Reflection principle

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.

Levy-Montague reflection

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. 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, 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 axiom or other new powerful axiom for set theory.

Azriel Levy proved (each instance of) the reflection principle over \(\mathrm{ZF}\). 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 \(V\) - 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 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)

This can be used to show there are arbitrarily large 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:^p.503

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:

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 \(V_\kappa\).

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

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 \(\Pi_2\)-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. 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.