Comment by the_fall
9 days ago
> I believe real numbers to be completely natural,
Most of real numbers are not even computable. Doesn't that give you a pause?
9 days ago
> I believe real numbers to be completely natural,
Most of real numbers are not even computable. Doesn't that give you a pause?
Why would we expect most real numbers to be computable? It's an idealized continuum. It makes perfect sense that there are way too many points in it for us to be able to compute them all.
Maybe I'm getting hung up on words, but my beef is with the parent saying they find real numbers "completely natural".
It's a reasonable assumption that the universe is computable. Most reals aren't, which essentially puts them out of reach - not just in physical terms, but conceptually. If so, I struggle to see the concept as particularly "natural".
We could argue that computable numbers are natural, and that the rest of reals is just some sort of a fever dream.
> It's a reasonable assumption that the universe is computable
Literally every elementary particle enters the chat to disagree. Also every cloud of smoke and each whisp of dissipated heat.
It feels like less of an expectation and more of a: the "leap" from the rationals to the reals is a far larger one than the leap from the reals to the complex numbers. The complex numbers aren't even a different cardinality.
> for us to be able to compute them all
It's that if you pick a real at random, the odds are vanishingly small that you can compute that one particular number. That large of a barrier to human knowledge is the huge leap.
The idea is we can't actually prove a non-computable real number exists without purposefully having axioms that allow for deriving non-computable things. (We can't prove they don't exist either, without making some strong assumptions).
> The idea is we can't actually prove a non-computable real number exists without purposefully having axioms that allow for deriving non-computable things.
Sorry, what do you mean?
The real numbers are uncountable. (If you're talking about constructivism, I guess it's more complicated. There's some discussion at https://mathoverflow.net/questions/30643/are-real-numbers-co... . But that is very niche.)
The set of things we can compute is, for any reasonable definition of computability, countable.
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You can go farther and say that you can't even construct real numbers without strong enough axioms. Theories of first order arithmetic, like Peano arithmetic, can talk about computable reals but not reals in general.