Buchholz's psi-functions: Difference between revisions
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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. |
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. |
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== Extension == |
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This was extended by Denis Maksudov like so. We let \(\Omega_0 = 1\) and, for \(\nu > 0\), \(\Omega_\nu = \aleph_\nu\). Then: |
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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