Various Intuitions and Conceptualizations of Measurable Cardinals.

[biffus22]

The concept of a "measurable cardinal" is rather difficult for many students of "Intermediate" Set Theory to grasp in terms of more basic set theoretic concepts -- as opposed say to concepts dealing with the relations among various "universes" or "models" etc. In fact, much of the problem may derive from the difficulty of imagining a "non-principal ultrafilter" within the P(X), the Power set of X. Whatever the difficulties involved, the concept is, it seems, also very difficult to teach. I am thus interested in how other people here understand, or better, "grasp," what a measurable cardinal is and is not, and how they attempt to teach their students the concept using more basic concepts such students already comprehend.