The Magidor function and diamond. (English) Zbl 1237.03031

Let \(\kappa\) be a regular uncountable cardinal and \(\lambda\) be a cardinal \(>\kappa\). The author shows that (Theorem 4) if \(2^{<\kappa}\leq M(\kappa,\lambda)\), then \(\diamondsuit_{\kappa, \lambda}\) holds, where \(M(\kappa,\lambda)\) is the Magidor function, which is equal to \(\lambda^\omega\) if \(\text{cf}(\lambda) \geq \kappa\), and \((\lambda^+)^\omega\) otherwise. Here the diamond principle \(\diamondsuit_{\kappa,\lambda}\) is a generalization of Jensen’s classical diamond, and it asserts that there exists a sequence \(\langle s_a: a \in P_\kappa(\lambda)\rangle\) with \(s_a \subseteq a\) such that for any \(X \subseteq \lambda\), \(\{a : s_a = X\cap a\}\) is a stationary subset of \(P_\kappa(\lambda)\).
In some sense, this result is optimal due to the well-known fact that, assuming \(\diamondsuit_{\kappa,\lambda}\), the least cardinality of any closed unbounded subset of \(P_\kappa(\lambda)\), \(c(\kappa,\lambda)\), equals \(\lambda^{<\kappa}\), and a result of Magidor that, given the nonexistence of \(\omega_1\)-Erdős cardinals in the core model \(K\), \(c(\kappa,\lambda)=M(\kappa,\lambda)\). Therefore, assuming \(\diamondsuit_{\kappa,\lambda}\) and there is no \(\omega_1\)-Erdős cardinal in the core model \(K\), \(2^{<\kappa}\leq M(\kappa,\lambda)\).
However, the author also exhibits (relative to a large cardinal) a (Cohen) forcing model in which \(2^{<\kappa}> M(\kappa,\lambda)\) and \(\diamondsuit_{\kappa,\lambda}\) holds. In fact, a strictly stronger variation of \(\diamondsuit_{\kappa,\lambda}\), \(\diamondsuit_{\kappa,\lambda}[NG_{\kappa,\lambda}]\), is instead discussed throughout the paper. A further variation of the form \(\diamondsuit_{\kappa,\lambda,\lambda}[J]\) is discussed at the end of the paper.


03E05 Other combinatorial set theory
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