set is unbounded but not closed, while every finite set of ordinals is closed but not bounded.


Question: What is an infinite ordinal?

Ordinal

A class is stationary if its intersection with any closed unbounded class is non-empty.


Question: What is an infinite ordinal?

Ordinal

All classes containing some closed bounded class as their subclass are stationary and all stationary classes are unbounded, but there are also stationary classes that are not closed or do not have a closed unbounded subclass; such is for example, the category of all boundary ordinals with numberable cofinality.


Question: What is an infinite ordinal?

Ordinal

Since the intersection of two closed unbounded classes is closed and unbounded, the intersection of the stationary class and the closed unbounded class is stationary.


Question: What is an infinite ordinal?

Ordinal

However, the intersection of two stationary classes can also be empty; this is the case, for example, when forming a cut from the class of all ordinals whose cofinality is ω and the class of all ordinals whose cofinality is overnumberable.


Question: What is an infinite ordinal?

Ordinal

Instead of formulating these definitions for (genuine) classes of ordinals, they can also be formulated for classes of ordinals smaller than some given ordinal α {\ displaystyle\ alpha}.


Question: What is an infinite ordinal?

Ordinal

A subset of the boundary ordinal α {\ displaystyle\ alpha} is said unbounded (i.e., cofinal) among α {\ displaystyle\ alpha} if every α {\ displaystyle\ alpha} smaller than ordinal is smaller than any ordinal of that set.


Question: What is an infinite ordinal?

Ordinal

More generally, subsets of any ordinal α {\ displaystyle\ alpha} can be said to be cofinal in α {\ displaystyle\ alpha} if every ordinal, yoa is less than α {\ displaystyle\ alpha }, is less than or equal to any ordinal belonging to this set.


Question: What is an infinite ordinal?

Ordinal

A subset is said to be closed in α {\ displaystyle\ alpha} if it is closed in the order topology of α {\ displaystyle\ alpha}, that is, if the limit value of ordinals belonging to the set is either contained in this set or is α {\ displaystyle\ alpha} itself.


Question: What is an infinite ordinal?

Ordinal

Ordinals Calculations Ordinals have three general calculations: aggregation, multiplication, and potency increase.


Question: What is an infinite ordinal?

Ordinal

Each of them can be defined in substantially two ways: either by constructing an explicitly defined well-ordered set presenting this calculation, or by transfinite recursion.


Question: What is an infinite ordinal?

Ordinal

Based on the definition of Neumann ordinal, the calculations can be defined as: [1] Summary α + 0 = α + (β+1) = (α + β) + 1 α + β = γ < β (α + β) {\ displaystyle\ alpha +\ beta =\ cup _ {\ gamma <\ beta} (\ alpha +\ beta)} when β is the boundary ordinal Multiplication α · 0 = 0 α · (β+1) = α · β + α ⋅ β = γ < β (α ⋅ γ) {\ displaystyle\ alpha\ cdot\ beta =\ cup _ {\ gamma <\ beta} (\ alpha\ cdot\ cdot)\ gamma)} when β is the boundary ordinal to the potency increase α0 = 1 0β = 0 when β is the boundary ordinal and > 0 αβ+1 = αβ · α β = g a m m a < β α γ {\ displaystyle\ alpha ^ {\ beta } =\ cup _ {gamma<\ beta}\ alpha ^ {\ gamma}}, jos α > 0 and β is the boundary ordinal.


Question: What is an infinite ordinal?

Ordinal

Each ordinal can be represented in a uniquely defined manner in Cantor normal form.


Question: What is an infinite ordinal?

Ordinal

In it, each ordinal is represented as a finite sum of ordinal potencies of ωs.


Question: What is an infinite ordinal?

Ordinal

However, it cannot be used as the basis for the general mnotation of ordinals, since, for example, ε0 should be defined by means of itself: ε0 = ωε0 However, these do not follow all laws of calculation valid by natural and also real numbers.


Question: What is an infinite ordinal?

