Bachmann-Howard ordinal: Difference between revisions

The Bachmann-Howard ordinal is a key milestone in the set of recursive ordinals. It is equal to the limit of the [[Large Veblen ordinal|dimensional Veblen]] function, as well as the Buchholz hydra with only zero and one labels. It was originally discovered as the limit of a basic ordinal collapsing function, namely Bachmann's psi, which was used in ordinal-analysis. In particular, the Bachmann-Howard ordinal is exactly the proof-theoretic ordinal of basic Kripke-Platek set theory, which has the same strength as \( \mathrm{ID}_1 \). This is a system of arithmetic augmented by inductive definitions. The ordinal collapsing function used to give this ordinal analysis had the Bachmann-Howard ordinal as its limit, and it can be represented as the countable collapse of \( \varepsilon_{\Omega+1} \). Buchholz further extended this to [[Buchholz's psi-functions|his famous set]] of collapsing functions, whose limit is the much larger [[Buchholz ordinal]].