Topic summary
Cantor normal form
Extracted from the Wikipedia article Ordinal arithmetic.
Cantor normal form
Every ordinal number α can be uniquely written as ω ⋅ c1 + ω ⋅ c2 + ⋯ + ω ⋅ ck, where k is a natural number, c1, c2, ..., ck are nonzero natural numbers, and β1 > β2 > ... > βk ≥ 0 are ordinal numbers. The degenerate case α = 0 occurs when k = 0 which requires there be no βs nor cs. This decomposition of α is called the Cantor normal form of α, and can be considered the base-ωpositional numeral system. The highest exponent β1 is called the degree of α, and satisfies β1 ≤ α. The equality β1 = α applies if and only if α = ω. In that case Cantor normal form fails to express the ordinal in terms of smaller ones; this can happen as explained below.