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vi21maobk9vp comments on Which fields of learning have clarified your thinking? How and why? - Less Wrong Discussion
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Comments (64)
What signature do we need for it? Because in the first-order theory of real numbers without sets you cannot express functions or sequences.
For example, full theory of everything expressible about real numbers using "+, *, =, 0, 1, >" can be reolved algorithmically.
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 symbols in formulas into the theory is one of the easy-to-reach interesting things in set theory.
About natural numbers - direct set representation is quite unnatural; underlying idea of well-ordered set is just an expression of the idea that natural numbers are the numbers we can use for counting.
The true all-mathematical value of set theory is, of course to be a universal measure of weirdness: if your theory can be modelled inside ZFC, you can stop explaining why it has no contradictions.