1.
Corps
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A corps is a military unit usually consisting of several divisions. Some military service branches are also called corps, such as the Military Police Corps, Royal Logistic Corps, Quartermaster Corps, a few civilian organizations use the name corps to imply a similar service level, such as the Peace Corps. In many armies, a corps is a formation composed of two or more divisions, and typically commanded by a lieutenant general. During World War I and World War II, due to the scale of combat. In Western armies with numbered corps, the number is indicated in Roman numerals. II Corps was also formed, with Militia units, to defend south-eastern Australia, sub-corps formations controlled Allied land forces in the remainder of Australia. I Corps headquarters was assigned control of the New Guinea campaign. In early 1945, when I Corps was assigned the task of re-taking Borneo, the Canadian Corps consisted of four Canadian divisions. After the Armistice, the peacetime Canadian militia was organized into corps and divisions. Early in the Second World War, Canadas contribution to the British-French forces fighting the Germans was limited to a single division, after the fall of France in June 1940, a second division moved to England, coming under command of a Canadian corps headquarters. This corps was renamed I Canadian Corps as a corps headquarters was established in the UK. I Canadian Corps eventually fought in Italy, II Canadian Corps in NW Europe, after the formations were disbanded after VE Day, Canada has never subsequently organized a Corps headquarters. The Chinese Republic had 133 Corps during the Second Sino-Japanese War, the Corps became the basic tactical unit of the NRA having strength nearly equivalent to an allied Division. The French Army under Napoleon used corps-sized formations as the first formal combined-arms groupings of divisions with reasonably stable manning, Napoleon first used the Corps dArmée in 1805. The use of the Corps dArmée was an innovation that provided Napoleon with a significant battlefield advantage in the early phases of the Napoleonic Wars. The Corps was designed to be an independent military group containing cavalry, artillery and infantry and this allowed Napoleon to mass the bulk of his forces to effect a penetration into a weak section of enemy lines without risking his own communications or flank. This innovation stimulated other European powers to adopt similar military structures, the Corps has remained an echelon of French Army organization to the modern day. As fixed military formation already in peace-time it was used almost in all European armies after Battle of Ulm in 1805, in Prussia it was introduced by Order of His Majesty from November 5,1816, in order to strengthen the readiness to war

2.
Ordinal number
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In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a collection of objects in order, one after another. Any finite collection of objects can be put in order just by the process of counting, labeling the objects with distinct whole numbers, Ordinal numbers are thus the labels needed to arrange collections of objects in order. An ordinal number is used to describe the type of a well ordered set. Whereas ordinals are useful for ordering the objects in a collection, they are distinct from cardinal numbers, although the distinction between ordinals and cardinals is not always apparent in finite sets, different infinite ordinals can describe the same cardinal. Like other kinds of numbers, ordinals can be added, multiplied, a natural number can be used for two purposes, to describe the size of a set, or to describe the position of an element in a sequence. When restricted to finite sets these two concepts coincide, there is one way to put a finite set into a linear sequence. This is because any set has only one size, there are many nonisomorphic well-orderings of any infinite set. Whereas the notion of number is associated with a set with no particular structure on it. A well-ordered set is an ordered set in which there is no infinite decreasing sequence, equivalently. Ordinals may be used to label the elements of any given well-ordered set and this length is called the order type of the set. Any ordinal is defined by the set of ordinals that precede it, in fact, the most common definition of ordinals identifies each ordinal as the set of ordinals that precede it. For example, the ordinal 42 is the type of the ordinals less than it, i. e. the ordinals from 0 to 41. Conversely, any set of ordinals that is downward-closed—meaning that for any ordinal α in S and any ordinal β < α, β is also in S—is an ordinal. There are infinite ordinals as well, the smallest infinite ordinal is ω, which is the type of the natural numbers. After all of these come ω·2, ω·2+1, ω·2+2, and so on, then ω·3, now the set of ordinals formed in this way must itself have an ordinal associated with it, and that is ω2. Further on, there will be ω3, then ω4, and so on, and ωω, then ωωω, then later ωωωω and this can be continued indefinitely far. The smallest uncountable ordinal is the set of all countable ordinals, in a well-ordered set, every non-empty subset contains a distinct smallest element. Given the axiom of dependent choice, this is equivalent to just saying that the set is ordered and there is no infinite decreasing sequence