Cardinal Arithmetic


I just came across Saharon Shelah’s website. I had heard of him in the context of the Whitehead problem before (is every abelian group $A$ such that $Ext^1(A,\mathbb{Z})=0$ free?). Shelah proved, amazingly (at least to me), that the Whitehead problem is independent of ZFC.

But it appears that Shelah did many other very interesting things. Using his theory of possible cofinalities, one can show that $\aleph_\omega^{\aleph_0}\le 2^{\aleph_0}+\aleph_{\omega_4}$, which I think is very surprising. I assume the point is that ZFC cannot decide if $\aleph_\omega^{\aleph_0}\le 2^{\aleph_0}$ or not, and ZFC cannot decide if $\aleph_\omega\le\aleph_{\omega_4}$, but it can show that one must hold.

If I didn’t have so much stuff I need to get done this week I’d look into this more.

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About Simon

Hi. I'm Simon Rubinstein-Salzedo. I'm a mathematics postdoc at Dartmouth College. I'm also a musician; I play piano and cello, and I also sometimes compose music and study musicology. I also like to play chess and write calligraphy. This blog is a catalogue of some of my thoughts. I write them down so that I understand them better. But sometimes other people find them interesting as well, so I happily share them with my small corner of the world.
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One Response to Cardinal Arithmetic

  1. mad_emperor says:

    I just read about Shelah on Wikipedia, and what says about him is simply amazing. I don’t know why I haven’t heard of him before. Good luck with applying to grad school.

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