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I definitely wasn't trying to indicate that there was a problem with this, just pointing out that the path of using the Surreals (or anything else) to give an "
by eigenket 6y ago
I definitely wasn't trying to indicate that there was a problem with this, just pointing out that the path of using the Surreals (or anything else) to give an "algorithmic" description of real addition or multiplication is probably a bad idea.
- SAI_Peregrinus 6y agoI wasn't even attempting to give an "algorithmic" description. Just a formulaic one, that depends on the axiom of infinity (and accepting transfinite induction as a valid process). Since even at least one of the usual process for constructing the Reals (Dedekind cuts) needs this I don't feel it's much of a stretch in reasoning. And the Surreals have a nicer recursive formula for multiplication that eventually turns into repeated sums, and they're a strict superset of the Reals, so it does apply there. If you want an algorithmic (no possible need an infinite number of steps) explicit construction of anything on the Reals you're going to be disappointed. They're an infinite set.
- eigenket 6y agoYou can get algorithmic explicit constructions of (for example) integer addition, rational addition and multiplication and even addition and multiplication of algebraic numbers on a Turing machine (all algebraic numbers are computable). All of these sets are infinite. The reals are particularly "badly behaved" even as far as infinite sets go.