Large Veblen Ordinal

	Large Veblen Ordinal In mathematics, the large Veblen ordinal is a certain large countable ordinal, named after Oswald Veblen. There is no standard notation for ordinals beyond the Feferman–Schütte ordinal Γ0. Most systems of notation use symbols such as ψ(α), θ(α), ψα(β), some of which are modifications of the Veblen function In mathematics, the Veblen functions are a hierarchy of normal functions ( continuous strictly increasing functions from ordinals to ordinals), introduced by Oswald Veblen in . If φ0 is any normal function, then for any non-zero ordinal α, φ ...s to produce countable ordinals even for uncountable arguments, and some of which are ordinal collapsing functions. The large Veblen ordinal is sometimes denoted by \phi_(0) or \theta(\Omega^\Omega) or \psi(\Omega^). It was constructed by Veblen using an extension of Veblen functions allowing infinitely many arguments. References * * {{Number-stub Ordinal numbers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Large Countable Ordinal In the mathematical discipline of set theory, there are many ways of describing specific countable ordinals. The smallest ones can be usefully and non-circularly expressed in terms of their Cantor normal forms. Beyond that, many ordinals of relevance to proof theory still have computable ordinal notations (see ordinal analysis). However, it is not possible to decide effectively whether a given putative ordinal notation is a notation or not (for reasons somewhat analogous to the unsolvability of the halting problem); various more-concrete ways of defining ordinals that definitely have notations are available. Since there are only countably many notations, all ordinals with notations are exhausted well below the first uncountable ordinal ω1; their supremum is called ''Church–Kleene'' ω1 or ω1CK (not to be confused with the first uncountable ordinal, ω1), described below. Ordinal numbers below ω1CK are the recursive ordinals (see below). Countable ordinals larger than this may ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Oswald Veblen Oswald Veblen (June 24, 1880 – August 10, 1960) was an American mathematician, geometer and topologist, whose work found application in atomic physics and the theory of relativity The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in .... He proved the Jordan curve theorem in 1905; while this was long considered the first rigorous proof of the theorem, many now also consider Camille Jordan's original proof rigorous. Early life Veblen was born in Decorah, Iowa. His parents were Andrew Anderson Veblen (1848–1932), Professor of Physics at the University of Iowa, and Kirsti (Hougen) Veblen (1851–1908). Veblen's uncle was Thorstein Veblen, noted economist and sociologist. Oswald went to school in Iowa City. He did his undergraduate studies at the University of Iowa, where he received a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Feferman–Schütte Ordinal In mathematics, the Feferman–Schütte ordinal Γ0 is a large countable ordinal. It is the proof-theoretic ordinal of several mathematical theories, such as arithmetical transfinite recursion. It is named after Solomon Feferman and Kurt Schütte, the former of whom suggested the name Γ0. There is no standard notation for ordinals beyond the Feferman–Schütte ordinal. There are several ways of representing the Feferman–Schütte ordinal, some of which use ordinal collapsing function In mathematical logic and set theory, an ordinal collapsing function (or projection function) is a technique for defining ( notations for) certain recursive large countable ordinals, whose principle is to give names to certain ordinals much larger ...s: \psi(\Omega^\Omega), \theta(\Omega), \varphi_\Omega(0), or \varphi(1,0,0). Definition The Feferman–Schütte ordinal can be defined as the smallest ordinal that cannot be obtained by starting with 0 and using the operations of ordinal addition an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Veblen Function In mathematics, the Veblen functions are a hierarchy of normal functions ( continuous strictly increasing functions from ordinals to ordinals), introduced by Oswald Veblen in . If φ0 is any normal function, then for any non-zero ordinal α, φα is the function enumerating the common fixed points of φβ for β<α. These functions are all normal. The Veblen hierarchy In the special case when φ₀(α)=ω^α this family of functions is known as the Veblen hierarchy. The function φ₁ is the same as the ε function: φ₁(α)= ε_α. If $\alpha < \beta \,,$ then $\varphi_(\varphi_(\gamma)) = \varphi_(\gamma)$ .M. Rathjen [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Ordinal Collapsing Functions In mathematical logic and set theory, an ordinal collapsing function (or projection function) is a technique for defining (notations for) certain recursive large countable ordinals, whose principle is to give names to certain ordinals much larger than the one being defined, perhaps even large cardinals (though they can be replaced with recursively large ordinals at the cost of extra technical difficulty), and then "collapse" them down to a system of notations for the sought-after ordinal. For this reason, ordinal collapsing functions are described as an impredicative manner of naming ordinals. The details of the definition of ordinal collapsing functions vary, and get more complicated as greater ordinals are being defined, but the typical idea is that whenever the notation system "runs out of fuel" and cannot name a certain ordinal, a much larger ordinal is brought "from above" to give a name to that critical point. An example of how this works will be detailed below, for an ordi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]

The Veblen hierarchy