# Veblen Hierarchy

Formalized Mathematics (2011)

• Volume: 19, Issue: 2, page 83-92
• ISSN: 1426-2630

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## Abstract

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The Veblen hierarchy is an extension of the construction of epsilon numbers (fixpoints of the exponential map: ωε = ε). It is a collection φα of the Veblen Functions where φ0(β) = ωβ and φ1(β) = εβ. The sequence of fixpoints of φ1 function form φ2, etc. For a limit non empty ordinal λ the function φλ is the sequence of common fixpoints of all functions φα where α < λ.The Mizar formalization of the concept cannot be done directly as the Veblen functions are classes (not (small) sets). It is done with use of universal sets (Tarski classes). Namely, we define the Veblen functions in a given universal set and φα(β) as a value of Veblen function from the smallest universal set including α and β.

## How to cite

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Grzegorz Bancerek. "Veblen Hierarchy." Formalized Mathematics 19.2 (2011): 83-92. <http://eudml.org/doc/267399>.

@article{GrzegorzBancerek2011,
abstract = {The Veblen hierarchy is an extension of the construction of epsilon numbers (fixpoints of the exponential map: ωε = ε). It is a collection φα of the Veblen Functions where φ0(β) = ωβ and φ1(β) = εβ. The sequence of fixpoints of φ1 function form φ2, etc. For a limit non empty ordinal λ the function φλ is the sequence of common fixpoints of all functions φα where α < λ.The Mizar formalization of the concept cannot be done directly as the Veblen functions are classes (not (small) sets). It is done with use of universal sets (Tarski classes). Namely, we define the Veblen functions in a given universal set and φα(β) as a value of Veblen function from the smallest universal set including α and β.},
author = {Grzegorz Bancerek},
journal = {Formalized Mathematics},
keywords = {Mizar; Veblen hierarchy; Veblen functions},
language = {eng},
number = {2},
pages = {83-92},
title = {Veblen Hierarchy},
url = {http://eudml.org/doc/267399},
volume = {19},
year = {2011},
}

TY - JOUR
AU - Grzegorz Bancerek
TI - Veblen Hierarchy
JO - Formalized Mathematics
PY - 2011
VL - 19
IS - 2
SP - 83
EP - 92
AB - The Veblen hierarchy is an extension of the construction of epsilon numbers (fixpoints of the exponential map: ωε = ε). It is a collection φα of the Veblen Functions where φ0(β) = ωβ and φ1(β) = εβ. The sequence of fixpoints of φ1 function form φ2, etc. For a limit non empty ordinal λ the function φλ is the sequence of common fixpoints of all functions φα where α < λ.The Mizar formalization of the concept cannot be done directly as the Veblen functions are classes (not (small) sets). It is done with use of universal sets (Tarski classes). Namely, we define the Veblen functions in a given universal set and φα(β) as a value of Veblen function from the smallest universal set including α and β.
LA - eng
KW - Mizar; Veblen hierarchy; Veblen functions
UR - http://eudml.org/doc/267399
ER -

## References

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14. [14] Bogdan Nowak and Grzegorz Bancerek. Universal classes. Formalized Mathematics, 1(3):595-600, 1990.
15. [15] Karol Pąk. The Nagata-Smirnov theorem. Part I. Formalized Mathematics, 12(3):341-346, 2004.
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