## 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.