I'm not sure, presumably to "+*=01>" one adds a bunch of special functions. The "o-minimal approach" to differential geometry requires no sequences. Functions are encodable as definable graphs, so are vector fields.
As you note, for completeness reasons an actual o-minimal theory is not as strong as set theory; one has to smuggle in e.g. natural numbers somehow, maybe with sin(x). I could have made a less tendentious point with Godel numbering.
Again, this is meant as a kind of joke, not as a natural way of looking at sets. The point is that I don't regard von Neumann's {{},{{}},{{},{{}}}} as a natural way of looking at the number three, either.
Once you say that functions are definable graphs, you are on a slippery slope. If you want to prove something about "all functions", you have to be able to quantify over all formulas. This means you have already smuggled natural numbers into the model without defining their properties well...
When you consider a usual theory, you are only interested in the formulas as long as you can write - not so here, if you want to say something about all expressible functions.
And studying (among other things) effects of smuggling natural numbers used to count...
Did computer programming make you a clearer, more precise thinker? How about mathematics? If so, what kind? Set theory? Probability theory?
Microeconomics? Poker? English? Civil Engineering? Underwater Basket Weaving? (For adding... depth.)
Anything I missed?
Context: I have a palette of courses to dab onto my university schedule, and I don't know which ones to chose. This much is for certain: I want to come out of university as a problem solving beast. If there are fields of inquiry whose methods easily transfer to other fields, it is those fields that I want to learn in, at least initially.
Rip apart, Less Wrong!