Singular cardinals hypothesis

In set theory, the singular cardinals hypothesis (SCH) arose from the question of whether the least cardinal number for which the generalized continuum hypothesis (GCH) might fail could be a singular cardinal.

According to Mitchell (1992), the singular cardinals hypothesis is:

Here, κ⁺ denotes the successor cardinal of κ.

Since SCH is a consequence of GCH which is known to be consistent with ZFC, SCH is consistent with ZFC. The negation of SCH has also been shown to be consistent with ZFC, if one assumes the existence of a sufficiently large cardinal number. In fact, by results of Moti Gitik, ZFC + the negation of SCH is equiconsistent with ZFC + the existence of a measurable cardinal κ of Mitchell order κ⁺⁺.

Another form of the SCH is the following statement:

where cf denotes the cofinality function. Note that κ^cf(κ)= 2^κ for all singular strong limit cardinals κ. The second formulation of SCH is strictly stronger than the first version, since the first one only mentions strong limits; from a model in which the first version of SCH fails at ℵ_ω and GCH holds above ℵ_ω+2, we can construct a model in which the first version of SCH holds but the second version of SCH fails, by adding ℵ_ω Cohen subsets to ℵ_n for some n.

Silver proved that if κ is singular with uncountable cofinality and 2^λ = λ⁺ for all infinite cardinals λ < κ, then 2^κ = κ⁺. Silver's original proof used generic ultrapowers. The following important fact follows from Silver's theorem: if the singular cardinals hypothesis holds for all singular cardinals of countable cofinality, then it holds for all singular cardinals. In particular, then, if ${\displaystyle \kappa }$ is the least counterexample to the singular cardinals hypothesis, then ${\displaystyle cf(\kappa )=\omega }$.

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