Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

As the conflict intensified, the king convened in 1563 the Reichstag to condemn John to death.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

The development led to the siege of Turku Castle in the summer of 1563.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

After skirmishes and bombardments, the castle surrendered on 12 August 1563, after which the splendour of the castle was destroyed, the pair of dukes were captured and taken to Sweden for imprisonment.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

In the 1560s, Swedish foreign policy was marked by warfare in the Baltics against Poland.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

In addition, Eerik XIV was mired in war against Denmark-Norway and Lübeck.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

This necessitated good relations with the East: the concentration of military forces elsewhere required good relations with the direction of Moscow.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Eerik XIV's term ended in 1568 with the rise of the nobility against the king.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

This time, Finland's top brass were also on the side of the rebels, as much the old supporters of Juhana as the creditmen of Eerik.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Turku Castle came into the possession of the rebels.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Eerik spent the next few years in captivity until he was sentenced to death and poisoned in 1577.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Long hatred during Juhana III's kingship, border skirmishes on the border of Käkisalmi County led to yet another war (1570—1595) known as the Nordic Five Thirteenth Years' War.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

For the most part, it was a brutal, mutual guerrilla war.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

In the early stages, however, warfare was organized and in 1573—1577 a time of armistice lived on the Finnish front.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Halfway through the decade, however, peasant rustles began: Karelians invaded Oulujärvi and Iijoki, southern Finland went beyond the Gulf of Finland.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

In the fold of the decade, organised warfare was triggered again with the invasion of Narva (1579) and the conquest of Käkisalmi in 1580.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Narva was overrun the following year, with subsequent negotiations leading to an intermediate peace lasting until 1590.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Despite the intermediate peace, guerrilla warfare continued on both sides of the eastern border, leading to the widespread desolation of Kainuu and northern Ostrobothnia in the 1580s.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Led by Pekka Vesainen, the Ostrobothians made retaliatory expeditions to Viena Karelia.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

The war ended with the Peace Treaty of Full Self in 1595.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

A new dispute over the throne upon the death of John III in 1592 left the throne without a holder.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

There were two successor candidates: Sigismund and Duke Charles.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

The kingship issue was also intertwined with church politics: in Counter-Reformation Europe, Sigismund was Catholic, making his position even more problematic in Lutheran Sweden.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

In 1593 Sigismund arrived in the country to be crowned.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

The ruler's assurance he issued in the autumn of the following year.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

In this case, he also adopted the Lutheran line of the enactment and ecclesiastical assembly of Uppsala (1593).


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

With the crowning of Sigismund, a personal union was created between Sweden and Poland that lasted some years; as King of Poland, Sigismund always remained until 1632.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

After the king's return to Poland, rose to the nobility among disagreements over Charles Duke's position as registrar.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

In this dispute, Klaus Fleming, the commandholder of Finland, sided with the king.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

In the Riksdag of Arboga of 1597, however, Charles was named regent.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

In this case, he also declared his opponents, above all Klaus Fleming, rebels.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Harshing justice Generally in Europe At the beginning of the New Age, in the 16th and 17th centuries, secular rights began to use the so-called “Law of Moses”, that is, some passages from the Bible.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Under the law of Moses, witchcraft and magic were prohibited and, among other things, blasphemy, swearing, infringement against one's parents, perjury, manslaughter, interest railing, false testimony, and several sexual offences could be sentenced to death.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

The Law of Moses became a general legal instruction in the Swedish Empire in 1608 and remained in force right up to the new Reich Law of 1734.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

However, the law was generally not applied in its full harshness and death sentences were commuted in about half of the cases.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

[36] [37] 16th century peasant unrest and mace war Peasant unrest increased during the reign of Gustav Vasa, inter alia, due to increased tax burdens and, on the other hand, the struggle for power of his sons.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

The culmination of the peasant unrest was the bludgeon war in 1596—1597.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

It ended bloody with the army defeating the Ostrobothians and Savonian peasants armed with mace, spears and bows.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

The revolt was directed against the nobility who exercised power in Finland and Klaus Fleming in particular.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

The mutineers sought help from Charles Duke, who was pursuing the Swedish crown.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

As causes of the rebellion, contemporary research attributes the war burden of the rap war, economic decline, and the suffering caused by the castle camp system.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Scholars have been divided as to the great influence the incitement against Klaus Fleming exercised by the Duke of Charles had on the genesis of the bludgeon war [38] [39].


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Great Power Period (1617—1721) Peace of Stolbova In 1604—1611 Duke Charles of Södermanland was King Charles IX of Sweden.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

During his reign, the kingdom was almost constantly waging war against Russia and Poland, especially over the ownership of the Baltic provinces Estonia and Livonia.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

In addition, in 1611—1613 there was a defeatist war against Denmark.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

On the death of Charles IX, his son Gustav II Adolf ascended to the throne.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

The state of the kingdom was poor and Sweden had to conclude in 1613 peace with Denmark.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

The Peace of Stolbova was concluded with Russia in 1617, as a result of which Käkisalmi County, Nutälinna and part of Inker were annexed by Sweden.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Modernisation and reforms under the leadership of the King and Reich Chancellor Axel Oxenstierna, Sweden and Finland began to implement substantial reforms after the foreign policy situation calmed down.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

The army and the war establishment were reorganised, in particular in accordance with the paragon given by the Dutch Morits Oranian.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

The navy and the administration of the Crown were also reformed.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

