Constructible hierarchy: Difference between revisions

Say a subset \(X\) of \(Y\) is definable if there are some \(z_0, z_1, \cdots, z_n \in Y\) and some formula \(\varphi\) in the language of set theory so that the elements of \(X\) are precisely the \(x\) so that \(Y\) satisfies \(\varphi(x, z_0, z_1, \cdots, z_n)\). For example, under the von Neumann interpretationdefinition of ordinal, the set of even numbers, the set of odd numbers, the set of prime numbers, the set of perfect squares greater than 17, and so on, are all definable. Using elementary cardinal arithmetic, youthere can note that thereare \(\max(\aleph_0, |Y|) = |Y|\) definable subsets of an infinite set \(Y\), and thus "almost all" subsets of an infinite set are not definable. The parameters \(\vec{z}\) aren'tare of importance inwhen the\(Y\) contextis ofuncountable, definableto subsetsensure ofthat thethere naturalare numbers,more sincethan all\(\aleph_0\) elementsdefinable subsets of the\(Y\), naturalbut numbersthey aredo definable,not buthave theyany willeffect be ifwhen \(Y\supseteq\mathbb N\) is uncountablecountable, becausesince noall uncountableelements beof pointwisethe definable,natural andnumbers ensure that there aren't just always \(\aleph_0\)are definable subsets of a set.

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>

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. Therefore, the axiom of constructibility, \(V = L\), has nice consequences such as \(\mathrm{AC}\), \(\mathrm{GCH}\), \(\diamond\), and more. Assuming the consistency of \(\mathrm{ZFC}\), this is independent, and thus seems like a reasonable axiom to add. 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. The holy grail of inner model theory is finding an inner model which satisfies the existence of supercompact cardinals, known as Ultimate-L. Although Ultimate-L has not yet been defined, Woodin has formulated an ideal version of the axiom "V = Ultimate-L" which implies \(\mathrm{GCH}\) and more and should ideally hold if V = Ultimate-L, with respect to an actual construction of Ultimate-L. This is inspired by the fact that, surprisingly, the axiom of constructibility can be formulated without any reference to the constructible hierarchy itself.