The Ulam matrix and the definition of measurable cardinals.

Question

I believe I have a pretty good handle on what an "Ulam matrix" is after reading Ulam's famous 1930 paper on the subject as well as looking at some of the previous answers to questions about that object here on MSE. What I still don't understand, however, is how being able to set up an Ulam matrix for a Real interval, as Ulam does in his original paper, allows one to describe a measurable cardinal, which in fact he does not do in that paper. It seems to me that all the Ulam matrix allows one to do is specify an uncountable number of pairwise disjoint subintervals on the line, but how does that in turn allow one to define a measurable cardinal, a set with a vastly higher cardinality than the set of all those subintervals. I am obviously missing something important here.

Answer 1

The Ulam matrix argument shows that if there is a $\sigma$-complete measure on the real numbers, then the real numbers are not of size $\aleph_1$, and more generally the measure concentrates on some $\kappa\leq2^{\aleph_0}$ which is weakly inaccessible.

I suggest Chapter 10 in Jech's Set Theory (3rd Millennium Edition) as a good introduction to the topic as a whole.