Ordinal

Thus, the following rules do not apply for ordinal calculations: [1] The interchange law of aggregation: α + β usually does not equal β + α The law of exchange of multiplication: α · β usually does not equal β · α The right-hand stratification law: for example (ω+1) · 2 is not equal to ω · 2 + 2 Input potency rule: (ω·2) 2 is not equal to ω2 · 4.


Question: What is an infinite ordinal?

Ordinal

For ordinals, so-called “natural” arithmetic operations can also be defined.


Question: What is an infinite ordinal?

Ordinal

These are interchangeable, but not continuous.


Question: What is an infinite ordinal?

Ordinal

Interpreted as nimbers, ordinals can also perform arithmetic calculations of nimbers.


Question: What is an infinite ordinal?

Ordinal

Ordinals and Cardinals Initial ordinals of cardinals Each ordinal is accompanied by a certain cardinal number, its cardinality.


Question: What is an infinite ordinal?

Ordinal

If there is a bijection between two ordinals (for example ω=1+ω and ω+1>ω), they are joined by the same cardinal.


Question: What is an infinite ordinal?

Ordinal

Every well-ordered ordinal that has some ordinal as its ordinal type has the same cardinality as this ordinal.


Question: What is an infinite ordinal?

Ordinal

The smallest ordinal associated with the given cardinal is said to be the initial ordinal of this cardinal.


Question: What is an infinite ordinal?

Ordinal

Every finite ordinal, or natural number, is initial, and no other ordinal has the same cardinality.


Question: What is an infinite ordinal?

Ordinal

But most infinite ordinals are not inital, as many infinite ordinals are accompanied by the same cardinality.


Question: What is an infinite ordinal?

Ordinal

The axiom of choice coincides with the fact that each set can be veryarranged, that is, every cardinal has an initial ordinal.


Question: What is an infinite ordinal?

Ordinal

In theories that include the axiom of choice, the cardinal number of each set has an initial ordinal.


Question: What is an infinite ordinal?

Ordinal

Cardinals can therefore be conceived as special cases of ordinals by identifying them with their initial ordinals, since each set is equally mighty with some (defined by Neumann) orbital.


Question: What is an infinite ordinal?

Ordinal

[1] Thus, an ordinal is cardinal if and only if it is not equally mighty with any ordinal smaller than itself.


Question: What is an infinite ordinal?

Ordinal

[1] In general, for the α:nth infinite ordinal, the notation ω α {\ displaystyle\ omega _ {\ alpha}} is used.


Question: What is an infinite ordinal?

Ordinal

Its cardinality is denoted by α {\ displaystyle\ aleph _ {\ alpha}}.


Question: What is an infinite ordinal?

Ordinal

For example, the cardinality of ω0, or ωs, is 0 {\ displaystyle\ aleph _ {0}}, which is also the cardinality of ω2 and ε0, as these too are numbering ordinals.


Question: What is an infinite ordinal?

Ordinal

Thus, if the axiom of choice is accepted, ω can be identified with 0 {\ displaystyle\ aleph _ {0}}, except that the notation 0 {\ displaystyle\ aleph _ {0}} is used when there is a question of cardinals and notation ω when there is a question of ordinals.


Question: What is an infinite ordinal?

Ordinal

(Different notations are used because, for example, 0 2 {\ displaystyle\ aleph _ {0} ^ {2}} = 0 {\ displaystyle\ aleph _ {0}} as cardinal, but ω 2 > ω {\ displaystyle\ omega ^ {2} >\ omega} as ordinal).


Question: What is an infinite ordinal?

