Reflection principle: Difference between revisions

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, for all \(N \subseteq V_\alpha\) 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.

<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 isso a cumulative hierarchy ifthat, 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>

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 reflecti#ngreflecting 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.