The Continuum Hypothesis - Logic at Harvard

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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
Theorem 1.6 (Shelah, 1978). If ℵ ω is a strong limit cardinal then2 ℵω < ℵ (2 ℵ 0) +.One drawback of this result is that the bound is sensitive to the actual sizeof 2 ℵ 0, which can be anything below ℵ ω . Remarkably Shelah was later ableto remedy this with the development of his pcf (possible cofinalities) theory.One very quotable result from this theory is the following:Theorem 1.7 (Shelah). If ℵ ω is a strong limit cardinal then (regardless ofthe size of 2 ℵ 0)2 ℵω < ℵ ω4 .In summary, although the continuum function at regular cardinals is relativelyunconstrained in ZFC, the continuum function at singular cardinalsis (provably in ZFC) constrained in significant ways by the behaviour of thecontinuum function on the smaller cardinals.Further Reading: For more cardinal arithmetic see Jech (2003). For more onthe case of singular cardinals and pcf theory see Abraham & Magidor (2010)and Holz, Steffens & Weitz (1999).2 Definable Versions of the Continuum Hypothesisand its NegationLet us return to the continuum function on regular cardinals and concentrateon the simplest case, the size of 2 ℵ 0. One of Cantor’s original approaches toCH was by investigating “simple” sets of real numbers. 4 One of the firstresults in this direction is the Cantor-Bendixson theorem that every infiniteclosed set is either countable or contains a perfect subset, in which case ithas the same cardinality as the set of reals. In other words, CH holds (inthis formulation) when one restricts one’s attention to closed sets of reals.In general, questions about “definable” sets of reals are more tractable thanquestions about arbitrary sets of reals and this suggests looking at definableversions of the continuum hypothesis.4 See Hallett (1984), pp. 3-5 and §2.3(b).7
Page 1 and 2: The Continuum HypothesisPeter Koell
Page 3 and 4: and Determinacy”, which contains
Page 5: infinite cardinals: Assuming GCH, i
Page 9 and 10: this version of CH holds for all se
Page 11 and 12: more generally, all universally Bai
Page 13 and 14: arithmetic (orabout L(R)) inthepres
Page 15 and 16: 3.2 Ω-LogicWe will now recast the
Page 17 and 18: We will need a strong form of this
Page 19 and 20: and T A ≠ T B .How then shall one
Page 21 and 22: Now to arrive at this conclusion on
Page 23 and 24: Theorem 4.1 (Woodin). Assume ZFC an
Page 25 and 26: Theorem 4.6 (Woodin). Assume ZFC an
Page 27 and 28: forced to fail. The challenge for t
Page 29 and 30: So, of the various theories T A inv
Page 31 and 32: Theorem 5.8 (Koellner and Woodin).
Page 33 and 34: the fact that under (∗) the theor
Page 35 and 36: in his work.Now, we have a great de
Page 37 and 38: Koellner, P. (2010). Strong logics

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