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The PCF conjecture and large cardinals. (English) Zbl 1153.03024
For details regarding PCF theory, reference may be made to S. Shelah’s papers of the 1980’s or his book [Cardinal arithmetic. Oxford: Clarendon Press (1994; Zbl 0848.03025)]. Shelah’s PCF conjecture says: If \(a\) is a set of regular cardinals such that \(|a| < \min(a)\), then \(|\text{pcf}(a)| = |a|\) where \(\text{pcf}(a)\) is the result of the application of Shelah’s topological closure operator pcf to the set \(a\).
From the abstract: “In this paper, the author proves that a combinatorial consequence of the negation of the PCF conjecture for intervals, involving free subsets relative to set mappings, is not implied by even the strongest known large cardinal axiom.”

MSC:
03E04 Ordered sets and their cofinalities; pcf theory
03E05 Other combinatorial set theory
03E10 Ordinal and cardinal numbers
03E35 Consistency and independence results
03E55 Large cardinals
Citations:
Zbl 0848.03025
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