On transfinite iteration

Transfinite Iteration Functionals and Ordinal Arithmetic

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 ...

Hyperations, Veblen progressions and transfinite iteration of ordinal functions

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...

Transfinite Equations in Transfinite Strings

On Transfinite Knuth-Bendix Orders

In this paper we discuss the recently introduced transfinite Knuth-Bendix orders. We prove that any such order with finite subterm coefficients and for a finite signature is equivalent to an order using ordinals below ω, that is, finite sequences of natural numbers of a fixed length. We show that this result does not hold when subterm coefficients are infinite. However, we prove that in this ge...

Transfinite Diameter

The transfinite diameter is a way of quantifying the size of compact sets in Euclidean space. This quantity is related to the Hausdorff dimension and the Lebesgue measure, but gives a slightly different perspective on the set than either of those do. In this paper, we introduce the transfinite diameter, and outline some attempts to calculate this quantity for three sets in R. For z1, z2, . . . ...