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If you did want to salvage a "calculus without real numbers" the computables are probably more promising. All of the criticisms of the article still entirely ap
by jerf 7y ago
If you did want to salvage a "calculus without real numbers" the computables are probably more promising. All of the criticisms of the article still entirely apply to them, but there's a better chance you could come up with substitute definitions that may yield some useful concepts; they're closed on a more useful operations. I believe this is an actual studied area of math. But it's certainly more complicated than using real numbers. We may not teach, ahem, "real" real numbers to high school students, but you'll get to it fairly early in dedicated college math courses at the undergrad level.
- roywiggins 7y agoThere's some work on it apparently. https://en.wikipedia.org/wiki/Computable_analysis https://en.wikipedia.org/wiki/Computable_analysis
- inflatableDodo 7y agoI've wondered before if there could be a well defined set of numbers between the integers and reals (perhaps with its own class of infinite set) that nicely excludes the uncomputables. If there is, I suspect that it might be what the universe is actually using and if I was going to go looking for it, I'd follow the breadcrumbs from umbral moonshine and go sniffing around the monster group.
- roywiggins 7y agoOther than the rationals? Those are all computable. Or the algebraics. Whether there's an infinity between the naturals and the reals depends on the continuum hypothesis, which is independent of the usual axioms of set theory, so you can pick.