Reflection principle: Difference between revisions

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: