Huge reflection
For an infinite cardinal \(\kappa\), a class \(P\), and a class \(\Gamma\) with \(\Gamma = \Sigma_n\) or \(\Pi_n\) for some \(n < \omega\), the \textit{Exact Structural Reflection principle} \(\Gamma(P)\)-ESR\((\kappa)\) asserts the existence of a cardinal \(\lambda > \kappa\) such that for any class \({\mathfrak C}\) of structures of the same type that is \(\Gamma\)-definable with parameters in \(P\), and any \(A \in {\mathfrak C}\) of rank \(\lambda\), there is an elementary embedding from some \(B \in {\mathfrak C}\) of rank \(\kappa\) into \(A\). The paper is a detailed study of this family of principles. To give a taste of the results, more definitions are needed. For \(m < \omega\), let \(C^{(m)}\) denote the \(\Pi_m\)-definable closed unbounded class of all ordinals \(\alpha\) such that \(V_\alpha\) is a \(\Sigma_m\)-elementary substructure of \(V\). For \(0 < n < \omega\) and two infinite cardinals \(\kappa < \lambda\), \(\kappa\) is \textit{weakly parametrically} \(n\)-\textit{exact for} \(\lambda\) if for any \(A \in V_{\lambda + 1}\), there are a transitive, \(\Pi_n (V_{\kappa + 1})\)-correct set \(M\) with \(V_\kappa \cup \{ \kappa \} \subseteq M\), a cardinal \(\lambda' \in C^{(n-1)}\) greater than \(\gimel_\lambda\), and an elementary embedding \(j : M \rightarrow H_{\lambda'}\) with \(j (\mathrm{crit}(j)) = \kappa\), \(j (\kappa) = \lambda\) and \(A \in\mathrm{ran}(j)\). One result states that for an infinite cardinal \(\kappa\) and \(0 < n < \omega\), \(\kappa\) is the least cardinal such that \(\Pi_n(V_\kappa)\)-ESR\((\kappa)\) holds if and only if \(\kappa\) is the least cardinal that is weakly parametrically \(n\)-exact for some \(\lambda > \kappa\). This is then used to show that exact structural reflection is consistencywise quite strong: On the one hand, if \(\kappa\) is weakly parametrically \(1\)-exact for \(\lambda\), then many cardinals \(\rho\) smaller than \(\kappa\) are almost huge with target \(\kappa\) (i.e. \(\rho\) is the critical point of an elementary embedding \(j : V \rightarrow M\) with \(j (\rho) = \kappa\), where \(M\) is transitive and closed under sequences of length less than \(\kappa\)). On the other hand, the existence of \(\kappa < \lambda\) such that \(\kappa\) is for all \(n > 0\) weakly parametrically \(n\)-exact for \(\lambda\) follows from that of an almost \(2\)-huge cardinal. The last part of the article is concerned with sequential forms of ESR that imply the existence of even larger cardinals.
scientific article; zbMATH DE number 7601553
scientific article