In mathematics, the Bachmann–Howard ordinal (also known as the Howard ordinal, or Howard-Bachmann ordinal) is a
large countable ordinal
In the mathematical discipline of set theory, there are many ways of describing specific countable ordinals. The smallest ones can be usefully and non-circularly expressed in terms of their Cantor normal forms. Beyond that, many ordinals of relev ...
.
It is the
proof-theoretic ordinal of several mathematical
theories
A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, ...
, such as
Kripke–Platek set theory
The Kripke–Platek set theory (KP), pronounced , is an axiomatic set theory developed by Saul Kripke and Richard Platek.
The theory can be thought of as roughly the predicative part of Zermelo–Fraenkel set theory (ZFC) and is considerably weak ...
(with the
axiom of infinity
In axiomatic set theory and the branches of mathematics and philosophy that use it, the axiom of infinity is one of the axioms of Zermelo–Fraenkel set theory. It guarantees the existence of at least one infinite set, namely a set containing ...
) and the system CZF of
constructive set theory
Constructivism may refer to:
Art and architecture
* Constructivism (art), an early 20th-century artistic movement that extols art as a practice for social purposes
* Constructivist architecture, an architectural movement in the Soviet Union in ...
.
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 (Ordinal notation, notations for) certain Recursive ordinal, recursive large countable ordinals, whose principle is to give n ...
:
*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
In the mathematical field of set theory, ordinal arithmetic describes the three usual operations on ordinal numbers: addition, multiplication, and exponentiation. Each can be defined in essentially two different ways: either by constructing an expl ...
, 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 φ
εΩ+1(0) 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 ''φ''0 is any normal function, then for any non-zero ordinal '' ...
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.
Citations
References
*
*
*
* (Slides of a talk given at Fischbachau.)
Proof theory
Ordinal numbers
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