Ordinal

Similarly ω 1 {\ displaystyle\ omega _ {1}} is the smallest overnumberable ordinal (to show that one exists, one can consider equivalence classes of well-orders of natural numbers: each such defines of the ordinal to be numbered, and ω 1 {\ displaystyle\ omega _ {1}} is the ordinal type of this set), ω 2 {\ displaystyle\ omega _ {2}} is the smallest ordinal with cardinality greater than 1 {\ displaystyle\ aleph _ {1}} and so on, and ω o m e g a {\ displaystyle\ omega _ {o} mega} is the ordinals ω n {\ displaystyle\ omega _ {n}} limit value when the values of n are natural numbers (every limit of cardinals is cardinal, and thus this limit value is actually the first cardinal greater than ω n {\ displaystyle\ omega _ {n}}).


Question: What is an infinite ordinal?

Ordinal

Cantor used cardinality to classify ordinals.


Question: What is an infinite ordinal?

Ordinal

He referred to natural numbers as the first reading class, ordinals with cardinality 0 {\ displaystyle\ aleph _ {0}}, i.e., numberingly infinite ordinals as the second reading class, and generally ordinals with cardinality cardinality n − 2 {\ displaystyle\ aleph _ {n-2}}, the nth reading class.


Question: What is an infinite ordinal?

Ordinal

[9] Cofinality The cofinality of the ordinal α {\ displaystyle\ alpha} is the smallest ordinal δ {\ displaystyle\ delta}, which is the cofinal subset of some α {\ displaystyle\ alpha} order type.


Question: What is an infinite ordinal?

Ordinal

In some works, the concept of cofinality is defined or used only for boundary ordinals.


Question: What is an infinite ordinal?

Ordinal

The cofinality of the set of ordinals, or any other well-ordered set, is the cofinality of the ordinal type of this set.


Question: What is an infinite ordinal?

Ordinal

Thus, for every boundary ordinal α {\ displaystyle\ alpha}, there exists a δ {\ displaystyle\ delta} -indexed genuinely increasing queue with a limit of α {\ displaystyle\ alpha}.


Question: What is an infinite ordinal?

Ordinal

For example, the cofinality of ω²is ω because the limit value of the queue ω·m, where the values of m are all natural numbers, is ω, and in general, the cofinality of every numbering boundary ordinal is ω.


Question: What is an infinite ordinal?

Ordinal

Also, the cofinality of the overnumbered boundary ordinal may be ω, for example the ordinal ω {\ displaystyle\ omega _ {\ omega}}, but such may also have overnumbering cofinality.


Question: What is an infinite ordinal?

Ordinal

The cofinality of the ordinal 0 is 0, the cofinality of each successor ordinal is 1.


Question: What is an infinite ordinal?

Ordinal

The cofinality of each boundary ordinal is at least ω {\ displaystyle\ omega}.


Question: What is an infinite ordinal?

Ordinal

An ordinal whose cofinality is equal to the ordinal itself is said to be regular, and all such are initial ordinals.


Question: What is an infinite ordinal?

Ordinal

The limit of regular ordinals is the limit of the initial ordinals and as such the initial, though neither necessarily nor general regular.


Question: What is an infinite ordinal?

Ordinal

If the selection axiom is accepted, it follows that ω α + 1 {\ displaystyle\ omega _ {\ alpha +1}} for all α, and further that ordinals 0, 1, ω {\ displaystyle\ omega}, ω 1 {\ displaystyle\ omega _ {1}} and ω 2 {\ displaystyle\ omega _ {2}} are regular, while, 2, 3, ω ω {\ displaystyle\ omega _ {\ omega}} and ω·2 are examples of initial of orbitals that are not regular.


Question: What is an infinite ordinal?

Ordinal

The cofinality of each ordinal α is a regular ordinal; that is, the cofinality of the cofinality of α is the same as the cofinality of α.


Question: What is an infinite ordinal?

Ordinal

In other words, the cofinality operation is idempotent.


Question: What is an infinite ordinal?

Ordinal

Some “large” numbering ordinals An example of a “very large” infinite but still numbering ordinal can be cited ε0.


Question: What is an infinite ordinal?

