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• # Artykuł - szczegóły

## Formalized Mathematics

2009 | 17 | 4 | 249-256

## Epsilon Numbers and Cantor Normal Form

EN

### Abstrakty

EN
An epsilon number is a transfinite number which is a fixed point of an exponential map: ωϵ = 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: [...] and for limit ordinal λ: [...] Tetration stabilizes at ω: [...] Every ordinal number α can be uniquely written as [...] where κ is a natural number, n1, n2, …, nk are positive integers, and β1 > β2 > … > βκ are ordinal numbers (βκ = 0). This decomposition of α is called the Cantor Normal Form of α.

249-256

wydano
2009-01-01
online
2010-07-08

### Twórcy

autor
• Białystok Technical University, Poland

### Bibliografia

• [1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.
• [2] Grzegorz Bancerek. Increasing and continuous ordinal sequences. Formalized Mathematics, 1(4):711-714, 1990.
• [3] Grzegorz Bancerek. König's theorem. Formalized Mathematics, 1(3):589-593, 1990.
• [4] Grzegorz Bancerek. Ordinal arithmetics. Formalized Mathematics, 1(3):515-519, 1990.
• [5] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
• [6] Grzegorz Bancerek. Sequences of ordinal numbers. Formalized Mathematics, 1(2):281-290, 1990.
• [7] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.
• [8] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.
• [9] Tetsuya Tsunetou, Grzegorz Bancerek, and Yatsuka Nakamura. Zero-based finite sequences. Formalized Mathematics, 9(4):825-829, 2001.
• [10] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.