Constructible hierarchy: Difference between revisions
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(How is the reflection principle stated for general cumulative hierarchies?) |
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Note that this is a cumulative hierarchy, and thus the [[reflection principle]] applies.{{citation needed}} |
Note that this is a cumulative hierarchy, and thus the [[reflection principle]] applies.{{citation needed}} |
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This is always contained in the respective rank of the von Neumann hierarchy: \(L_\alpha \subseteq V_\alpha\). This can be shown by a transfinite induction argument. It initially completely actually agrees with \(V\): all subsets of a finite set are definable, therefore \(L_\alpha = V_\alpha\) for \(\alpha \leq \omega\). However, while \(V_{\omega+1}\) is uncountable, there are (as we mentioned) only countably many subsets of a countable subset, and thus \(L_{\omega+1}\) is countable and a proper subset of \(V_{\omega+1}\). In general, \(|L_\alpha| = |\alpha|\) for \(\alpha \geq \omega\). |
This is always contained in the respective rank of the von Neumann hierarchy: \(L_\alpha \subseteq V_\alpha\). This can be shown by a transfinite induction argument. It initially completely actually agrees with \(V\): all subsets of a finite set are definable, therefore \(L_\alpha = V_\alpha\) for \(\alpha \leq \omega\). However, while \(V_{\omega+1}\) is uncountable, there are (as we mentioned) only countably many subsets of a countable subset, and thus \(L_{\omega+1}\) is countable and a proper subset of \(V_{\omega+1}\). In general, \(|L_\alpha| = |\alpha|\) for \(\alpha \geq \omega\).<ref>Most set theory texts</ref> |
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If \(\kappa = \beth_\kappa\), then \(|L_\kappa| = |V_\kappa|\). However, the existence of a \(\kappa > \omega\) so that \(L_\kappa = V_\kappa\) (they're equal, not just equinumerous) is independent from the axioms of \(\mathrm{ZFC}\), if they're consistent. This is because some models of \(\mathrm{ZFC}\) think it's true, and others think it's false, thus the completeness theorem applies. |
If \(\kappa = \beth_\kappa\), then \(|L_\kappa| = |V_\kappa|\). However, the existence of a \(\kappa > \omega\) so that \(L_\kappa = V_\kappa\) (they're equal, not just equinumerous) is independent from the axioms of \(\mathrm{ZFC}\), if they're consistent. This is because some models of \(\mathrm{ZFC}\) think it's true, and others think it's false, thus the completeness theorem applies. |
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