A new form of government was brought into effect in the country and the kingdom was divided into counties.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

In the 17th century, in addition, a Governor-General was appointed in Finland from time to time to develop the conditions


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

The office of Governor General of Finland was Nils Tuurenpoika Bielke and Count Peter Brahe, whose official residence was Turku Castle.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

During the Great Reign, new cities and the Royal Court of Appeal of Turku were also established in Finland, and in 1640 the Turku Academy, Finland's first university.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

During the Thirty Years' War in the Baltics, Sweden was still waging war against Poland-Lithuania and in 1629 the provisional Peace of Altmark was concluded.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

In the Thirty Years' War between the Catholic Emperor and Protestant princes that broke out in Germany, Gustav II Adolf joined in 1630 after coming ashore in northern Germany.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Sweden's goal in the war was to support the German Protestants and strengthen its own position.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Gustav II Adolf, however, died at the Battle of Lützen in 1632 and his daughter Kristiina ascended the throne as a minor.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

In practice, the kingdom was ruled at this point by a caretaker government led by Axel Oxenstierna.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

In the Peace of Westphalia, 1648, which ended the Thirty Years' War, Sweden established itself as a great power.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

However, the Thirty Years' War and other conflicts of the Great Power period severely consumed the resources of Sweden and Finland.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

A significant number of peasants had to serve in the army and navy, where mortality rates were high.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

From wars to peace period, during Queen Kristiina's reign 1632—1654 Sweden and Finland gave vast territories as fiefdoms to the nobility.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

The nobility had the right to collect tax on their fiefdoms, which complicated the financial position of the Crown.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

After Kristiina's abdication of the crown rose to power under Charles X Gustaf, who ruled the Swedish Empire from 1654 to 1660.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

During his reign, Sweden waged wars against both Poland, Russia, and Denmark.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Charles X died in 1660 and Charles XI ascended to the throne as his successor, in whose reign 1660—1697, a major reduction—the large-scale withdrawal of fiefdoms to the Crown took place in Sweden and Finland.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Charles XI diminished the power of the nobility and ruled the kingdom autocratically.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

For Finland, the reign of Charles XI meant a lengthy peace.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

The Protestant clergy were responsible for teaching literacy to the people and ecclesiastical life was dominated by pure-apprenticity.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

1695—1697 Finland was hit by disastrous years of disastrous roofing, which resulted in a significant portion of the population dying from starvation and disease.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

The Great Northern War under Charles XI's successor King Charles XII, in 1700 the Great Northern War broke out, resulting in Sweden losing its great power status.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

The cause of the war was an anti-Swedish alliance established by Sweden's enemies Denmark, Russia, Polish-Lithuania and Saxony.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

In the Peace of Stolbova of 1617, Russia had lost contact with the Baltic Sea.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

The reformist Tsar Peter I sought to reopen Russia a link with the Baltic Sea to facilitate its contacts and commerce with Western Europe.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Although Charles XII succeeded one by one in defeating Denmark, the Russian invasion army (at the Battle of Narva) and Poland, the Swedes ultimately suffered a decisive defeat to the Russians at the Battle of Poltava in 1709.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

The king then fled to Turkey and the kingdom was open to enemy attack.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Vyborg was taken in 1710 and the Russians occupied the rest of Finland after the Battle of Kostianvirta on Pälkäne, the Battle of Napue in Ostrobothnia and the Battle of Riilahti off Hanko by 1714.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

The Russian occupation period 1714—1721 is referred to as big-hatred.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

The occupation period was disastrous for Finland.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Thousands of people were killed and even more were taken to Russia, in addition, a large number of the country's officials and clergy fled to Sweden.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

The Great Northern War and at the same time big-hatred ended with the Peace of Uusikaupunki between Sweden and Russia of 1721.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

The peace treaty annexed Inkerinmaa, Estonia, Livonia and Käkisalmi County, as well as Vyborg.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Karelia annexed to Russia is often referred to as Old Finland.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

The Age of Freedom and the Gustavian Period (1721—1809) Power away from the king in the Swedish Empire After Charles XII, the Acts took power from the king in the reigns of 1719 and 1720 and the period of autocracy became the period of decree (the “time of freedom”).


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Economics and science developed during this period, but on the other hand, the power politics of the parties caused problems.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

France and Russia gained influence in Sweden by funding rival parties called jackets and hats.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

[40] In the 1720s, reconstruction began in Finland following the Great Northern War.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

After the peace of New Town, the Russian occupying forces began to leave the country more neatly than agreed.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

During 1721, new land lords were appointed in Finland and were responsible for rebuilding.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Repair work proceeded quickly, as as early as the mid-1730s the rebuilding can be considered to have finished.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

[41] Finnish-born Arvid Horn was Permanent President of Sweden from 1720 to 1738.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

[42] His policy does not particularly find features that emphasise Finnishness.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

The most significant acclamation of Finland of his time was the Law of 1734.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

It remained in force during the period of autonomy and is the basis for the legislation of an independent Finland.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

At the same time, attention began to be paid to Finnish as a language of administration as well.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

A post of translator was established at the Permanent College in 1734, translating documents into Finnish and assisting peasant Reichstag in Stockholm.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

The 1734 law was translated into Finnish by the county secretary Samuel Forseen.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

[43] The War of the Hats came to power in the Reichstag of 1738.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

The programme of the Hat Party included a war of reparation against the Russians, in which it was intended to conquer all the territories lost in 1721 and furthermore Russian Karelia.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

