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 will denote this α⊗β. We show that
α⊗(β⊕γ)=(α⊗β)⊗(α⊗γ) 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.Comment: 18 pages, 3 table