Ordinal collapsing function: Difference between revisions
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RhubarbJayde (talk | contribs) (Created page with "An ordinal collapsing function, typically abbreviated OCF, is a general method of constructing an ordinal representation system, by "collapsing" uncountable or nonrecursive ordinals such as \( \Omega \) or <nowiki>\( \omega_1^{\mathrm{CK}} \)</nowiki> to smaller, recursive ordinals such as the SVO. The primary idea is that, at the point of epsilon numbers and beyond, especially at the level of Stron...") |
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* Let \( \Omega = \aleph_1 \) and \( C_0^\Omega(\alpha, \beta) = \beta \cup \{0, \Omega\} \). |
* Let \( \Omega = \aleph_1 \) and \( C_0^\Omega(\alpha, \beta) = \beta \cup \{0, \Omega\} \). |
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* For all \( n \), \( C_{n+1}^\Omega(\alpha,\beta) = \{\gamma+delta, \omega^\gamma, \psi_\Omega(\eta): \gamma, \delta, \eta \in C_n^\Omega(\alpha, \beta) \land \eta < \alpha\} |
* For all \( n \), \( C_{n+1}^\Omega(\alpha,\beta) = \{\gamma+delta, \omega^\gamma, \psi_\Omega(\eta): \gamma, \delta, \eta \in C_n^\Omega(\alpha, \beta) \land \eta < \alpha\} \) |
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* \( C^\Omega(\alpha,\beta) \) is the union of \( C_n^\Omega(\alpha,\beta) \) for all finite \( n \). |
* \( C^\Omega(\alpha,\beta) \) is the union of \( C_n^\Omega(\alpha,\beta) \) for all finite \( n \). |
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* \( \psi_\Omega(\alpha) \) is the least \( \rho < \Omega \) so that \( C^\Omega(\alpha, \rho) \cap \Omega = \rho \). |
* \( \psi_\Omega(\alpha) \) is the least \( \rho < \Omega \) so that \( C^\Omega(\alpha, \rho) \cap \Omega = \rho \). |
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