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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 there is a measurable cardinal, and let \(\kappa\) isbe the least measurable cardinal, 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 \(\alpha < \kappa\), \(j(\alpha) = \alpha\). Let \(\varphi(x)\) be the formula "\(x\) is the least measurable cardinal", which is first-order expressible. Then, since \(V \models \varphi(\kappa)\), we have \(V \models \varphi(j(\kappa))\). Therefore, \(j(\kappa)\) is the least measurable cardinal. Contradiction!</nowiki>
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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