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• # Artykuł - szczegóły

## Formalized Mathematics

2011 | 19 | 2 | 83-92

## Veblen Hierarchy

EN

### Abstrakty

EN
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.

83-92

wydano
2011-01-01
online
2011-07-18

### Twórcy

autor
• Białystok Technical University, Poland

### Bibliografia

• [1] Grzegorz Bancerek. Increasing and continuous ordinal sequences. Formalized Mathematics, 1(4):711-714, 1990.
• [2] Grzegorz Bancerek. Köonig's theorem. Formalized Mathematics, 1(3):589-593, 1990.
• [3] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
• [4] Grzegorz Bancerek. Sequences of ordinal numbers. Formalized Mathematics, 1(2):281-290, 1990.
• [5] Grzegorz Bancerek. Tarski's classes and ranks. Formalized Mathematics, 1(3):563-567, 1990.
• [6] Grzegorz Bancerek. The well ordering relations. Formalized Mathematics, 1(1):123-129, 1990.
• [7] Grzegorz Bancerek. Zermelo theorem and axiom of choice. Formalized Mathematics, 1(2):265-267, 1990.
• [8] Grzegorz Bancerek. Epsilon numbers and Cantor normal form. Formalized Mathematics, 17(4):249-256, 2009, doi: 10.2478/v10037-009-0032-8.[Crossref]
• [9] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.
• [10] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
• [11] Czesław Byliński. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
• [12] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
• [13] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.
• [14] Bogdan Nowak and Grzegorz Bancerek. Universal classes. Formalized Mathematics, 1(3):595-600, 1990.
• [15] Karol Pąk. The Nagata-Smirnov theorem. Part I. Formalized Mathematics, 12(3):341-346, 2004.
• [16] Piotr Rudnicki and Andrzej Trybulec. Abian's fixed point theorem. Formalized Mathematics, 6(3):335-338, 1997.
• [17] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
• [18] Tetsuya Tsunetou, Grzegorz Bancerek, and Yatsuka Nakamura. Zero-based finite sequences. Formalized Mathematics, 9(4):825-829, 2001.
• [19] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.