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I get that the natural numbers, integers, rational numbers, algebraic numbers, and computable numbers are all countable. The reals are uncountable. If I wanted to understand arithmetic restricting all my discourse to only countable sets, what do I lose? To my knowledge, numbers like Chaitin's Omega are not relevant to arithmetic. Are there any arithmetical truths which are dependent on the existence of non-computable, or uncomputable numbers?

Edit: To clarify, I am asking whether we need non-computable numbers for arithmetic (a.k.a. number theory). I know that non-computable numbers are needed in real analysis and other fields.

To further clarify: Are there solutions to number theory problems, such as the distribution and properties of primes, which require uncomputable numbers? We definitely need irrational and imaginary numbers, since an answer to the Riemann Hypothesis will depend on them. I haven't seen a solution yet to a number theory problem which requires uncomputable numbers, nor am I entirely sure it is even possible for there to be one.

Quus
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    @HenningMakholm I think that this question is different---it isn't about the need for the reals in general, but the need for the reals in arithmetic, which seems a more specific kind of question. – Xander Henderson Nov 28 '17 at 03:12
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    Define "arithmetic". – Angina Seng Nov 28 '17 at 03:16
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    What do you mean by "need for arithmetic"? Are you familiar with the concepts of groups, rings and fields? These are the structures that are required to do certain arithmetic operations and there are countable subsets of the reals that do all of them. – Sriotchilism O'Zaic Nov 28 '17 at 03:18
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    In that case the question seems to be putting the cart rater solidly before the horse. We don't "need numbers" because we want something to do arithmetic on; rather arithmetic is something we use to handle numbers we already have a reason to be interested in. – hmakholm left over Monica Nov 28 '17 at 03:18
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    More formally, the first-order theory of real closed fields is complete (and decidable), so if by "arithmetic truth" you mean a first-order property of $(\mathbb R,0,1,{+},{\cdot})$, then all of those are also satisfied by the algebraic reals. – hmakholm left over Monica Nov 28 '17 at 03:23
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    What do you mean by "need"? Do you need $\pi$ or $e$ for arithmetic? What about $\sqrt{2}$? – Noah Schweber Nov 28 '17 at 03:56
  • This reminds me of the phrase one of my algebraist friends used in the beginning of his talk: "Let $\varepsilon$ be a small positive rational number...". On a more serious note, what's the advantage of restricting yourself to just countable structures? The disadvantages are numerous: you immediately lose many cute things like, say https://en.wikipedia.org/wiki/Hasse_principle , you cannot even talk about families of subsets (say of the same integers) without asking yourself whether you have included too many at every step, etc. – fedja Nov 28 '17 at 05:09
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    This is not a duplicate of the question: "do we really need reals" since this is specific to number theory and focuses on the difference not between rationals and reals but on the difference between computables and reals. I prefer to work in countable numbers because it is more intuitive, for the same reason that I hold a neo-Lorentzian interpretation of relativity. I would rather construct my reals from Cauchy sequences with well-defined formulas instead of having to take leaps of faith that arbitrary sequences will eventually converge. That's all. – Quus Nov 29 '17 at 19:50
  • Perhaps more concrete than the different kind of real numbers : intuitively what we need in maths are consistent effectively axiomatized theories capable to formalize the arithmetic (so we can try enumerating its theorems until it finds a proof of RH) – reuns Nov 29 '17 at 21:07

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