We study the relationship between the cofinality c(Sym(ω)) of the infinite symmetric group and the cardinal invariants u and g. In particular, we prove the following two results. Theorem 0.1. It is consistent with ZFC that there exists a simple Pω1-point and that c(Sym(ω)) = ω2 = 2ω. Theorem 0.2. If there exist both a simple Pω1-point and a Pω2-point, then c(Sym(ω)) = ω1.
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