Reflection principle: Difference between revisions

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, ''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]].

: It may be helpful to give sinesome 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., \(\kappa=\theta_\kappa\)).