Meeting numbers and pseudopowers

Abstract

We study the role of meeting numbers in pcf theory. In particular, Shelah's Strong Hypothesis is shown to be equivalent to the assertion that for any singular cardinal σ of cofinality ω, there is a size σ+ collection Q of countable subsets of σ with the property that for any infinite subset a of σ, there is a member of Q meeting a in an infinite set.

Narrow systems revisited

“…The second and third facts below are both implicit in [22]; the cited references provide more explicit explanations. (1)[22, §2, Claim 2.4] If $$\mu$$ is a singular cardinal of uncountable cofinality and $$\lbrace \nu < \mu \mid \mathrm{pp}(\nu) = \nu ^+\rbrace$$ is stationary in $$\mu$$, then $$\mathrm{pp}(\mu) = \mu ^+$$. (2)[19, Observation 4.4] Suppose that $$\mu$$ is a singular cardinal and $$\mathrm{pp}(\mu) > \mu ^+$$. Then, there is an increasing sequence of regular cardinals $$\vec{\mu } = \langle \mu _i \mid i < \mathrm{cf}(\mu) \rangle$$ converging to $$\mu$$ such that $$\mathrm{cf}(\prod \vec{\mu }, <^*) > \mu ^+$$. (3)[13, Proposition 4.18] Let $$\kappa$$ be an infinite cardinal such that $$\sf SSH$$ holds above $$\kappa$$.…”