When Russia had throne disputes, Hats assumed the country was weakened.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Sweden declared war in June 1741.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Three Swedish infantry regiments were sent to Finland, but the warfleet was virtually crippled by pestilence.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Irish Field Marshal Peter de Lacy, who led the Russian forces, took his troops to the Battle of Lappeenranta, where a badly underpowered Swedish force led by Carl Henrik Wrangel suffered a wicked defeat in August 1741.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Charles Emil Lewenhaupt, who served as Commander-in-Chief in Finland, agreed at the end of 1741 to a truce proposed by Emperor Elizabeth.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

[44] After the end of the armistice, the Russians marched to southern Finland in February 1742.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

After the Swedish army surrendered, the Russians occupy western Finland.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Elizabeth circulated a manifesto in March 1742 in which she promised to support the Finns if they wanted to secede from Sweden and establish an independent state.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

[44] However, after the Finnish occupation, talk of independence ceased.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

[45] The Russian occupation period has been called petty hatred, but the designation is misleading to some extent, as Russian troops behaved well.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

[44] On 7 August 1743 Sweden and Russia concluded the Peace of Turku, in which the border was moved to Kymijoki, Mäntyharju and Savonlinna.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Savonlinna city and Olavinlinna remained on the Russian side.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

[44] New Era Thoughts After the war, mercantilist principles of trade led to economic gains from tar and shipbuilding remaining in Stockholm.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

In the 1760s, Anders Chydenius, a vicar in Kokkola, started calling for freedom of business and expression.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

[40] In the last decades of the 18th century, the Turku Academy was particularly awakened by the influence of Henrik Gabriel Porthan, the “Father of Finnish History”, an interest in Finnish history and Finnish folk poetry.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

[46] Of the scholars, Eino Jutikkala says “In the second half of the 18th century, the inhabitants of the different provinces of Finland and members of different adjustments became more consciously Finns as opposed to the Swedes living behind the sea.” [47] The troubled reign of Gustav III In 1772 Gustav III After the coup, a new form of government was enacted by which power reverted back to the king.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

The era sought to get rid of mercantilism.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Freedom of speech and religion expanded.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

The confidence of Finnish officers in the king was weakened because the nobility lost their authority to the king, who was supported by the people.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Some high-ranking soldiers transferred to Russian service.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

[40] In 1788—1790, the so-called Gustav III War started by Gustav III took place between Sweden and Russia.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Sweden was also matched by Denmark.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

The war was opposed by officers who formed the Anjala Union, among others.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Despite its maritime victory in the Swedish Strait, Sweden did not gain additional territories in the peace of Värälä.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

The 1789 Book of Association and Collateral increased the king's power even further.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

With the war, the bitterness of the nobility towards the king increased, and this eventually led to the assassination of the king in 1792.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

[40] Sweden loses Finland The Finnish War was fought in 1808—1809 between Russia and Sweden.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

The war was prompted by the peace established by Russia and France on 7 July 1807 in Tilsit.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

In it, France and Russia allied and Russia pledged to pressure, if necessary by force of arms, including Sweden to join the continental exclusion against Britain, an embargo by which France would have strengthened its position as a sea power against Britain.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

The last Grand Duke of Finland during Swedish rule was the second son of Gustav IV Adolf, Charles Gustaf, who died 1802—1805 as a child.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

[48] As a result of the Finnish War, Finland passed to Russian rule and the administration was organized on the basis of autonomy.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

The Russian Emperor became Grand Duke of the Autonomous Grand Duchy of Finland.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Sources Citation error, replace with {{cite book| author = Linna, Martti (ed.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

) | title = Sources of the Early Middle Ages | publisher = History Aitta| year = 1989| isbn = 9789519600611}} Citation error, replace with {{cite book| author = Tarkiainen, Kari| title = Swedish Oriental| publisher = Svenska litteratursällskapet i Finland| year = 2010| isbn = 9798-5551-853-212-2}} Citation error, replace with {{cite book| author = Vahtola, Jouse| title = History of Finland — from the Stone Age to the 21st century| location = Helsinki| publisher = Otava| year = 2017| isbn = 978-951-1-31574-2}} Citation error, replace with {{cite book| author = Virrankoski, Pentti| title = History of Finland — country and people| location = Helsinki| publisher = Society of Finnish Literature | year = 2012| isbn = 978-952-222-377-7}} References Literature Citation error, replace with {{cite book| author = Vahtola, Jouse| title = History of Finland: From the Ice Age to the European Union| location = in Helsinki | publisher = Otava| year = 2003 (4. edition 2005) | id = ISBN 951-1-17397-9}} Citation error, replace with {{cite book| author = Zetterberg, Seppo (ed.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

) | title = Little Giant of Finnish History | location = Helsinki| publisher = WSOY| year = 1987| id = ISBN 951-0-14253-0}} Handbook of Finnish History.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Previous section, ed.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Arvi Korhonen.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

WSOY 1949, Porvoo — Helsinki.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

Citation error, replace with {{cite book| author = Karonen, Petri| title = Northern Great Power: Sweden and Finland 1521-1809| location = Helsinki| publisher = WSOY| year = 1999| id = ISBN 951-0-23739-6}} Helge Pohjolan-Pirhonen, History of Finland 1523—1617.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

WSOY 1960, Porvoo — Helsinki.


Question: How long did it take for Lutheranism to become established in Sweden?

