My mathematical prejudices: 1. I don't accept choice on the continuum 2. I accept choice on countable collections, as well as countable infinity. 3. I don't think of the continuum as small enough to be a set, 4. I think every subclass (subset) of the real line is measurable, 5. I prefer constructions to abstractions. In this, I hope I am following Paul Cohen in spirit.
my dream theorem is a proof of the consistency of ZFC from a countable computable ordinal, as explicitly describable as possible, which plays the same role for ZFC as $\epsilon_0$ does for PA. This would complete Hilbert's program, giving what I would consider a finitary proof of the consistency of set theory. If you think this is impossible because of Godel's theorem, you haven't understood Godel's theorem fully. The problem is that ordinal naming schemes crap out much too early for this to work with an explicit construction with today's methods.