I think we may have something of a clash of backgrounds here. The reason I'm inclined to take the real continuum seriously is that there are numerous physical quantities that seem to be made of real or complex numbers. The reason I take mathematical induction seriously is that it looks like you might always be able to add one minute to the total number of minutes passed. The reason I take second-order logic seriously is that it lets me pin down a single mathematical referent that I'm comparing to the realities of space and time.

The reason I'm not inclined to take the least uncountable ordinal seriously is because, occupying as it does a position above the Church-Kleene ordinal and all possible hypercomputational generalizations thereof, it feels like talking about the collection of all collections - the supremum of an indefinitely extensible quality that shouldn't have a supremum any more than I could talk about a mathematical object that is the supremum of all the models a first-order set theory can have. If set theory makes the apparent continuum from physics collide with this first uncountable ordinal, my inclination is to distrust set theory.