[xelxebar]

> What do "real" numbers buy you?

They're well-known and have a simpler implementation, and we are familiar with their quirks. There is a giant body of useful knowledge built up around standard real analysis. That doesn't really exist if you insist on using only computable numbers.

The computables are also more fiddly in many ways. Because equality is undecidable, you can't have discontinuous functions, you need to carry around error epsilons all over the place, and we lose useful tools like the Heine-Borel theorem, I think.

Try proving some results in PDE theory, and I think you might change your mind.

In general, I find clarity in thinking of numbers as the system that implements them, rather than as platonic objects with individual reality. What does using Old Boring tech buy you over using Shiny New Thing?

[testaccount28]

> equality is undecidable

equality is always undecidable until you see the light of intuition. consider the rational number whose numerator is 0 if $theorem is true, and 1 if it is false, and whose denominator is 1.

[xelxebar]

Okay, theorem=generalized-continuum hypothesis. If you use exotic axioms to give that a definite result, the go eat a Gödel.

We define computable numbers to be Turing machines, lambda reduction processes, or whatever your favorite model of computation happens to be. If you don't like this kind of definition, then we need to talk philosophy of computation.

To decide equality, we let your machines clunk along until they both produce a result, which we then compare (using another machine). Hello Mr. Halting Problem. Specific programs are fine, but comparing against arbitrary classes of program is the bugger. This is why discontinuous functions cannot exist in a hardline computable analysis theory.

[btilly]

That is not a number in constructivism.

But there are numbers in constructivism for which it is unknown whether they are zero. Some of which must remain unknown, if mathematics is consistent. This is a rather important and weird edge case.