Period of Swedish rule in Finland

On topic elsewhere in Category:History of Finland Category:History


Question: Can we grow olives in Finland?

Oilwood

The oil tree (Olea europaea) is a small tree belonging to the family of olive plants (Oleaceae), also referred to as olive tree, olive and oil olive.


Question: Can we grow olives in Finland?

Oilwood

The plant occurs indigenous to areas east of the Mediterranean Sea in Lebanon and Syria, coastal regions of Asia Minor, and along the southern coast of the Caspian Sea.


Question: Can we grow olives in Finland?

Oilwood

Today it is human-related spread throughout the Mediterranean.


Question: Can we grow olives in Finland?

Oilwood

Its fruit, olive (or oilberry [1]), is an important agricultural product in the Mediterranean region, used for nutrition and for the preparation of olive oil.


Question: Can we grow olives in Finland?

Oilwood

Description The oil tree is an evergreen tree that rarely grows to 15 meters taller.


Question: Can we grow olives in Finland?

Oilwood

Its greyish-green leaves are lanceolate, 4—10 cm long and 1—3 cm wide.


Question: Can we grow olives in Finland?

Oilwood

The trunk of the tree is usually lumpy and warped.


Question: Can we grow olives in Finland?

Oilwood

The small white flowers of the tree have four petals and sepals, two stamens and a bifurcated stigma on the pistil.


Question: Can we grow olives in Finland?

Oilwood

They usually form on the part that has grown in the previous year of the tree and form raceme-like inflorescences.


Question: Can we grow olives in Finland?

Oilwood

Olive berry Fruit, olive or oilberry, is a small oil-rich plum berry 1—2.5 cm long.


Question: Can we grow olives in Finland?

Oilwood

It is smaller in wild forms of the species than in processed cultivars.


Question: Can we grow olives in Finland?

Oilwood

The olives are either picked raw, green, or allowed to ripen, giving them a dark blue colour and yielding so-called black olives.


Question: Can we grow olives in Finland?

Oilwood

Raw olives cannot be eaten as such, but are either salted or leavened to remove acrimony and improve shelf life.


Question: Can we grow olives in Finland?

Oilwood

However, canned black olives are usually treated with chemicals that have been artificially used to make them turn black.


Question: Can we grow olives in Finland?

Oilwood

They are first soaked in lye, then treated with acid, which changes the color.


Question: Can we grow olives in Finland?

Oilwood

Finally, they are obtained with iron gluconate (E579) a uniform black color.


Question: Can we grow olives in Finland?

Oilwood

Once pasteurized, they are packed in jars.


Question: Can we grow olives in Finland?

Oilwood

[2] History Oilwood is a very old crop.


Question: Can we grow olives in Finland?

Oilwood

It is not known when it was first cultivated.


Question: Can we grow olives in Finland?

Oilwood

The earliest evidence of its domestication dates from the Chalcolithic period in Jordan.


Question: Can we grow olives in Finland?

Oilwood

Also in Crete it is known to have been cultivated as early as around 3000 BCE.


Question: Can we grow olives in Finland?

Oilwood

They have possibly been a source of wealth for Minoan culture.


Question: Can we grow olives in Finland?

Oilwood

[3] Olive is one of the most frequently mentioned plants in literature.


Question: Can we grow olives in Finland?

Oilwood

In Homer's Odyssey, Odysseus crawled under a bifurcated olive tree grown from the same trunk.


Question: Can we grow olives in Finland?

Oilwood

[4] The Roman poet Horatius mentions it when he recounts his own, very simple eating habits.


Question: Can we grow olives in Finland?

Oilwood

According to that, olives, endives and Althaeas would have been sufficient for his sustenance.


Question: Can we grow olives in Finland?

Oilwood

[5] In 1779, olives were mentioned by Lord Monboddo as belonging to the mind foods of ancient peoples and as being one of the most perfect foods.


Question: Can we grow olives in Finland?

Oilwood

[6] The olive tree is mentioned several times in the Bible as well.


Question: Can we grow olives in Finland?

Oilwood

According to it, the dove brought an olive tree leaf to Noah's Ark, indicating that the deluge was over.


Question: Can we grow olives in Finland?

Oilwood

The tree is mentioned in other holy books such as the Quran and the Book of Mormon.


Question: Can we grow olives in Finland?

Oilwood

The leafy branch of the oil tree is a symbol of prosperity, glory and peace.


Question: Can we grow olives in Finland?

Oilwood

It has been used as a victory sign in sport, but also in war.


Question: Can we grow olives in Finland?

Oilwood

They have also been used for ritual purposes and sacrificed to the gods.


Question: Can we grow olives in Finland?

Oilwood

They have been found in Tutankhamon's grave as well.


Question: Can we grow olives in Finland?

Oilwood

Olive oil has long been considered sacred.


Question: Can we grow olives in Finland?

Oilwood

In ancient Greece, it was used to anoint kings.


Question: Can we grow olives in Finland?

Oilwood

The same was also done in ancient Israel, which is also referred to by the title Messiah (anointed) of the savior king promised by the prophets.


Question: Can we grow olives in Finland?

Oilwood

Olive oil has also been burned in the sacred lamps of temples, and also in the “eternal fire” of the ancient Olympic Games.


Question: Can we grow olives in Finland?

Oilwood

The winners of the Games were crowned with olive tree foliage.


Question: Can we grow olives in Finland?

