# Epsilon Numbers and Cantor Normal Form

Formalized Mathematics (2009)

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

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topGrzegorz 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

top- [1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990. Zbl06213858
- [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.

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