Ordinal

It is the smallest ordinal that implements the equation ω α = α {\ displaystyle\ omega ^ {\ alpha} =\ alpha}, and at the same time the queue 0, 1, ω {\ displaystyle\ omega}, ω {\ displaystyle\ omega ^ {\ omega}}, ω ω ω {\ displaystyle\ omega ^ {\ omega ^ {\ omega}}, etc.


Question: What is an infinite ordinal?

Ordinal

Many ordinals can be defined similarly as the fixpoints of some ordinal function.


Question: What is an infinite ordinal?

Ordinal

Thus, ι {\ displaystyle\ iota} for an ordinal for which ω α = α {\ displaystyle\ omega ^ {\ alpha} =\ alpha} is denoted ε ι {\ displaystyle\ varepsilon _ {\ iota}}, and in the same way one could try to find ι {\ displaystyle\ iota} :s such ordinal for which ε α = α {\ displaystyle\ varepsilon _ {\ alpha} =\ alpha} and so on.


Question: What is an infinite ordinal?

Ordinal

This could be attempted systematically, but however tahasa ordinals are defined and constructed, there is always an ordinal which is just above all ordinals constructed in this way Perhaps the most important ordinal which, in this way, limits the constructive system, is Churchin—Kleene ordinal ω 1 C K {\ displaystyle\ omega _ {1} ^ {\ mathrm {CK}}} (which is numberable despite appearing in its name ω 1 {\ displaystyle\ omega _ {1}} ).


Question: What is an infinite ordinal?

Ordinal

It is the smallest ordinal that, in a sense, cannot possibly be represented as a countable function.


Question: What is an infinite ordinal?

Ordinal

Also, below ω 1 C K {\ displaystyle\ omega _ {1} ^ {\ mathrm {CK}}}, however, one can define remarkably broad ordinals that measure the “proof theoretical strength” of some formal systems; thus for example ε 0 {\ displaystyle\ varepsilon _ {0}} measures the strength of arithmetics based on Peano axioms.


Question: What is an infinite ordinal?

Ordinal

One can also define ordinals that are larger than the Churchin—Kleene ordinal and which they too have significance in several fields of logic.


Question: What is an infinite ordinal?

Ordinal

Topology and ordinals Every ordinal number can be formed into a topological space by equipping it with an ordinal topology.


Question: What is an infinite ordinal?

Ordinal

This topology is discrete if and only if the ordinal is a numbering cardinal, that is, at most ω.


Question: What is an infinite ordinal?

Ordinal

A subset of ω+1 of a set is open in order topology if and only if it either is cofinite (i.e., its complement is a finite set) or ω is not included in it as an embryo.


Question: What is an infinite ordinal?

Ordinal

Downward closed sets of ordinals A subset B of an ordered set is downwardly closed if all of A's embryos smaller than any embryo of B are also embryos of B.


Question: What is an infinite ordinal?

Ordinal

If any set of ordinals is downwardly bounded, it is the ordinal itself, the smallest ordinal not belonging to that set.


Question: What is an infinite ordinal?

Ordinal

Examples: The set of ordinals smaller than ordinal 3 is 3 = {0, 1, 2], the smallest ordinal not less than 3.


Question: What is an infinite ordinal?

Ordinal

The set of finite ordinals is infinite, and the smallest infinite ordinal is ω. The set of ordinals to be numbered is overnumberable, and the smallest overnumbering ordinal is ω1.


Question: What is an infinite ordinal?

Ordinal

Sources Citation error, replace with {{cite book| title = The Book of Numbers| number = Cantor's Ordinal Numbers| location = New York| publisher = Springer-Verlag| year = 1996| pages = 266—267, 274}} Citation error, replace with {{cite book| author = Hamilton, A. G.| title =, Numbers, Sets and Axioms: the Apparatus of Mathematics| number = Ch.


Question: What is an infinite ordinal?

Ordinal

6, Ordinal and cardinal numbers| publisher = Cambridge University Press| year = 1982| id = ISBN 0-521-24509-5}} Citation error, replace with {{cite book| author = Hazewinkel, Michiel (ed.


Question: What is an infinite ordinal?

