Buchholz's psi-functions: Difference between revisions
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* \(C_\nu^{n+1}(\alpha) = C_\nu^n(\alpha) \cup \{\gamma + \delta, \psi_mu(\xi): \gamma, \delta, \xi \in C_\nu^n(\alpha) \land \xi < \alpha \land \xi \in C_\mu(\xi) \land \mu \leq \omega\}\) |
* \(C_\nu^{n+1}(\alpha) = C_\nu^n(\alpha) \cup \{\gamma + \delta, \psi_mu(\xi): \gamma, \delta, \xi \in C_\nu^n(\alpha) \land \xi < \alpha \land \xi \in C_\mu(\xi) \land \mu \leq \omega\}\) |
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* \(C_\nu(\alpha) = \bigcup\{C_\nu^n(\alpha): n < \omega\}\) |
* \(C_\nu(\alpha) = \bigcup\{C_\nu^n(\alpha): n < \omega\}\) |
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* \(\psi_\nu(\alpha) = \operatorname{mex}(C_\nu(\alpha))\), where \(\ |
* \(\psi_\nu(\alpha) = \operatorname{mex}(C_\nu(\alpha))\), where \(\operatorname{mex}\) denotes minimal excludant. |
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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]]. 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]]. 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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* \(C_\nu^{n+1}(\alpha) = C_\nu^n(\alpha) \cup \{\gamma + \delta, \psi_mu(\xi): \gamma, \delta, \mu, \xi \in C_\nu^n(\alpha) \land \xi < \alpha\}\) |
* \(C_\nu^{n+1}(\alpha) = C_\nu^n(\alpha) \cup \{\gamma + \delta, \psi_mu(\xi): \gamma, \delta, \mu, \xi \in C_\nu^n(\alpha) \land \xi < \alpha\}\) |
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* \(C_\nu(\alpha) = \bigcup\{C_\nu^n(\alpha): n < \omega\}\) |
* \(C_\nu(\alpha) = \bigcup\{C_\nu^n(\alpha): n < \omega\}\) |
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* \(\psi_\nu(\alpha) = \operatorname{mex}(C_\nu(\alpha))\), where \(\ |
* \(\psi_\nu(\alpha) = \operatorname{mex}(C_\nu(\alpha))\), where \(\operatorname{mex}\) denotes minimal excludant. |
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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 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 \(\Omega_\Lambda = \Lambda\). |
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