Constructible hierarchy: Difference between revisions
→Alternate characterisations
RhubarbJayde (talk | contribs) (Created page with "The constructible hierarchy is a way of "building up" the constructible universe, the smallest ideal model of set theory which contains the ordinals. Therefore, it is important in inner model theory, as well as in the study of \(\alpha\)-recursion theory, stability, Gandy ordinals and reflection principles. == Definition == Say a subset \(X\) of \(Y\) is definable if there are some \(z_0, z_1, \cdots, z_n \i...") |
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Work of Jensen<ref>The fine structure of the constructible hierarchy, R. Björn Jensen, ''Annals of Mathematical Logic'', 1972</ref> showed that, within \(L\), various fine structure and combinatorics hold. This includes the generalized continuum hypothesis and the diamond principle. However, Scott proved that [[measurable]] cardinals can not exist in \(L\) (if \(\kappa\) is measurable, \(\kappa\) is still an element of \(L\), but the necessary measure witnessing its measurability can't be in \(L\), and thus \(L\) doesn't realize it's measurable). This is because \(L\) thinks \(V = L\), yet the existence of a measurable cardinal implies \(V \neq L\):
<nowiki>Assume \(\kappa\) is measurable, and let \(\mathcal{U}\) witness this. Assume \(V = L\). Set \(\mathcal{M} = (V^\kappa / \mathcal{U}, \in_{\mathcal{U}})\) be the ultrapower. By \(\kappa\)-completeness, the relation \(\in_{\mathcal{U}}\) is well-founded, extensional and set-like. Therefore, the Mostowski collapse lemma implies that there is some transitive \(M\) so that \((M, \in) \cong \mathcal{M}\). Let \(\pi: V^\kappa / \mathcal{U} \to M\) be the isomorphism, and \(\tilde{j}: V \to V^\kappa / \mathcal{U}\) be the canonical ultrapower embedding. Set \(j = \pi \circ \tilde{j}\). Then \(j: V \to M\). Clearly, \(M\) is an inner model, thus \(L \subseteq M\), and since \(V = L\), \(V = M\). Thus, \(j: V \to V\) is an elementary embedding. You can see that the critical point is \(\kappa\): for all \(\alpha < \kappa\), \([\alpha, \alpha, \cdots] \in_{\mathcal{U}} [0, 1, 2, \cdots]\) and thus \(\pi([0, 1, 2, \cdots]) = \kappa\), and \([0, 1, 2, \cdots] \in_{\mathcal{U}} [\kappa, \kappa, \kappa, \cdots]\). Thus, \(j(\kappa) > \kappa\) and, for all \(\alpha < \kappa\), \(j(\alpha) = \alpha\).
Inner model theory is the practice of finding canonical inner models which are defined in a similar way to \(L\) and have the same fine structure but are able to accomodate large cardinals.
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