Oilwood

Even today it is used in many religious ceremonies.


Question: Can we grow olives in Finland?

Oilwood

According to Greek myth, Athene received the status of patron goddess of Attica and its capital Athens, for which she had competed with Poseidon because she gave the city an olive tree as a gift.


Question: Can we grow olives in Finland?

Oilwood

[3] As a symbol of peace, the olive tree branch also appears on the UN flag around the world map.


Question: Can we grow olives in Finland?

Oilwood

Old olive trees Pliny the Elder told of a sacred olive tree growing in Greece that would have been 1600 years old.


Question: Can we grow olives in Finland?

Oilwood

A few of the olive trees still growing in the Garden of Gethsemane in Jerusalem are claimed to date back to the time of Jesus.


Question: Can we grow olives in Finland?

Oilwood

[7] A few olive trees growing in Italy are believed to date from the Roman period, though it is uncertain whether they are the same as those mentioned in some old written sources.


Question: Can we grow olives in Finland?

Oilwood

At least one growing in Crete has been found to be over 2000 years old.


Question: Can we grow olives in Finland?

Oilwood

Its age has been defined dendrochronologically, that is, by counting annual rings.


Question: Can we grow olives in Finland?

Oilwood

[8] There is also a known olive tree on the island of Brijun in Istria, Croatia, which has been calculated to be about 1600 years old.


Question: Can we grow olives in Finland?

Oilwood

It still produces about 30 kilograms of fruit a year, yielding premium olive oil from them.


Question: Can we grow olives in Finland?

Oilwood

[9] A tree growing in Santu Baltolu di Carana, Sardinia, known to the inhabitants of the region as Ozzastru, has been claimed by some studies to be 3000 or even 4000 years old.


Question: Can we grow olives in Finland?

Oilwood

There are other millennial olive trees growing in the same garden.


Question: Can we grow olives in Finland?

Oilwood

Cultivation and use Oilwood has been cultivated since ancient times for olive oil, valuable wood material, foliage and edible olives.


Question: Can we grow olives in Finland?

Oilwood

Its fruit is naturally acrid, but through fermentation and marination it is made edible.


Question: Can we grow olives in Finland?

Oilwood

Green and black olives are completely cleaned to remove oleuropein, an acrid tasting carbohydrate.


Question: Can we grow olives in Finland?

Oilwood

Occasionally, they are also soaked in sodium hydroxide solution, that is, lye, to speed up the process.


Question: Can we grow olives in Finland?

Oilwood

In antiquity, farmers claimed that an oil tree would not prosper if one tried to grow it far from the sea.


Question: Can we grow olives in Finland?

Oilwood

According to Theofrastos, it could be cultivated no more than 300 stadia (55.6 kilometers) away from the coast.


Question: Can we grow olives in Finland?

Oilwood

Modern experience does not fully confirm such a claim, and although it is best done on coasts, climate-appropriate regions such as Greece (Kalamata olives), the Pyrenees Peninsula and north-west Africa, where winters are mild, it is cultivated far from its coast.


Question: Can we grow olives in Finland?

Oilwood

Most of the olives are produced in the vicinity of the Mediterranean Sea.


Question: Can we grow olives in Finland?

Oilwood

Some of it is cultivated in other regions with similar Mediterranean climates, such as South Africa, Chile, Australia, New Zealand and California.


Question: Can we grow olives in Finland?

Oilwood

Studies have shown olives and olive oil to have health-beneficial effects.


Question: Can we grow olives in Finland?

Oilwood

Also, the leaves of the tree are used for medicinal purposes.


Question: Can we grow olives in Finland?

Oilwood

Subspecies There are at least five widespread natural subspecies: Olea europaea subsp.


Question: Can we grow olives in Finland?

Oilwood

europaea — orchard olive (in Europe) Olea europaea subsp.


Question: Can we grow olives in Finland?

Oilwood

cuspidata (in Eritrea, Ethiopia and other parts of East Africa, including Iran, China and the area between them) Olea europaea subsp.


Question: Can we grow olives in Finland?

Oilwood

guanchica (Canary Islands) Olea europaea subsp.


Question: Can we grow olives in Finland?

Oilwood

maroccana (in Morocco) Olea europaea subsp.


Question: Can we grow olives in Finland?

Oilwood

Laperrinei (in Algeria, Sudan, Niger and India) There are thousands of varieties of the olive tree.


Question: Can we grow olives in Finland?

Oilwood

There are 300 varieties in use in Italy alone, but only a few are widely cultivated.


Question: Can we grow olives in Finland?

Oilwood

The main Italian varieties are “Leccino”, “Frantoio” and “Carolea”.


Question: Can we grow olives in Finland?

Oilwood

It is not sure if any of these can be identified with descriptions from antiquity, though it is entirely possible that some of these narrow-leaved varieties are descended from the then Licinian olive tree.


Question: Can we grow olives in Finland?

Oilwood

Since many varieties cannot be fertilized from pollen of the same variety, they are often cultivated in pairs so that some secondary variety is cultivated among one of the primary varieties to provide fertilization.


Question: Can we grow olives in Finland?

Oilwood

Such a pair is formed by, for example, “Frantoio” and “Leccino”.


Question: Can we grow olives in Finland?

Oilwood

Recently, attempts have been made to develop hybrid cultivars that would have better resistance to plant diseases or that would grow faster or produce a larger or more uniform crop.


Question: Can we grow olives in Finland?

