×

zbMATH — the first resource for mathematics

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.

MSC:
03E05 Other combinatorial set theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Axiomatic set theory 13 pp 397– (1971)
[2] Set theory (1980)
[3] Set theory (2002)
[4] DOI: 10.1016/0003-4843(73)90014-4 · Zbl 0262.02062
[5] DOI: 10.1090/S0002-9947-04-03445-2 · Zbl 1082.03034
[6] DOI: 10.1007/BF02401842 · Zbl 1017.03022
[7] DOI: 10.1007/BF02764635 · Zbl 0798.03047
[8] DOI: 10.1090/S0002-9947-1992-1041044-9
[9] DOI: 10.1007/978-1-4020-5764-9_15 · Zbl 1198.03053
[10] DOI: 10.1007/BF02785587 · Zbl 0955.03047
[11] Around classification theory of models pp 224– (1986)
[12] DOI: 10.1016/j.apal.2008.09.022 · Zbl 1173.03036
[13] DOI: 10.4153/CMB-2008-057-5 · Zbl 1167.03030
[14] Commentationes Universitatis Carolinae 48 pp 211– (2007)
[15] Set theory and its applications pp 119– (1989)
[16] DOI: 10.1090/S0002-9947-1990-0939805-5
[17] Bounds for covering numbers 71 pp 1303– (2006)
[18] DOI: 10.1142/9789812796554_0015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.