The Continuum Hypothesis - Logic at Harvard

1.2 Singular CardinalsThe case of cardinal arithmetic on singular cardinals is much more subtle.Forthesakeofcompletenesswepausetobrieflydiscussthisbeforeproceedingwith the continuum hypothesis.It was generally believed that, as in the case for regular cardinals, thebehaviour of the function κ ↦→ 2 κ would be relatively unconstrained withinthe setting of ZFC. But then Silver proved the following remarkable result: 3Theorem 1.4 (Silver, 1974). If ℵ δ is a singular cardinal of uncountablecofinality, then, if GCH holds below ℵ δ , then GCH holds at ℵ δ .It turns out that (by a deep result of Magidor, published in 1977) GCHcan first fail at ℵ ω (assuming the consistency of a supercompact cardinal).Silver’s theorem shows that it cannot first fail at ℵ ω1 and this is provable inZFC.This raises the question of whether one can “control” the size of 2 ℵ δwith a weaker assumption than that ℵ δ is a singular cardinal of uncountablecofinality such that GCH holds below ℵ δ . The natural hypothesis to consideris that ℵ δ is a singular cardinal of uncountable cofinality which is a stronglimit cardinal, that is, that for all α < ℵ δ , 2 α < ℵ δ . In 1975 Galvin andHajnal proved (among other things) that under this weaker assumption thereis indeed a bound:Theorem 1.5 (Galvin and Hajnal, 1975). If ℵ δ is a singular strong limitcardinal of uncountable cofinality then2 ℵ δ< ℵ (|δ| cf(δ) ) +.It is possible that there is a jump—in fact, Woodin showed (again assuminglarge cardinals) that it is possible that for all κ, 2 κ = κ ++ . What the abovetheorem shows is that in ZFC there is a provable bound on how big the jumpcan be.The next question is whether a similar situation prevails with singularcardinals of countable cofinality. In 1978 Shelah showed that this is indeedthe case. To fix ideas let us concentrate on ℵ ω .3 To say that GCH holds below δ is just to say that 2 ℵα = ℵ α+1 for all ω α < δ andto say that GCH holds at δ is just to say that 2 ℵ δ= ℵ δ+1 ).6