http://www.pnas.org/content/110/33/13300.full ## Abstract

01:30:15

Thanks Jonathan

I can't help it, I was about to log off, but I love this post, I could do this for my entire life, can't someone grant me a scholarship, fellows??!

Cantor proved in 1874 [Cantor G (1874) *J Reine Angew Math* 77:258–262] that the continuum is uncountable, and Hilbert’s first problem asks whether it is the smallest uncountable cardinal. A program arose to study cardinal invariants of the continuum, which measure the size of the continuum in various ways. By Gödel [Gödel K (1939) *Proc Natl Acad Sci USA* 25(4):220–224] and Cohen [Cohen P (1963) *Proc Natl Acad Sci USA* 50(6):1143–1148], Hilbert’s first problem is independent of ZFC (Zermelo-Fraenkel set theory with the axiom of choice). Much work both before and since has been done on inequalities between these cardinal invariants, but some basic questions have remained open despite Cohen’s introduction of forcing. The oldest and perhaps most famous of these is whether “,” which was proved in a special case by Rothberger [Rothberger F (1948) *Fund Math* 35:29–46], building on Hausdorff [Hausdorff (1936) *Fund Math* 26:241–255]. In this paper we explain how our work on the structure of Keisler’s order, a large-scale classification problem in model theory, led to the solution of this problem in ZFC as well as of an a priori unrelated open question in model theory.