Reflection principle: Difference between revisions
→Justification and motivation for large cardinal axioms
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==Levy-Montague reflection==
One of the most common reflection principles is the assertion that first-order properties of the universe of all sets are "reflected" down to a rank \(V_\alpha\) of the von Neumann hierarchy. Formally, for every formula \(\varphi\) and sequence of parameters \(x_0, x_1, \ldots, x_n\), there is some ordinal \(\alpha\) where \(x_0, x_1, \cdots, x_n \in V_\alpha\), and \(\varphi(x_0, x_1, \cdots, x_n)\) is true in \(V_\alpha\) iff it is really true. This is known as the Levy-Montague reflection principle.<ref>N. Barton, A. E. Caicedo, G. Fuchs, J. D. Hamkins, J. Reitz, R. Schindler, [https://arxiv.org/abs/1708.06669 Inner-model reflection principles] (2018). Accessed 4 September 2023.</ref> Because of the use of "\(\varphi(x_0, x_1, \cdots, x_n)\) is true" for an arbitrary first-order formula \(\phi\), but Tarski's undefinability theorem this may not be stated as a single first-order formula. This may be appear to be a formalization of Cantor's notion of Cantor's [[Absolute infinity|Absolute]], however there are ranks below which this principle holds. Each instance of this schema is actually provable in \(\mathrm{ZF}\), rather than being a candidate for a [[Large cardinal|large cardinal axiom]] or other new powerful axiom for set theory.
Azriel Levy proved (each instance of) the reflection principle over \(\mathrm{ZF}\).<ref>A. Kanamori,
An even more general form of the reflection principle is as follows. Say a cumulative hierarchy is a family of sets \(W_\alpha\) indexed by ordinals so that, for all \(\alpha\), we have \(W_\alpha \subseteq W_{\alpha+1} \subseteq \mathcal{P}(W_\alpha)\); and for all limit ordinals \(\lambda\), \(W_\lambda = \bigcup_{\alpha < \lambda} W_\alpha\). Let \(W = \bigcup_{\alpha \in \mathrm{Ord}} W_\alpha\). Then, for every formula \(\varphi\), there are arbitrarily large \(\alpha\) so that, for all \(x_0, x_1, \cdots, x_n \in W_\alpha\), \(\varphi(x_0, x_1, \cdots, x_n)\) is true in \(W_\alpha\) iff it is true in \(W\). (Citation needed? I have seen this before too somewhere)
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As a result, if it is assumed that \(V\) "should" satisfy reflection, if one wants to generate "true" large cardinal axioms, a good place to start is by using reflection of \(V\) to guarantee existence of sets with strong reflective properties. Reinhardt gives the following example:<ref>W. N. Reinhardt, "[https://canvas.eee.uci.edu/courses/8140/files/2966398/download?verifier=2cFyXQWmyjg7qKTxIbBasupeSVEkZQZJaE1yGkWz&download_frd=1 Remarks on reflection principles, large cardinals, and elementary embeddings]". In ''Axiomatic Set Theory, Part 2'' (1974), edited by T. Jech, ISBN 978-0-8218-9298-5. MathSciNet ID [https://mathscinet.ams.org/mathscinet/relay-station?mr=0401475 0401475].</ref>
: It may be helpful to give
: The simplest is perhaps: the universe of sets is inaccessible (i.
Reflection may also be used to justify the axioms of existence of indescribable cardinals (soon following). Specifically, a stronger reflection principle may be obtained from Levy-Montague reflection by allowing use of \(\Pi^1_n\) formulae with second-order parameters. Then if the universe satisfies this property, in the style of the above argument, there is a \(V_\kappa\) that satisfies this reflection property, this \(\kappa\) is a \(\Pi^1_n\)-indescribable cardinal. In this way \(\Pi^1_n\) may be seen as a localization of the previous reflecting property to a \(V_\kappa\).<ref name="Welch17" />
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A cardinal \(\kappa\) is \(\Gamma\)-indescribable, for \(\Gamma \subseteq \mathcal{L}^*(\mathbf{U})\) if, for every \(\varphi \in \Gamma\) and \(A \subseteq V_\kappa\), if \((V_\kappa, \in, A) \models \varphi\), then \((V_\alpha, \in, A) \models \varphi\) for some \(\alpha < \kappa\). In particular, \(\kappa\) is \(\Sigma^1_1\)-indescribable iff it is \(\Pi^1_0\)-indescribable iff it is \(\Pi_2\)-indescribable iff it is [[Inaccessible cardinal|strongly inaccessible]]. Also, \(\kappa\) is \(\Pi^1_1\)-indescribable iff it is [[Weakly compact cardinal|weakly compact]]. Being \(\Pi^1_n\)-indescribable is \(\Pi^1_{n+1}\)-describable, and thus any weakly compact cardinal is a limit of strongly inaccessible cardinals. Furthermore, this characterisation of weak compactness implies every weakly compact cardinal is also [[Mahlo cardinal|strongly Mahlo]], strongly hyper-Mahlo, and more. Notice that, unlike reflecting ordinals, the reflection principle does not imply the existence of \(\Pi_n\)-indescribable cardinals for all \(n\). Ord is Mahlo does, but is itself weaker than the existence of a \(\Pi^1_1\)-indescribable.
==References==
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