Ordinal

) | title = Encyclopedia of Mathematics| number = Ordinal number| publisher = Springer| year = 2001| id = ISBN 978-1-55608-010-4| format = [online version]}} Citation error, replace with {{cite book| author = Kanamori, Akihiro (letter)


Question: What is an infinite ordinal?

Ordinal

& Irvine, Andrew & Woods, Joh H. (ed) | title = The Handbook of the Philosophy of Science, vol.


Question: What is an infinite ordinal?

Ordinal

4| number = Set Theory from Cantor to Cohen| publisher = Cambridge University Press| year =}} Citation error, replace with {{cite book| author = Sierpiński, W.| title = Cardinal and Ordinal Numbers (2nd ed) | location = Warsaw | publisher = Państwowe Wydawnictwo Naukowe| year = 1965}} error, replace with {{cite book| author = Suppes, Patrick| title = Axiomatic Set Theory| publisher = D.Van Nostrand Company Inc.| year = 1960| id = ISBN 0-486-61630-4}} Citation error, replace with {{cite book| author = von Neumann, John & van Heijenoort, Jean (ed.


Question: What is an infinite ordinal?

Ordinal

) | title = From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931| number = On the introduction of transfinite numbers| publisher = Harvard University Press| year = 2002 (3rd edition; originally 1923) | id = ISBN 0-674-32449-8| format = [online version] | pages = 346-354}} Citation error, replace with {{cite book| author = Devlin, Keith| title = The Joy of Sets, Fundamentals of Contemporary Set Theory, 2nd ed| location =, U.S.A.| publisher = Springer-Verlag, Inc.| year = 1992| id = ISBN 0-387-94094-4| language = Transclusion error: {{En}} is only for use in File namespace.


Question: What is an infinite ordinal?

Ordinal

Use {{lang-en}} or {{en icon}} instead.}


Question: What is an infinite ordinal?

Ordinal

Check date values in: |accessdate= (help) References on the topic elsewhere MathWorld | urlname=ordinalNumber| title=Ordinal Number}} Category:Mass Doctrine


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

Ancient Greek technique refers to the technique of ancient Greece and the Hellenistic world under its influence during the period beginning around 500 BCE.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

and then gradually merging with the technique of ancient Rome.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

Ancient Greek Society and Technology The attitude of the Ancient Greeks towards technology was duplicitous.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

In production and commerce, the technique was applied abundantly.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

Well-known are amphoras developed for the transport of oils and wines, and ceramics made especially intricately.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

In farming, a plow with an iron tip was introduced.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

Despite this, the most respected citizens were theorists who were fascinated by the development of various ideas, theorems and mathematics.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

Instead, applying these to practice did little to intrigue them.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

The reason was largely the structure of Greek society, where production work belonged to slaves.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

Natural philosophy, sometimes considered a precursor to science, originated in Greece during the year 1000 BCE.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

around.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

It was then that mathematics, geometry and the property of fire began to be considered.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

Aristotle is well known for this, but he too exploited Thaleen (600 BCE)


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

and the earlier ideas of other pre-democratics.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

Pythagorean (582—500 BCE)


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

the school developed the basics of geometry, after which came to Euclide (about 300 BCE.)


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

works.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

Demokritos (circa 470 BCE)


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

contemplated the atomic essence of matter.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

[1] [2] Trade, shipping and warfare were characterised by the culture of Greece and the Aegean Sea, as well as with the peoples of the North, which is marked by jewellery made from Baltic amber found in Greece.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

Coins were minted to promote commerce.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

The war success against the Persians was largely based on the fleet.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

[3] Thales taught seafarers orienteering using stars, and Thalee's pupil Anaksimandros drew the first then known map of the world.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

In the 7th century, the first threeweights were built.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

Following the conquests of Alexander the Great, a unified Hellenic culture emerged with direct effects on later Western culture.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

[4] Hellenistic period Technological progress was at its fastest in the Hellenistic period.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

