Epsilon Numbers and Cantor Normal Form

An epsilon number is a transfinite number which is a fixed point of an exponential map: ω ε = ε. The formalization of the concept is done with use of the tetration of ordinals (Knuth's arrow notation, ↑↑). Namely, the ordinal indexing of epsilon numbers is defined as follows: ε0 = ω ↑↑ ω, εα+1 = εα ↑↑ ω, and for limit ordinal λ: ε λ = lim α<λ εα = α<λ εα. Tetration stabilizes at ω: α ↑↑ β = α ↑↑ ω for α = 0 and β ≥ ω. Every ordinal number α can be uniquely written as n1ω β 1 + n2ω β 2 + · · · + n k ω β k , where k is a natural number, n1, n2,. . ., n k are positive integers, and β1 > β2 >. .. > β k are ordinal numbers (β k = 0). This decomposition of α is called the Cantor Normal Form of α. The notation and terminology used here are introduced in the following papers: