What are some of the most advenced results in cardinal arithmetic? For the sake of discussion, I'm not intrested in consistency Results (e.g: "It's consistent that $2^{\aleph_0}=\aleph_1$"). A good example of what I search for is Silver's theorem. Thanks.
Results in cardinal arithmetic
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$\begingroup$
set-theory
cardinals
1 Answers
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A lot of modern set theory has roughly had the theme, "Nothing about cardinal arithmetic can be proved in ZFC." My favorite source of counterexamples to this idea is PCF (possible cofinalities) theory. This was invented by Shelah, and used by him and others to prove a number of ZFC-results; see also this review of Shelah's book on cardinal arithmetic.
One of the most famous of these is the following statement: $$\mbox{If $\aleph_\omega$ is a strong limit, then $2^{\aleph_\omega}<\aleph_{\omega_4}$}.$$ It is not currently known whether this is sharp, and indeed Shelah is on record as asking "Why the hell is it four?".
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0Are there any more results like this? or this+silver are the only strong results known? – 2017-01-07
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0@YfYf If you google around for PCF theory, you'll find **lots** of statements in a similar vein. Or, look at Shelah's book. – 2017-01-07
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0ok, i will, thanks. – 2017-01-07
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0How difficult is that proof? Could it be presented in a 90 minute seminar? – 2017-09-04
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0@Mingus It's extremely difficult, and absolutely not, respectively - unless your audience already knows some PCF theory. – 2017-09-09