In mathematics, the Bachmann–Howard ordinal (also known as the Howard ordinal, or Howard-Bachmann ordinal) is a
large countable ordinal.
It is the
proof-theoretic ordinal
In proof theory, ordinal analysis assigns ordinals (often large countable ordinals) to mathematical theories as a measure of their strength.
If theories have the same proof-theoretic ordinal they are often equiconsistent, and if one theory has ...
of several mathematical
theories, such as
Kripke–Platek set theory (with the
axiom of infinity) and the system CZF of
constructive set theory.
It was introduced by and .
Definition
The Bachmann–Howard ordinal is defined using an
ordinal collapsing function
In mathematical logic and set theory, an ordinal collapsing function (or projection function) is a technique for defining (notations for) certain recursive large countable ordinals, whose principle is to give names to certain ordinals much larger t ...
:
*''ε''
''α'' enumerates the
epsilon numbers, the ordinals ''ε'' such that ω
''ε'' = ''ε''.
*Ω = ω
1 is the
first uncountable ordinal
In mathematics, the first uncountable ordinal, traditionally denoted by \omega_1 or sometimes by \Omega, is the smallest ordinal number that, considered as a set, is uncountable. It is the supremum (least upper bound) of all countable ordinals. Whe ...
.
*''ε''
Ω+1 is the first epsilon number after Ω = ''ε''
Ω.
*''ψ''(''α'') is defined to be the smallest ordinal that cannot be constructed by starting with 0, 1, ω and Ω, and repeatedly applying
ordinal addition, multiplication and exponentiation, and ''ψ'' to previously constructed ordinals (except that ''ψ'' can only be applied to arguments less than ''α'', to ensure that it is well defined).
*The Bachmann–Howard ordinal is ''ψ''(''ε''
Ω+1).
The Bachmann–Howard ordinal can also be defined as
for an extension of the
Veblen function
In mathematics, the Veblen functions are a hierarchy of normal functions ( continuous strictly increasing functions from ordinals to ordinals), introduced by Oswald Veblen in . If φ0 is any normal function, then for any non-zero ordinal α, φ ...
s ''φ''
''α'' to certain functions ''α'' of ordinals; this extension was carried out by Heinz Bachmann and is not completely straightforward.
[M. Rathjen]
The Art of Ordinal Analysis
(2006), p.11. Accessed 21 February 2023.
References
*
*
*
* (Slides of a talk given at Fischbachau.)
Citations
Proof theory
Ordinal numbers
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