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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:
 30000 Ordered sets and their cofinalities; pcf theory 300000 Other combinatorial set theory 3e+10 Ordinal and cardinal numbers 3e+35 Consistency and independence results 3e+55 Large cardinals
##### Keywords:
PCF theory; PCF conjecture; large cardinals
Zbl 0848.03025
Full Text:
##### References:
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