Bachmann–Howard ordinal
E890897
The Bachmann–Howard ordinal is a large countable ordinal that serves as a key benchmark in proof theory, marking the strength of powerful formal systems extending predicative arithmetic.
Statements (30)
| Predicate | Object |
|---|---|
| instanceOf | countable ordinal ⓘ |
| appearsIn | ordinal analyses of theories stronger than predicative analysis ⓘ |
| characterizes | the proof-theoretic strength of powerful systems extending predicative arithmetic ⓘ |
| comparedWith | proof-theoretic ordinals of systems of analysis ⓘ |
| field |
mathematical logic
ⓘ
proof theory ⓘ set theory ⓘ |
| greaterThan | Feferman–Schütte ordinal Γ₀ NERFINISHED ⓘ |
| hasNotation | often denoted by ψ(Ω_ω) or similar collapsing-function terms ⓘ |
| hasProperty |
can be defined without reference to uncountable sets
ⓘ
can be described using collapsing functions ⓘ countable but not representable by simple Veblen hierarchy alone ⓘ is a limit of smaller recursive ordinals ⓘ serves as a boundary for many systems of second-order arithmetic ⓘ strictly less than any uncountable ordinal ⓘ well-founded under ordinal comparison ⓘ |
| is |
a benchmark for the strength of formal systems
ⓘ
a key example in the study of large countable ordinals ⓘ a recursive ordinal ⓘ a well-ordered type of a recursive well-ordering ⓘ |
| namedAfter |
Heinz Bachmann
NERFINISHED
ⓘ
William Alvin Howard NERFINISHED ⓘ |
| relatedTo |
Veblen hierarchy
NERFINISHED
ⓘ
impredicative comprehension ⓘ ordinal collapsing functions ⓘ predicative arithmetic ⓘ |
| usedIn |
analysis of impredicative theories
ⓘ
ordinal analysis of formal theories ⓘ |
| usedToMeasure |
consistency strength of certain formal systems
ⓘ
proof-theoretic strength of subsystems of second-order arithmetic ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.