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* Let <math>\Omega</math> represent an uncountable ordinal such as <math>\omega_1</math>;
* Then define <math>C^{\Omega}(\alpha, \beta)</math> as the closure of <math>\beta \cup \{ 0, \Omega\}</math> under addition, <math>(\xi \rightarrow \omega^\xi)</math> and <math>(\xi \rightarrow \psi_\Omega(\xi))</math> for <math>\xi < \alpha</math>.
* <math>\psi_\Omega(\alpha)</math> is the smallest countable ordinal ρ such that <math>C^\Omega(\alpha, \rho) \cap \Omega= \rho</math>
<math>\psi_\Omega(\varepsilon_{\Omega+1})</math> is the Bachmann–Howard ordinal, the proof-theoretic ordinal of Kripke–Platek set theory with the axiom of infinity (KP).
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