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In mathematical logic and set theory, an ordinal notation is a finite sequence of symbols from a finite alphabet that names an ordinal number according to some scheme that gives meaning to the language. Given such a scheme, one should be able to define a recursive well-ordering of a subset of the natural numbers by associating a natural number with each finite sequence of symbols via a Gödel numbering. There are many such schemes of ordinal notations, including schemes by Wilhelm Ackermann, Heinz Bachmann, Wilfried Buchholz, Georg Cantor, Solomon Feferman, Gerhard Jäger, Isles, Pfeiffer, Wolfram Pohlers, Kurt Schütte, Gaisi Takeuti (called ordinal diagrams), Oswald Veblen. Stephen Cole Kleene has a system of notations, called Kleene's O, which includes ordinal notations but it is not as well behaved as the other systems described here. Usually one proceeds by defining several functions from ordinals to ordinals and representing each such function by a symbol. In many systems, such as Veblen's well known system, the functions are normal functions, that is, they are strictly increasing and continuous in at least one of their arguments, and increasing in other arguments. Another desirable property for such functions is that the value of the function is greater than each of its arguments, so that an ordinal is always being described in terms of smaller ordinals. There are several such desirable properties. Unfortunately, no one system can have all of them since they contradict each other. ==A simplified example using a pairing function== As usual, we must start off with a constant symbol for zero, "0", which we may consider to be a function of arity zero. This is necessary because there are no smaller ordinals in terms of which zero can be described. The most obvious next step would be to define a unary function, "S", which takes an ordinal to the smallest ordinal greater than it; in other words, S is the successor function. In combination with zero, successor allows one to name any natural number. The third function might be defined as one that maps each ordinal to the smallest ordinal that cannot yet be described with the above two functions and previous values of this function. This would map β to ω·β except when β is a fixed point of that function plus a finite number in which case one uses ω·(β+1). The fourth function would map α to ωω·α except when α is a fixed point of that plus a finite number in which case one uses ωω·(α+1). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Ordinal notation」の詳細全文を読む スポンサード リンク
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