Reflection principle: Difference between revisions
→Justification and motivation for large cardinal axioms
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A reflection principle is a principle stating that sets can be found which "behave like" the universe of all sets. There are various reflection principles of differing strengths, ranging from provable in ZFC up to the large cardinal axioms.
The reflection principle is the assertion that properties of the universe of all sets are "reflected" down to a smaller set. Formally, for every formula \(\varphi\) and set \(N\), there is some limit ordinal \(\alpha\) so that \(N \subseteq V_\alpha\) and, for all \(x_0, x_1, \cdots, x_n \in V_\alpha\), \(\varphi(x_0, x_1, \cdots, x_n)\) is true in \(V_\alpha\) iff it is really true. This may be considered a guarantee of the existence (be it mathematical or metaphysical) of Cantor's [[Absolute infinity|Absolute]], however, this 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.▼
==Levy-Montague reflection==
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<nowiki>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 \(W\).</nowiki>▼
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 cardinal|correct]].
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This can be used to show there are arbitrarily large [[Stability|stable]] ordinals, for example.
== Justification and motivation for large cardinal axioms ==
Reflection principles are often justified using Cantor's description of the class of all ordinals as incomprehensible:<ref>P. Maddy, "[https://www.cs.umd.edu/~gasarch/BLOGPAPERS/belaxioms1.pdf Believing the Axioms I]". Journal of Symbolic Logic vol. 53, no. 2 (1988), pp.481--511.</ref><sup>p.503</sup>
: Hallet ... traces reflection to Cantor's theory that the sequence of all transfinite numbers is absolutely infinite, like God. As such, it is incomprehensible to the finite human mind, not subject to mathematical manipulation. Thus nothing we can say about it, no theory or description, could single it out; in other words, anything true of \(V\) is already true of some [\(V_\alpha\)].
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 some informal arguments illustrating the use of reflection principles.
: The simplest is perhaps: the universe of sets is inaccessible (i.e., satisfies the replacement axiom), ''therefore'' there is an inaccessible cardinal. This can be elaborated somewhat, as follows. Let \(\theta_\nu\) enumerate the inaccessible cardinals. By the same sort of reasoning, \(\theta_\nu\) is not bounded; the Cantor absolute \(\Omega\) (all ordinals) is an inaccessible above any proposed bound \(\beta\), ''therefore'' there is an inaccessible cardinal above \(\beta\). Clearly, then, there are \(\Omega\) inaccessibles above below \(\Omega\); ''therefore'' there is an inaccessible \(\kappa\) such that there are \(\kappa\) inaccessibles below it (i.e., \(\kappa=\theta_\kappa\)).
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" />
===Examples of ordinal properties from reflection principles===
Let \(\Gamma\) be a set of formulae. Then an ordinal \(\alpha\) is \(\Gamma\)-reflecting if, for every \(b \in L_\alpha\) and \(\varphi \in \Gamma\) so that \(L_\alpha \models \varphi(b)\), there is some \(\beta < \alpha\) so that \(b \in L_\beta\) and \(L_\beta \models \varphi(b)\).
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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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