Heron Alexandrian (circa 10—70) developed the first steam engines (a rotary steam turbine), cranes, a rectangular measuring device called a dioptra, presses, a fountain and many other devices.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

[5] Records of these have survived in Heron's own writings, such as Pneumatiká, which describes, among other things, the organ and the fire syringe.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

In Automatopoietike (Finnish. 'Automatic manufacture') multiplies moving miniature models and dolls.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

The Mekhaniká work describes pillars, fleece arrangements, winch, screw, lever and wedge.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

[6] With their constant warfare, the Greeks developed a variety of war technologies, known of which is the incinerator, invented by Archimedes.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

The most famous of Archimedes's machines is the Archimedes screw.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

[7] Galenos (130—210) developed the principles of medicine that were applied for a thousand years.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

Calculation methods were used, among other things, to calculate the calibers of throwing machines, but this formula was based more on experience than on understanding physical majors.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

While advances in the development of mechanical machinery were, these were not much used to raise people's standard of living.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

[8] However, the importance of Greek science and engineering to later developments was significant as theories and skills moved through the Romans to Europe.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

[9] The organization of the Library of Alexandria in Alexandria formed a concentration of science and engineering when King Ptolemy I established a library there in the 3rd century BCE.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

An institution dedicated to the creation of knowledge was established in connection with the Library of Alexandria Museion.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

Ptolemy III ordered everyone visiting the city to hand over their books to be copied.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

The library's first attendant, Zenodotos Ephesus, arranged the books in alphabetical order.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

In terms of information management, alphabetizing was an important invention.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

In the Chinese writing system, the organization of information is done according to the number of horizontal and vertical lines contained in the writing characters.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

Kallimakhos Kyreneläinen, Zenodotos' successor, began cataloguing the library and divided the authors into dozens of subjects, of which an alphabetized list was drawn up.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

He thus initiated the development of indexing methods.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

[10] See also Ancient Roman Engineering Ancient Science Sources Citation error, replace with {{cite book| author = Adams, James| title = Engineer's World| others = Translated by Kimmo Pietian | publisher = Art House| year = 1995| id = ISBN 951-884-163-2}} Citation error, replace with {{cite book| author = Kero, Reino & Kujanen, Hannu (ed.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

) | title = From Stone Axe to Information Technology: the Social History of Technology from the Stone Age to the Today| number = Technology and Science in the Middle East and Greece (Eng.


Question: What was developed as a successor to the Antikythera machine?

Ancient Greek technique

Miika Huhta) | publisher = University of Turku, Department of History| year = 1989| id = ISBN 951-880-296-3| pages = 40-54}} Citation error, replace with {{cite book| author = Mahajan, Shobhit| title = Inventions before and now| others = Translated by Eetu Hiltunen| publisher = Tandem Vermblag Gmblag h| year = 2008| id = ISBN 978-3-8331-4819-4}} Citation error, replace with {{cite book| author = Wiio, Antti| title = When Information Technology Changed the World, Revolutionary IT Inventions from the Stone Age to the Today| location = Espoo| publisher = Tecnologos Oy/Delta Books | year = 2007| id = ISBN 978-951-961-96141-2-0| accessdate = 3.3.2012}} References Category:Ancient Greece Category:History of Technology


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

The period of Swedish rule in Finland is a period in Finnish history when Finland was part of the Kingdom of Sweden.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

The period of Swedish rule was not a continuous era and its inception was a long-term process involving ambiguity.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

The period of Swedish rule ended as a result of the Finnish War on 17 September 1809.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

The period of Swedish rule began with the crusades to Finland.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

With the new powers, the Finnish Grand Peasants of Moisioi were forced to give up their positions and their estates fell into the ownership of the Swedish Crown and the Catholic Church.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

[1] In connection with the crusades, as a result of the colonization of the Finnish coast by the Swedes, its Finnish inhabitants essentially lost their fishing rights in the area.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

Conflicts between newcomers and Finns also arose over farming rights.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