Oilwood

Major varieties include: 'Manzanillo', which has a large, roundish fruit with a purple or green surface.


Question: Can we grow olives in Finland?

Oilwood

Cultivated worldwide.


Question: Can we grow olives in Finland?

Oilwood

'Moraiolo', the nearest from Tuscany and today widely cultivated in central Italy, especially on the mountain slopes due to its good tolerance to wind and cold.


Question: Can we grow olives in Finland?

Oilwood

Moraiolo is strong and aromatic in taste.


Question: Can we grow olives in Finland?

Oilwood

Cultivated abundantly in the province of Umbria on the slopes of the


Question: Can we grow olives in Finland?

Oilwood

“Frantoio” and “Leccino”.


Question: Can we grow olives in Finland?

Oilwood

There are varieties native to Tuscany, from which olive oil is mainly prepared.


Question: Can we grow olives in Finland?

Oilwood

Leccino has a mild sweet taste, while Frantoio has a stronger aftertaste.


Question: Can we grow olives in Finland?

Oilwood

Because of their prized taste, these are now cultivated in other countries.


Question: Can we grow olives in Finland?

Oilwood

'Arbequina' a small brown Catalan olive variety, used for both edible and oil preparation.


Question: Can we grow olives in Finland?

Oilwood

'Empeltre', a medium black Spanish olive.


Question: Can we grow olives in Finland?

Oilwood

'Kalamata', a large black olive with a meaty flavor, was named after the Greek city of Kalamata.


Question: Can we grow olives in Finland?

Oilwood

Usually preserved in vinegar or olive oil; Kalamata olive has the EU Protected Designation of Origin status' Koroneiki ', derived from Pelponnesus.


Question: Can we grow olives in Finland?

Oilwood

A small elusive olive with special quality olive oil “Pecholine”, or “picholine”, originating in southern France, green, medium and elongated.


Question: Can we grow olives in Finland?

Oilwood

'Lucques', originated in the French department of Aude.


Question: Can we grow olives in Finland?

Oilwood

Green, large and elongated.


Question: Can we grow olives in Finland?

Oilwood

'Souri', originated in Lebanon and spread widely across the Middle East.


Question: Can we grow olives in Finland?

Oilwood

High oil content and especially aromatic taste.


Question: Can we grow olives in Finland?

Oilwood

“Nabali”, or “Baladi”, originated in Palestine.


Question: Can we grow olives in Finland?

Oilwood

[10], along with the Souri and Malissi varieties, produces olive oil considered to be the highest quality in the world.


Question: Can we grow olives in Finland?

Oilwood

[11] 'Barnea', a modern variety developed in Israel that is resistant to plant diseases and produces a bountiful harvest.


Question: Can we grow olives in Finland?

Oilwood

Used both for edible and oil production.


Question: Can we grow olives in Finland?

Oilwood

Cultivated in Israel, Australia and New Zealand.


Question: Can we grow olives in Finland?

Oilwood

'Maalot', another modern Israeli plant disease-resistant variety.


Question: Can we grow olives in Finland?

Oilwood

Developed from the North African variety 'Chemlali'.


Question: Can we grow olives in Finland?

Oilwood

'Mission', refined in California.


Question: Can we grow olives in Finland?

Oilwood

Produces edible black olives.


Question: Can we grow olives in Finland?

Oilwood

Growth and reproduction The oil tree grows best in calcareous land in a marine climate.


Question: Can we grow olives in Finland?

Oilwood

Due to its wide root system, it is very resistant to drought.


Question: Can we grow olives in Finland?

Oilwood

The tree can reach several hundred years of age and still produce crops at that time.


Question: Can we grow olives in Finland?

Oilwood

The oil tree grows very slowly, but its trunk can eventually become quite thick.


Question: Can we grow olives in Finland?

Oilwood

A.


Question: Can we grow olives in Finland?

Oilwood

P. de Candolle mentions the circumference of one tree as having been ten metres.


Question: Can we grow olives in Finland?

Oilwood

The tree rarely grows 15 metres taller, and due to grafting it tends to remain even shallower on cultivations.


Question: Can we grow olives in Finland?

Oilwood

Its yellow to greenish-brown woody matter often contains darker streaks.


Question: Can we grow olives in Finland?

Oilwood

Because it is quite loud, it is highly appreciated in carpentry.


Question: Can we grow olives in Finland?

Oilwood

Oil wood is propagated in many ways, most commonly, however, from cuttings.


Question: Can we grow olives in Finland?

Oilwood

The tree takes root in suitable ground easily.


Question: Can we grow olives in Finland?

Oilwood

Occasionally, it is also necessary to use a caper.


Question: Can we grow olives in Finland?

Oilwood

Sometimes it is propagated from the seed, which is then soaked in an alkaline solution to allow it to germinate faster.


Question: Can we grow olives in Finland?

Oilwood

In general, temperatures below -10 °C severely damage trees, but smaller sporadic frosts it except young specimens can withstand well.


Question: Can we grow olives in Finland?

Oilwood

As a fruit crop and its processing in southern Europe, olives are usually harvested in winter.


Question: Can we grow olives in Finland?

Oilwood

Harvesting takes several weeks, but the timing varies in different regions and also depends on the variety.


Question: Can we grow olives in Finland?

Oilwood

The oil content of different varieties varies considerably.


Question: Can we grow olives in Finland?

Oilwood

Its pericarp most commonly contains 60— 70% oil.


