Bird ordinal

The Bird ordinal (sometimes called Bird's ordinal) is an intermediate ordinal between the Takeuti-Feferman-Buchholz ordinal and Extended Buchholz ordinal which occurs occasionally in apeirological notations such as BMS. It was named by the apeirological community in honor of Chris Bird. This is because it is believed to correspond to the limit of his final system of array notations, and thus the growth rate of a natural extension of his U function. It can be written as \( \psi_0(\Omega_\Omega) \) (not to be confused with \( \psi_0(\Omega_\omega) \)) in Denis Maksudov's extension of Wilfried Buchholz's system of ordinal collapsing functions. It is believed to correspond to the proof-theoretic ordinal of \( \mathrm{Aut}(\mathrm{ID}) \), the minimal extension of Peano arithmetic so that, if it proves transfinite induction along a recursive well-order of order-type \( \alpha \), then it also is able to deal with iterated inductive definitions of length \( \alpha \). This makes it essentially the maximal possible extension of the notion of iterated inductive definitions, without the addition of second-order schemata, and shows is much greater than the Takeuti-Feferman-Buchholz ordinal, which is only able to deal with iterated inductive definitions of length \( \omega \). One could possibly consider the analogy that the Bird ordinal is to the Buchholz ordinal as \( \varphi(2,0,0) \) is to \( \Gamma_0 = \varphi(1,0,0) \), although this could potentially be seen as underestimating the size of the Bird ordinal.

Alternatively, using the correspondence between identical expressions in extended Buchholz's function and nothing OCF, the Bird ordinal can be seen as an analog of \( \Gamma_0 \) while the Buchholz ordinal is analogous to \( \varepsilon_0 \).