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Epsilon Numbers and Cantor Normal Form

By Grzegorz Bancerek


Summary. 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: ελ = lim α<λ εα = εα. α<λ Tetration stabilizes at ω: α ↑ ↑ β = α ↑ ↑ ω for α ̸ = 0 and β ≥ ω. Every ordinal number α can be uniquely written as n1ω β1 + n2ω β2 + · · · + nkω β k where k is a natural number, n1, n2,..., nk are positive integers, and β1> β2>...> βk are ordinal numbers (βk = 0). This decomposition of α is called th

Year: 2009
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