[2] Swedish became the language of the nobility and the upper bourgeoisie, and priests and scholars were obliged to use it as their language.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

During Swedish rule, the Finnish language was in danger of being lost, because it could not be used with the authorities, and newspapers were not written or produced enough non-fiction.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

Throughout the countries' connection, Finns were considered a separate people and, due to the language issue, did not have the same rights as the Swedes.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

Swedish kings visited Finland rarely and Finns were considered more primitive than Swedes or Germans and their language inferior.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

[3] During Swedish rule, Finland became part of Western Christendom and the cultural, social and economic order of Western Europe, on the basis of which, among other things, a market economy, constitutional statesmanship and legalistic principles tied to legalistic principles of law emerged. administration.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

The Finnish language and some features of Finnish antiquity and culture remained despite conscious Swedishisation during Swedish rule, although they changed when adapting to new circumstances.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

[3] [4] Finland's location as the frontier country of the kingdom brought numerous wars and raiding expeditions to the region.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

Swedish Crown Prince Adolf Fredrik stated in 1746 that Finland was Sweden's warehouse and protective wall.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

[3] During this period the Swedish Empire has also been referred to as Swedish-Finland.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

The term was coined retrospectively to describe the kingdom which included the main parts of present-day Sweden and Finland.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

Historian Peter Englund has stated that Finland during Swedish rule was not part or province of confederation but “... the eastern half of that kingdom which was practically destroyed in 1809, after which both parts went their separate ways.” In particular, Englund considers the Swedish Great Power period to be the common “property” of present-day Sweden and Finland, since the rise to superpower would have been impossible for a parched state without the resources of the eastern part of the empire.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

[5] Early stages of Swedish rule The beginning of the period of Swedish rule was a long-standing process involving different regions of Finland at varying degrees at different times.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

Finland, or Sweden, cannot be spoken of at this stage as a unified administrative entity.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

The Kingdom of Sweden spread to Finland in the midst of its own process of birth and consolidation.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

As recently as the beginning of the 13th century, the law of West Götaland constantly referred to the East Götans and the Uplantians as foreigners.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

[6] Sweden's first written output, the Eerikinkronika, written in the 1320s, also speaks not of the Swedes but, for example, the Sveans, the Gothenites and the Uplantians.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

[7] The Kingdom of Sweden can be said to have arisen around the year 1350.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

[6] The early stages of the period of Swedish rule are fundamentally related to the missionary work of the Catholic Church in Finland.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

The northern Baltic region was a pagan area from the point of view of the Catholic Church and its goal was the conversion of the region to Catholicism.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

In practice, the aim of the church was driven by the kings who professed the Catholic faith in the region, as well as by the crusaders, who were allowed in the name of Christianity to conquer new territories under their control.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

Crusades to Finland were made at least by Swedes and Danes, possibly including Germans and Norwegians.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

[8] [9] [10] Orthodox Novgorod also fought a typical war against the Finnish tribes.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

Since at least 1042, the Hämäläs, or more commonly the Western Finns, had been waging almost constant war against Novgorod and the Baltic-Finnish peoples of the Karelian region, such as the Vatyans and Ingerians.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

These conflicts contributed to the Swedes” chances of conquering territories from Finland.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

A contemporaneous description of the Baltic conversion, Henry's Livonian Chronicle, gives a bloody picture of the conversion work of late 12th century and early 13th century German Knights and Danes.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

Christianity was well established in Sweden by the 12th century and its nascent kingdom sought to compete with Germany and Denmark for influence in the Baltics and around the Gulf of Finland by fighting the pagans and conquering lands from these.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

However, the Baltic war expedition ended for the Swedes in 1220 with the destruction of the Swedish forces, including the earl and bishop, by the Islanders, and the fortress they had established at Lihula.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

[11] [12] The first historical reference to the rule of Swedes in Finland is the earliest legend of St. Eerik, dating from the 1270s, according to which King Eerik the Saint and the English bishop Henry made the so-called First Crusade to Finland in the 1150s.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

Legend tells that the king returned after an expedition to Sweden with the bishop remaining to arrange the ecclesiastical conditions in Finland.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

However, the war expedition is not considered a real event today.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

[13] The following reference in historical sources to Swedish influence in Finland is the statement in the Lübeck chronicle of 1266: “Duke Birger forced Finland to belong to Sweden”.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

[14] Possibly the same event is dealt with by the Eerikinchronicle, written in the 1320s, which describes the Uplantians' expedition against the Hazy people.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

The war expedition in question has been considered to have possibly been the Second Crusade to Finland made between 1249 and 1250.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

According to the Eerikinchronika, as a result of the crusade, construction of a castle was started in Häme.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

The Swedes also began colonization of the Finnish coast in stages around the time of the Crusades, starting in Åland and Southwest Finland.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

The colonization of the Finnish coast by Christian people was a key means of spreading the faith.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

Of these emigrants, Finnish-Swedes emerged over time.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

[15] The Bishop of Turku is first mentioned in connection with the bishops of Sweden on 11 February 1253 in connection with the exchange of land between the Earl of Birger and Gudheim Abbey.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

[16] The Swedish Church spread to Finland practically only in the 14th century.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

[17] Since about the 12th century, the Catholic Church had operated independently in Southwest Finland.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

The influence of Western and Eastern Christianity had spread to Finland even earlier in the prehistoric period, with indications such as tomb finds made in archaeological excavations.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

The Eerikinchronicle recounts the destruction of Sigtuna by the Karelians in 1187, the largest and most important city in Sweden at that time.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

The Karelians made a war expedition to Sweden also in 1257.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

[18] In 1259 Pope Alexander IV approved the marriage of King Valdemar of Sweden and Sophia, granddaughter of King Christopher of Denmark, so that Sweden would be better and more effectively able to counter the constant attacks on the kingdom's villages and churches by pagans living nearby.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

[19] In spite of this, even in 1275 the Christians of the Swedish Empire were tormented by the pagans of Karelia and Lapland.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

[20] The first chronicle of Novgorod states that Dmitry Alexandrovich conquered Karelia in 1278.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

The pagan Karelians had been, up to that point, trading partners for Gotland, among others.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

[21] Regardless of trade links, according to Eerikinkronika, the Uplantians made the so-called Third Crusade into Finland in 1293, conquering Karelia with 14 betrothals.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

[22] Birger Maununpoika's letter of authorisation of 4 March 1295 to Lübeck and other Baltic trading cities mentions as the motive for the conversion of Karelians the long long practice of banditry and looting by the Karelians and the taking captive of Swedes and other visitors to the Baltic Sea and torturing.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

[23] The first historical reference of the Catholic Church to Swedish rule outside Finland dates back to about 1284.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

In this act of Maunu Ladonluko on the general peace of land, Pentti Birgerinpoika is referred to as the Duke of Finland, that is, present-day Southwest Finland.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

[24] However, he may have never visited Finland.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

Despite the expansion of Swedish power, the conflict between Hämäläs and Novgorod lasted for hundreds of years.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

The First Chronicle of Novgorod states that Novgorod made a war expedition against the Hazorites still in 1292.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

Conflicts between Finns and Swedes after the Crusades, the colonization of the Swedes in their wake also led to conflicts between Finns and Swedes.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

The Ostrobothian stories tell of the need to help the colonists, while other stories describe Sweden as mainly criminals and evildoers who came to the country to wage war against the Finns.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

[25] In the 16th century, Mikael Agricola wrote that pagan Finns had harassed Swedish colonists, for which reason they had received help from their relatives in Sweden.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

[25] The original Finnish settlement of Kimito Island completely disappeared from the site after the Swedes entered the area, which may contribute to the conflict between the groups.


Question: When did Sweden rule Finland?

Period of Swedish rule in Finland

[26] Conflicts between Finns and colonists also arose regarding the rights to use the territories.