Transfinite Function Iteration and Surreal Numbers
نویسندگان
چکیده
منابع مشابه
Integration on Surreal Numbers
The thesis concerns the (class) structure No of Conway’s surreal numbers. The main concern is the behaviour on No of some of the classical functions of real analysis, and a definition of integral for such functions. In the main texts on No, most definitions and proofs are done by transfinite recursion and induction on the complexity of elements. In the thesis I consider a general scheme of defi...
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Let No be Conway’s class of surreal numbers. I will make explicit the notion of a function f on No recursively defined over some family of functions. Under some ‘tameness’ and uniformity conditions, f must satisfy some interesting properties; in particular, the supremum of the class ̆ x ∈ No : f (x) ≥ 0 ̄ is actually an element of No. As an application, I will prove that concatenation function x ...
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An open problem posed by John H. Conway in [2] was whether one could, on his system of numbers and games, ' . . . define operations of addition and multiplication which will restrict on the ordinals to give their usual operations'. Such a definition for addition was later given in [4], and this paper will show that a definition also exists for multiplication. An ordinal exponentiation operation...
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Although transfinite iteration functionals have been used in the past to construct ever-larger initial segments of the ordinals ([5],[1]), there appears to be little investigation into the nature of the functionals themselves. In this note, we investigate the relationship between (countable) transfinite iteration and ordinal arithmetic. While there is a nice connection between finite iteration ...
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In this paper we introduce hyperations and cohyperations, which are forms of transfinite iteration of ordinal functions. Hyperations are iterations of normal functions. Unlike iteration by pointwise convergence, hyperation preserves normality. The hyperation 〈f 〉ξ∈On of a normal function f is a sequence of normal functions so that f 0 = id, f 1 = f and for all α, β we have that f = ff. These co...
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ژورنال
عنوان ژورنال: Advances in Applied Mathematics
سال: 1997
ISSN: 0196-8858
DOI: 10.1006/aama.1996.0513