Question: Can we grow olives in Finland?

Oilwood

Per tree, oil is most commonly obtained between 1.5 and 2.2 kilograms per year.


Question: Can we grow olives in Finland?

Oilwood

[12] Traditional fermentation When picked directly from wood, olives contain phenolic compounds and oleuropein, which makes them bitter tasting.


Question: Can we grow olives in Finland?

Oilwood

There are many methods to make them edible.


Question: Can we grow olives in Finland?

Oilwood

The oldest methods take advantage of microorganisms found in fruit, which provoke fermentation reactions.


Question: Can we grow olives in Finland?

Oilwood

In this case, oleuropein and phenolic compounds are broken down.


Question: Can we grow olives in Finland?

Oilwood

At the same time, lactic acid is formed, which is a natural preservative, as well as several flavoring substances.


Question: Can we grow olives in Finland?

Oilwood

Olives are edible after two weeks or a month, but sometimes processing takes up to three months.


Question: Can we grow olives in Finland?

Oilwood

Many substances are used to flavour them, including herbs, olive oil, feta cheese, bell peppers, chilli, lemon juice, wine, vinegar, juniper berries and anchovies.


Question: Can we grow olives in Finland?

Oilwood

Pests, plant diseases and weather influence A fungus called Cycloconium oleaginum can sicken oil trees for several consecutive years, causing damage to crops.


Question: Can we grow olives in Finland?

Oilwood

Also destructive is the bacterial species Pseudomonas savastaoi pv.


Question: Can we grow olives in Finland?

Oilwood

oleae [13] There are also beetle species found in oil trees that can wreak havoc.


Question: Can we grow olives in Finland?

Oilwood

[14] Hares eat oil tree bark and can damage young trees in particular.


Question: Can we grow olives in Finland?

Oilwood

If the bark is lost all around the tree, it is likely to die.


Question: Can we grow olives in Finland?

Oilwood

In France and northern and central Italy, frosts occasionally damage oil trees.


Question: Can we grow olives in Finland?

Oilwood

Hail and prolonged rains can also be a detriment.


Question: Can we grow olives in Finland?

Oilwood

Economy Production Over the past 44 years, the area under cultivation of olives has tripled from 2.6 million to 8.5 million hectares.


Question: Can we grow olives in Finland?

Oilwood

According to FAO statistics, the ten main producing countries, all of which are in the Mediterranean region, produce 95% of the world's olives.


Question: Can we grow olives in Finland?

Oilwood

Olea europaea has spread from cultivation to the wild even where it does not grow.


Question: Can we grow olives in Finland?

Oilwood

Deposits of its original wild form in southern Europe are now widely marginalized in the way of strains that have spread, re-wilded from cultivation.


Question: Can we grow olives in Finland?

Oilwood

[15] In a few areas where the tree has been planted, particularly in South Australia, it has spread to the surrounding wildlife to the extent that it has directly become a weed displacing the native vegetation of the region.


Question: Can we grow olives in Finland?

Oilwood

There its seeds are spread by foxes that have spread there as newcomers, as well as many birds such as the emu and also the European starling.


Question: Can we grow olives in Finland?

Oilwood

[16] Sources Elsewhere Includes studies on health benefits Transclusion error: {{En}} is only for use in File namespace.


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Oilwood

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Oilwood

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Oilwood

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Question: Can we grow olives in Finland?

Oilwood

Category:Oilwood plants Category:Trees Category:Fruit Category:Food crops to be cultivated Category:Key articles in screening


Question: What is the boundary 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 the boundary 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 the boundary 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 the boundary ordinal?

Ordinal

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


Question: What is the boundary ordinal?

Ordinal

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


Question: What is the boundary ordinal?

Ordinal

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


Question: What is the boundary 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 the boundary 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 the boundary ordinal?

Ordinal

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


Question: What is the boundary 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 the boundary 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 the boundary 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 the boundary ordinal?

Ordinal

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


Question: What is the boundary ordinal?

Ordinal

Similar to other numbers, ordinals can 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 the boundary 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 the boundary 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 the boundary 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 the boundary 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 the boundary 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 the boundary 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 the boundary 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 the boundary 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 the boundary 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 the boundary 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 the boundary ordinal?

Ordinal

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


Question: What is the boundary 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 the boundary 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 the boundary 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 the boundary ordinal?

Ordinal

There are also infinite ordinals.


Question: What is the boundary ordinal?

Ordinal

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


Question: What is the boundary 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 the boundary ordinal?

Ordinal

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


Question: What is the boundary 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 the boundary 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 the boundary 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 the boundary 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 the boundary 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 the boundary 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 the boundary ordinal?

Ordinal

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


Question: What is the boundary ordinal?

Ordinal

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


Question: What is the boundary 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 the boundary 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 the boundary 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 the boundary 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 the boundary ordinal?

Ordinal

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


Question: What is the boundary 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 the boundary ordinal?

Ordinal

The order somorphism of sets is obviously an equivalence relation.


Question: What is the boundary 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 the boundary 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 the boundary 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 the boundary 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 the boundary 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 the boundary 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 the boundary ordinal?

Ordinal

However, this difficulty is overcoming.


Question: What is the boundary ordinal?

Ordinal

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


Question: What is the boundary 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 the boundary 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 the boundary 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 the boundary 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 the boundary 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 the boundary 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 the boundary ordinal?

Ordinal

By mathematical notation, λ = [0, λ).


