Buchholz's Ordinal

In mathematics, ψ₀(Ω_ω), widely known as Buchholz's ordinal, is a large countable ordinal that is used to measure the proof-theoretic strength of some mathematical systems. In particular, it is the proof theoretic ordinal of the subsystem $\Pi_1^1$-CA₀ of second-order arithmetic; this is one of the "big five" subsystems studied in reverse mathematics (Simpson 1999). It is also the proof-theoretic ordinal of $\mathsf$, the theory of finitely iterated inductive definitions, and of $KP\ell_0$,T. Carlson
Elementary Patterns of Resemblance
(1999). Accessed 12 August 2022. a fragment of Kripke-Platek set theory extended by an axiom stating every set is contained in an admissible set. Buchholz's ordinal is also the order type of the segment bounded by $D_0D_\omega0$ in Buchholz's ordinal notation $\mathsf$. Lastly, it can be expressed as the limit of the sequence: $\varepsilon_0 = \psi_0(\Omega)$, $\mathsf = \psi_0(\Omega_2)$, $\psi_0(\Omega_3)$, ...

Definition

* $\Omega_0 = 1$, and $\Omega_n = \aleph_n$ for ''n'' > 0.

* $C_i(\alpha)$ is the closure of $\Omega_i$ under addition and the $\psi_\eta(\mu)$ function itself (the latter of which only for $\mu < \alpha$ and $\eta \leq \omega$).
* $\psi_i(\alpha)$ is the smallest ordinal not in $C_i(\alpha)$.
*Thus, ''ψ''₀(Ω_''ω'') is the smallest ordinal not in the closure of $1$ under addition and the $\psi_\eta(\mu)$ function itself (the latter of which only for $\mu < \Omega_\omega$ and $\eta \leq \omega$).

References

* G. Takeuti, ''Proof theory'', 2nd edition 1987
* K. Schütte, ''Proof theory'', Springer 1977

Ordinal numbers
Proof theory

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