A note on Conway multiplication of ordinals.
John L. Hickman · Notre Dame Journal of Formal Logic · 1983
We denote by 'ω' the first transfmite ordinal, and by ' cv + β\ 'αβ', and 6 a β \ respectively, the usual ordinal sum, product, and exponentiation of an ordinal a by an ordinal β.We assume that the reader is familiar with Cantor's co-normal form theorem, which uniquely represents a nonzero ordinal as a sum of powers of ω.Given a nonzero ordinal α, we let β(αθ be the number of summands in the co-normal form of α, and express this form as Σ\ω e ^Oί ' ι ^c(aJ); i<i(a)lRegrettably, there seems to be no "nice" way of formulating an adequate definition of "natural" ordinal addition, and so we shall have to use the following not-so-nice way.Let α, β be ordinals.If aβ = 0, set a 4β -a + β\ otherwise set a + β equal to the unique ordinal y whose co-normal form has the following properties:(1) ie(yj); i < £(γ)! = ie(aj);j < £(α)J U [e(β,k)\ k < i(β)\.(2) (a) If e(y,i) = e(oc,j) for some / < β(ce) but e(y, i) = e(β,k) for no k < t(β), then c(y, i) = c(a,j).(b) If e(y, i) = e(β, k) for some k < i(β) but e(y, i) = e(a,j) for no / < ί(α), then c{y, ϊ) = c(β, k).(c) If e(y,i) = e(a t j) = e(β,k) for some / < £(α) and k < ί(j8), then c(y,i) = c(oιj) + c(β,k).We can now define Conway multiplication, denoted by ς X', which was introduced by Gonshor in [ 1 ] and attributed by him to Conway.If oiβ = 0, then we set ot X β = 0; otherwise we set a X β = (a X δ) + a if β = δ + 1 for some δ, and a X β -suρ{α X δ; δ < β\ if β is a limit ordinal.