<abstract xml:lang="en" xmlns="http://www.w3.org/1999/xhtml">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:<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="forma.2009.17.issue-4_figure1.gif"></inline-graphic>and for limit ordinal λ:<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="forma.2009.17.issue-4_figure2.gif"></inline-graphic>Tetration stabilizes at ω:<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="forma.2009.17.issue-4_figure3.gif"></inline-graphic>Every ordinal number α can be uniquely written as<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="forma.2009.17.issue-4_figure4.gif"></inline-graphic>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.</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 aswhere 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.

Epsilon Numbers and Cantor Normal Form

Formalized Mathematics

{"article-title":"Epsilon Numbers and Cantor Normal Form"}

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...

Epsilon Numbers and Cantor Normal Form

Publicado en línea: 08 jul 2010

Páginas: 249 - 256

DOI: https://doi.org/10.2478/v10037-009-0032-8

This content is open access.