Question: What is the boundary 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 the boundary ordinal?

Ordinal

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


Question: What is the boundary 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 the boundary 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 the boundary ordinal?

Ordinal

All embryos of each ordinal are ordinals too.


Question: What is the boundary 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 the boundary 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 the boundary ordinal?

Ordinal

In addition, each set of ordinals is veryarranged.


Question: What is the boundary ordinal?

Ordinal

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


Question: What is the boundary 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 the boundary 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 the boundary ordinal?

Ordinal

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


Question: What is the boundary ordinal?

Ordinal

The class of all ordinals is not a set.


Question: What is the boundary 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 the boundary ordinal?

Ordinal

This is the Buralin—Fort paradox.


Question: What is the boundary ordinal?

Ordinal

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


Question: What is the boundary 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 the boundary ordinal?

Ordinal

Alternative definitions are currently presented as a definition of ordinal.


Question: What is the boundary 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 a subset perfectly ordered transitive set x is the transitive set of transitive sets transitive set.


Question: What is the boundary ordinal?

Ordinal

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


Question: What is the boundary 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 the boundary 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 the boundary ordinal?

Ordinal

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


Question: What is the boundary 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 the boundary 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 the boundary 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 the boundary 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 the boundary ordinal?

Ordinal

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


Question: What is the boundary ordinal?

Ordinal

Such a definition is normally referred to as transinitic recursion.


Question: What is the boundary ordinal?

Ordinal

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


Question: What is the boundary ordinal?

Ordinal

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


Question: What is the boundary 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 the boundary 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 the boundary 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 the boundary ordinal?

Ordinal

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


Question: What is the boundary 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 the boundary ordinal?

Ordinal

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


Question: What is the boundary 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 the boundary 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 the boundary ordinal?

Ordinal

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


Question: What is the boundary ordinal?

Ordinal

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


Question: What is the boundary ordinal?

Ordinal

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


Question: What is the boundary ordinal?

Ordinal

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


Question: What is the boundary 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 the boundary 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 the boundary ordinal?

Ordinal

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


Question: What is the boundary 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 the boundary 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 the boundary ordinal?

Ordinal

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


Question: What is the boundary ordinal?

Ordinal

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


Question: What is the boundary ordinal?

Ordinal

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


Question: What is the boundary ordinal?

Ordinal

There's always such a thing.


Question: What is the boundary ordinal?

Ordinal

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


Question: What is the boundary ordinal?

Ordinal

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


Question: What is the boundary 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 the boundary 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 the boundary ordinal?

Ordinal

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


Question: What is the boundary ordinal?

Ordinal

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


Question: What is the boundary 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 the boundary ordinal?

Ordinal

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


Question: What is the boundary ordinal?

Ordinal

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


Question: What is the boundary ordinal?

Ordinal

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


Question: What is the boundary ordinal?

Ordinal

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


Question: What is the boundary ordinal?

Ordinal

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


Question: What is the boundary 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 the boundary 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 the boundary 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 the boundary 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 the boundary 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 the boundary 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 the boundary 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 the boundary ordinal?

Ordinal

These are called epsilon figures.


Question: What is the boundary 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 the boundary ordinal?

Ordinal

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


Question: What is the boundary 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 the boundary 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 the boundary 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 the boundary 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 the boundary 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 the boundary 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 the boundary ordinal?

Ordinal

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


Question: What is the boundary 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 the boundary 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 the boundary 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 the boundary 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 the boundary 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 the boundary 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 the boundary 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 the boundary ordinal?

Ordinal

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


Question: What is the boundary 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 the boundary 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 the boundary ordinal?

Ordinal

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


Question: What is the boundary ordinal?

Ordinal

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


Question: What is the boundary 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 the boundary 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 the boundary ordinal?

Ordinal

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


Question: What is the boundary ordinal?

Ordinal

These are interchangeable, but not continuous.


Question: What is the boundary ordinal?

Ordinal

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


Question: What is the boundary ordinal?

Ordinal

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


Question: What is the boundary 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 the boundary ordinal?

Ordinal

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


Question: What is the boundary ordinal?

Ordinal

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


Question: What is the boundary ordinal?

Ordinal

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


Question: What is the boundary ordinal?

Ordinal

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


Question: What is the boundary 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 the boundary ordinal?

Ordinal

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


Question: What is the boundary 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 the boundary 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 the boundary ordinal?

Ordinal

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


Question: What is the boundary ordinal?

Ordinal

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


Question: What is the boundary 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 the boundary 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 the boundary 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 the boundary 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 the boundary ordinal?

Ordinal

Cantor used cardinality to classify ordinals.


Question: What is the boundary 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 the boundary 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 the boundary ordinal?

Ordinal

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


Question: What is the boundary 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 the boundary 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 the boundary 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 the boundary 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 the boundary ordinal?

Ordinal

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


Question: What is the boundary ordinal?

Ordinal

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


Question: What is the boundary 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 the boundary 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 the boundary 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 the boundary 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 the boundary ordinal?

Ordinal

In other words, the cofinality operation is idempotent.


Question: What is the boundary ordinal?

Ordinal

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


Question: What is the boundary 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 the boundary ordinal?

Ordinal

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


Question: What is the boundary 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 the boundary 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 the boundary ordinal?

Ordinal

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


Question: What is the boundary 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 the boundary 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 the boundary 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 the boundary ordinal?

Ordinal

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


Question: What is the boundary 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.