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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.

by the_fall1 hours ago|

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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.

by egorelik3 hours ago|

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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).

by creata3 hours ago|

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> 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.

by aeneasmackenzie16 minutes ago|

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I hold that the discovery of computation was as significant as the set theory paradoxes and should have produced a similar shift in practice. No one does naive set theory anymore. The same should have happened with classical mathematics but no one wanted to give up excluded middle, leading to the current situation. Computable reals are the ones that actually exist. Non-computable reals (or any other non-computable mathematical object) exist in the same way Russel’s paradoxical set exists, as a string of formal symbols.

Formal reasoning is so powerful you can pretend these things actually exist, but they don’t!

I see you are already familiar with subcountability so you know the rest.

by egorelik3 hours ago|

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I am talking about constructivism, but that's not entirely the same as saying the reals are not uncountable. One of the harder things to grasp one's head around in logic is that there is a difference between, so to speak, what a theory thinks is true vs. what is actually true in a model of that theory. It is entirely possible to have a countable model of a theory that thinks it is uncountable. (In fact, there is a theorem that countable models of first order theories always exist, though it requires the Axiom of Choice).

by creata2 hours ago|

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I think that what matters here (and what I think is the natural interpretation of "not every real number is computable") is what the theory thinks is true. That is, we're working with internal notions of everything.

by egorelik2 hours ago|

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I'd agree with that for practical purposes, but sometimes the external perspective can be enlightening philosophically.

In this case, to actually prove the statement internally that "not every real number is computable", you'd need some non-constructive principle (usually added to the logical system rather than the theory itself). But, the absence of that proof doesn't make its negation provable either ("every real number is computable"). While some schools of constructivism want the negation, others prefer to live in the ambiguity.