Sasha Volokh has pointed out to me in an email both that the analysis which I went through is done in the beginning of Ec grad school and that

In fact, there's been a whole effort (which we covered) to rederive all of microeconomics without using utility functions and only using the Weak
Axion of Revealed Preference. (You can derive *almost* everything that way.) Also, social choice theory (Arrow etc.) generally uses preferences,
not utility functions.

His email and my thinking about this makes me almost wish that I'd taken an economics class at some point... I say almost cause I did try at least once and didn't make it through one class... The way the math was dealt with just annoyed me too much. However, if there were an Ec class specifically for math majors (you know, like math for ec majors, except the other way around) I think I really would have loved it.

Thanks to Sasha for the quick and insightful reply.

Speaking of thanks, thanks to Cooped Up for linking to our site, and welcome to anyone who is following that link. Anyone else should follow our link to them, he currently has a fascinating post about the 9th ammendment which is definitely worth reading.