Reflection principle

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 \(N \subseteq V_\alpha\), \(x_0, x_1, \cdots, x_n \in V_\alpha\), \(\varphi(x_0, x_1, \cdots, x_n)\) 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, however, this is actually provable in \(\mathrm{ZF}\), rather than being a candidate for a large cardinal axiom or other new powerful axiom for set theory.

Azriel Levy proved both the truth of the reflection principle, over \(\mathrm{ZF}\), and that the truth predicate for \(\Sigma_n\)-formulae is \(\Sigma_{n+1}\), implying we can find a club of cardinals \(\kappa\) so that each \(V_\kappa\) is a \(\Sigma_n\)-elementary substructure of \(V\) - such cardinals are called \(\Sigma_n\)-correct.

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 is a cumulative hierarchy if, 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 \(W\).

This can be used to show there are arbitrarily large stable ordinals, for example.

Alternate meaning

An alternate type of reflection principle instead asserts that, instead of properties of an inner model reflecting down to a level of a cumulative hierarchy, properties of a single level of a cumulative hierarchy reflect down to a lower level of the same cumulative hierarchy. In particular, this gives rise to stability, reflecting ordinals and indescribable or shrewd cardinals.

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 \(L_\beta \models \varphi(b)\).

By downwards absoluteness, an ordinal is \(\Pi_0\)-, \(\Sigma_0\)-, \(\Delta_0\)-, \(\Delta_1\)- or \(\Sigma_1\)-reflecting (notice the first three are synonymous) iff it is a limit ordinal. Also, the Tarski-Vaught test implies that an ordinal is \(\Sigma_{n+1}\)-reflecting iff it is \(\Pi_n\)-reflecting, and one can easily check that being \(\Pi_2\)-reflecting and admissible are equivalent, by converting any \(\Pi_2\)-formula into a formula equivalent to "\(f\) is total" for some \(\Delta_0(L_\alpha)\)-definable \(f\). This leads to some connections between reflecting ordinals and \(\alpha\)-recursion theory.

Using the Tarski-Vaught test and \(\Sigma_{n+1}\)-truth predicate for \(\Sigma_n\), notice that the reflection principle for \(W = L\) implies the existence of arbitrarily large \(\Pi_n\)-reflecting ordinals.

A large cardinal axiom based off of this alternate reflection principle is the following: extend the language of set theory to \(\mathcal{L}^*(\mathbf{U})\) by adding a unary predicate symbol \(\mathbf{U}\). For a structure of the form \(\langle X, E, \mathcal{A} \rangle\) and a \(\mathcal{L}^*(\mathbf{U})\)-formula \(\varphi\), we define \(\langle X, E, \mathcal{A} \rangle \models \varphi\) by interpreting \(\mathbf{U}(t)\) as \(t \in \mathcal{A} \cap X\). In old historical terms, a cardinal \(\kappa\) was said to be \(\Gamma\)-indescribable, for \(\Gamma \subseteq \mathcal{L}^*(\mathbf{U})\) if, for every \(\varphi \in \Gamma\) and \(A \subseteq \kappa \times \kappa\), if \((\kappa, <, A) \models \varphi\), then \((\alpha, <, A \upharpoonright \alpha) \models \varphi\) for some \(\alpha < \kappa\). In particular, if \(\kappa\) is \(\mathcal{L}^*(\mathbf{U})\)-reflecting then it is regular, and so on. However, the more modern definition is the following:

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 \(\Pi^0_2\)-indescribable iff it is strongly inaccessible. Also, \(\kappa\) is \(\Pi^1_1\)-indescribable iff it is 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 strongly Mahlo, strongly hyper-Mahlo, and more.