# Epsilon Numbers and Cantor Normal Form

Formalized Mathematics (2009)

• Volume: 17, Issue: 4, page 249-256
• ISSN: 1426-2630

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## Abstract

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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.

## How to cite

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Grzegorz Bancerek. "Epsilon Numbers and Cantor Normal Form." Formalized Mathematics 17.4 (2009): 249-256. <http://eudml.org/doc/266687>.

@article{GrzegorzBancerek2009,
abstract = {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.},
author = {Grzegorz Bancerek},
journal = {Formalized Mathematics},
language = {eng},
number = {4},
pages = {249-256},
title = {Epsilon Numbers and Cantor Normal Form},
url = {http://eudml.org/doc/266687},
volume = {17},
year = {2009},
}

TY - JOUR
AU - Grzegorz Bancerek
TI - Epsilon Numbers and Cantor Normal Form
JO - Formalized Mathematics
PY - 2009
VL - 17
IS - 4
SP - 249
EP - 256
AB - 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.
LA - eng
UR - http://eudml.org/doc/266687
ER -

## References

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1. [1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990. Zbl06213858
2. [2] Grzegorz Bancerek. Increasing and continuous ordinal sequences. Formalized Mathematics, 1(4):711-714, 1990.
3. [3] Grzegorz Bancerek. König's theorem. Formalized Mathematics, 1(3):589-593, 1990.
4. [4] Grzegorz Bancerek. Ordinal arithmetics. Formalized Mathematics, 1(3):515-519, 1990.
5. [5] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
6. [6] Grzegorz Bancerek. Sequences of ordinal numbers. Formalized Mathematics, 1(2):281-290, 1990.
7. [7] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.
8. [8] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.
9. [9] Tetsuya Tsunetou, Grzegorz Bancerek, and Yatsuka Nakamura. Zero-based finite sequences. Formalized Mathematics, 9(4):825-829, 2001.
10. [10] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.

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