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In mathematics, the Bachmann–Howard ordinal (also known as the Howard ordinal, or Howard-Bachmann ordinal [1]) is a large countable ordinal.It is the proof-theoretic ordinal of several mathematical theories, such as Kripke–Platek set theory (with the axiom of infinity) and the system CZF of constructive set theory.
Concerning the second statement, a precise version is given by ordinal analysis: for example, Kripke–Platek set theory can prove [4] that the process terminates for any given less than the Bachmann–Howard ordinal, but it cannot do this uniformly, i.e., it cannot prove the termination starting from the Bachmann–Howard ordinal.
The Bachmann–Howard ordinal (sometimes just called the Howard ordinal, ψ 0 (ε Ω+1) with the notation above) is an important one, because it describes the proof-theoretic strength of Kripke–Platek set theory. Indeed, the main importance of these large ordinals, and the reason to describe them, is their relation to certain formal systems ...
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 a larger proof-theoretic ordinal than another it can often prove the consistency of the second theory.
Fundamental sequences arise in some settings of definitions of large countable ordinals, definitions of hierarchies of fast-growing functions, and proof theory.Bachmann defined a hierarchy of functions in 1950, providing a system of names for ordinals up to what is now known as the Bachmann–Howard ordinal, by defining fundamental sequences for namable ordinals below . [9]
The slow-growing hierarchy grows much more slowly than the fast-growing hierarchy. Even g ε 0 is only equivalent to f 3 and g α only attains the growth of f ε 0 (the first function that Peano arithmetic cannot prove total in the hierarchy) when α is the Bachmann–Howard ordinal.
Updated April 5, 2023. Haley Katz Bachmann is joining Hearst Magazines as senior vice president of fashion and luxury. She will have oversight of U.S. advertising sales for the luxury and fashion ...
The Howard ordinal (also known as the Bachmann–Howard ordinal) was named after him. He was the first to carry out an ordinal analysis of the intuitionistic theory () of inductive definitions. [2] p.27. He was elected to the 2018 class of fellows of the American Mathematical Society. [3]
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