Reflection principle: Difference between revisions
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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>https://arxiv.org/abs/1708.06669</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. |
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>https://arxiv.org/abs/1708.06669</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. |
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Azriel Levy proved (each instance of) the reflection principle over \(\mathrm{ZF}\).<ref>A. Kanamori, |
Azriel Levy proved (each instance of) the reflection principle over \(\mathrm{ZF}\).<ref>A. Kanamori, ''The Higher Infinite'', p.58. Springer Monographs in Mathematics (2003). ISBN 978-3-540-88866-6.</ref> Since the truth predicate for a certain class of \(\Sigma_n\)-formulae is itself \(\Sigma_n\), there is a club of cardinals \(\kappa\) so that each \(V_\kappa\) is a \(\Sigma_n\)-elementary substructure of \(V\)<ref name="Welch17">P. D. Welch, "[https://research-information.bris.ac.uk/ws/portalfiles/portal/132496875/CLMPS_Helsinki_2015.pdf Global Reflection Principles]", pp.8--10. In ''Logic, methodology and philosophy of science: proceedings of the fifteenth international congress''.</ref> - such cardinals are called \(\Sigma_n\)-correct. |
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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) |
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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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. |
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. |
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==References== |
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