Buchholz's psi-functions: Difference between revisions
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* \(\psi_\nu(\alpha) = \operatorname{mex}(C_\nu(\alpha))\), where \(\operatorname{mex}\) denotes minimal excludant.
The limit of this system is \(\psi_0(\varepsilon_{\Omega_\omega+1})\), which is equal to the [[Takeuti-Feferman-Buchholz ordinal]]. The [[Buchholz ordinal]] is also defined in terms of this function, namely as \(\psi_0(\Omega_\omega)\). This ordinal collapsing function admits a canonical associated ordinal notation, which was used to give an ordinal-analysis of the theory of \(\nu\)-times iterated inductive definitions for \(\nu \leq \omega\). Also, the ordinal notation admits a natural isomorphism to the set of Buchholz hydras.
This was extended by Denis Maksudov like so. We let \(\Omega_0 = 1\) and, for \(\nu > 0\), \(\Omega_\nu = \aleph_\nu\). Then:
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* \(\psi_\nu(\alpha) = \operatorname{mex}(C_\nu(\alpha))\), where \(\operatorname{mex}\) denotes minimal excludant.
The small difference is that we replace \(\mu \leq \omega\) with \(\mu \in C_\nu^n(\alpha)\), and remove \(\xi \in C_\mu(\xi)\). The limit of this new system is \(\psi_0(\Lambda)\), where \(\Lambda\) is the least ordinal so that \(\Omega_\Lambda = \Lambda\). The [[Bird ordinal]] and [[extended Buchholz ordinal]] are defined with this function.
This admits an ordinal notation too, as well as a canonical set of fundamental sequences.
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