Intermediate arithmetic operations on ordinal numbers

Abstract

There are two well‐known ways of doing arithmetic with ordinal numbers: the “ordinary” addition, multiplication, and exponentiation, which are defined by transfinite iteration; and the “natural” (or “Hessenberg”) addition and multiplication (denoted ⊕ and ⊗), each satisfying its own set of algebraic laws. In 1909, Jacobsthal considered a third, intermediate way of multiplying ordinals (denoted × ), defined by transfinite iteration of natural addition, as well as the notion of exponentiation defined by transfinite iteration of his multiplication, which we denote α×β. (Jacobsthal's multiplication was later rediscovered by Conway.) Jacobsthal showed these operations too obeyed algebraic laws. In this paper, we pick up where Jacobsthal left off by considering the notion of exponentiation obtained by transfinitely iterating natural multiplication instead; we shall denote this α⊗β. We show that α⊗(β⊕γ)=false(α⊗βfalse)⊗false(α⊗γfalse) and that α⊗(β×γ)=(α⊗β)⊗γ; note the use of Jacobsthal's multiplication in the latter. We also demonstrate the impossibility of defining a “natural exponentiation” satisfying reasonable algebraic laws.

Some transfinite natural sums

“…Ideas from might be relevant to the problem. An order‐theoretical characterization of the countably infinite natural product appears in .…”

“…In particular, # γ <ζ α γ is also the largest mixed sum of (α γ ) γ <ζ which can be realized in such a way that both (1) and (2) hold.…”

“…There are similarities with the countable case: the transfinitely iterated natural sum can be computed in a way similar to the more usual transfinite ordinal sum, except just for a finite number of steps, in which we should take the finite natural sum in place of the ordinal sum; cf. Corollary 3.10 (1). Moreover, in the same spirit of [20], the iterated natural sum has an order-theoretical characterization: it is the largest mixed sum which satisfies a finiteness condition; cf.…”

“…In contrast with the more usual ordinal sum, the resulting natural operation is commutative and cancellative. Some historical remarks about the natural sum, including some variations and further references, can be found, e.g., in and [, pp. 24–25]; a list of references to mathematical applications appears in the introduction of .…”

An Infinite Natural Product

“…See Berline and Lascar [3] and Simpson [22] for interdisciplinary applications. Further references, including some variants and historical remarks, can be found in the quoted papers and in Altman [1], Blass and Gurevich [4, in particular, Section 8] and Ehrlich [9, pp. 24-25].…”