Ordinal collapsing function: Difference between revisions

Revision as of 01:47, 1 August 2022

Revision as of 08:53, 1 August 2022

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* Let <math>P(\alpha)</math> be the set of distinct terms in the Cantor normal form of <math>\alpha</math> (with each term of the form <math>\omega^\xi</math> for <math>\xi \in \mathsf{On}</math>, see [[Cantor normal form theorem]])

* <math>C^0_\nu(\alpha) = \Omega_\nu</math>

* <math>C^{n+1}_\nu(\alpha) = C^{n}_\nu(\alpha) \cup \{\psi_\nu(\xi) \mid \xi \in \alpha \cap C^{n}_\nu(\alpha) \andland \xi \in C_u(\xi) \andland u \leq \omega \}</math>

* <math>C_\nu(\alpha) = \bigcup\limits_{n < \omega} C^n_\nu(\alpha)</math>

* <math>\psi_\nu(\alpha) = \min(\{\gamma \mid \gamma \notin C_\nu(\alpha)\})</math>

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* Define <math>\Omega_0 = 1</math> and <math>\Omega_\nu = \aleph_\nu</math> for <math>\nu > 0</math>.

* <math>C^0_\nu(\alpha) = \{\beta \mid \beta < \Omega_\nu\}</math>

* <math>C^{n+1}_\nu(\alpha) = \{\beta + \gamma, \psi_\mu(\eta) \mid \mu, \beta, \gamma, \eta \in C^{n}_\nu(\alpha) \andland \eta < \alpha\}</math>

* <math>C_\nu(\alpha) = \bigcup\limits_{n < \omega} C^n_\nu(\alpha)</math>

* <math>\psi_\nu(\alpha) = \min(\{\gamma \mid \gamma \notin C_\nu(\alpha)\})</math>

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* <math>C_0(\alpha, \beta) = \beta \cup \{0\}</math>

* <math>C_{n + 1}(\alpha, \beta) = \{\gamma + \delta \mid \gamma, \delta \in C_n(\alpha, \beta)\} \cup \{I(\gamma, \delta) \mid \gamma, \delta \in C_n(\alpha, \beta)\} \cup \{\psi_\pi(\gamma) \mid \pi, \gamma, \in C_n(\alpha, \beta) \andland \gamma < \alpha\}</math>

* <math>C(\alpha, \beta) = \bigcup\limits_{n < \omega} C_n(\alpha, \beta)</math>

* <math>\psi_\pi(\alpha) = \min(\{\beta < \pi \mid C(\alpha, \beta) \cap \pi \subseteq \beta \})</math>

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* M⁰ = <math>K \cap \mathsf{Lim}</math>, where Lim denotes the class of limit ordinals.

* For α > 0, M^α is the set <math>\{\pi < K \mid C(\alpha, \pi) \cap K = \pi \andland \forall \xi \in C(\alpha, \pi) \cap \alpha, M^\xi \mathsf{} </math> is stationary in <math>\pi \andland \alpha \in C(\alpha, \pi)\} </math>

* <math>C(\alpha, \beta) </math> is the closure of <math>\beta \cup \{0, K\} </math> under addition, <math>(\xi, \eta) \rightarrow \varphi(\xi, \eta) </math>, <math>\xi \rightarrow \Omega_\xi </math> given ξ < K, <math>\xi \rightarrow \Xi(\xi) </math> given ξ < α, and <math>(\xi, \pi, \delta) \rightarrow \Psi^\xi_\pi(\delta) </math> given <math>\xi \leq \delta < \alpha </math>.

* <math>\Xi(\alpha) = \min(M^\alpha \cup \{K\}) </math>.

* For <math>\xi \leq \alpha </math>, <math>\Psi^\xi_\pi(\alpha) = \min(\{\rho \in M^\xi \cap \pi: C(\alpha, \rho) \cap \pi = \rho \andland \pi \andland \alpha \in C(\alpha, \rho)\} \cup \{\pi\}) </math>.

=== Collapsing large cardinals ===