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Let's imagine we're only using rational numbers for everything in mathematics. Problems arise quite soon when you try to calculate diagonals of squares or perhaps roots of something like $f(x)=x^2-2$.

What we are then introduced to is continuity and real numbers as a solution to those problems. They are a wonderfully dense set and are unaccountably infinite.

So now my question is, is that really necessary? I mean, isn't adding ALL the irrational numbers a huge overkill for any possible mathematical use except for maybe the more philosophical branches of it?

Let's say we make a set that contains all the rational numbers but then add in every number that can be expressed in any ways using a finite string of mathematical symbols. Since we can't really work with equations or functions (or anything, really) that have an infinite amount of symbols, this set should cover all of our needs. There will never be a root that we can't calculate or a diagonal of a length that isn't in our set. All of our favorite constants like $\pi$ or $e$ are there.

But this set is still absolutely nothing compared to $\mathbb{R}$ since we kept our countableness.

Now, I do appreciate mathematics and all of it's aspects and I do realize that everything is worth studying. It just seems that real numbers should only be a part of some specific niche, and not an ever present part of everything.

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  • $\begingroup$ Your statement "All of our favorite constants like $\pi$ or $e$ are there" is totally opposed to your initial philosophy: what if someone is fond of every possible constant? $\endgroup$ – Matemáticos Chibchas Oct 22 '14 at 2:52
  • $\begingroup$ @MatemáticosChibchas Well, I'd say it's pretty difficult to be fond of a number that you have absolutely no way of expressing. $\endgroup$ – Luka Horvat Oct 22 '14 at 10:26
  • $\begingroup$ See Why do we need the real numbers? Part IV discusses definable real numbers, which is what you are asking about. $\endgroup$ – Rahul Nov 13 '14 at 12:23
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I mean, isn't adding ALL the irrational numbers a huge overkill for any possible mathematical use except for maybe the more philosophical branches of it?

No. Completeness is useful. Compactness of closed and bounded subsets is useful. Connectedness of intervals is useful. The point is not to have access to some particular obscure numbers, it is to have access to the set in its entirely, and all the structure that goes with it, with all the holes filled in. Limits are important in many areas of math, and by restricting to some countable subset of $\mathbb R$, you lose most of the ability to use limits.

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  • $\begingroup$ What I'm trying to get at is that we might as well give that countable set all those useful properties and none would be the wiser since, by definition, all the limits would also be in that set. I mean, I get that you can prove something like "only an uncountable set is..." at a higher level, but is there actually anything you CAN'T do with this set that you can do with reals? $\endgroup$ – Luka Horvat Nov 13 '14 at 9:36
  • $\begingroup$ @Luka: "give that countable set all those useful properties": No, it doesn't have those properties, and we cannot change that. "all the limits would also be in that set": All the limits you're willing to acknowledge perhaps, but that isn't good enough. The consequence is that there would be far fewer things that converge; Cauchy sequences wouldn't converge. Monotone bounded sequences wouldn't converge. Power series wouldn't converge. Newton's method wouldn't converge. $\endgroup$ – Jonas Meyer Nov 13 '14 at 16:06
  • $\begingroup$ And regardless of whether the particular limits of particular sequences are in there, we can't change the fact that sequences in our set will still exist that should converge but don't. We can't use compactness, completeness, connectedness, etc. because none of those properties are satisfied by a closed and bounded interval in a countable dense set of real numbers. (Another : The fundamental theorem of algebra is useful, and your proposal would eliminate it.) $\endgroup$ – Jonas Meyer Nov 13 '14 at 16:11
  • $\begingroup$ Yeah. I definitely see what you mean. Every property that uses the fact that $\mathbb{R}$ is uncountable wouldn't work for our set. I also understand that to some problems we can say "the limit exists" without actually being able to calculate it, and those also wouldn't work with our set. What I'm saying is that in the majority of cases (the ones I've dealt with so far at least), we don't really need those high level properties. Every problem where the task is to calculate an actual value would be valid with our restricted set. $\endgroup$ – Luka Horvat Nov 13 '14 at 17:34
  • $\begingroup$ I guess that I'm underestimating how much I use those facts that arise from uncountableness. It just seems silly that we have sets like $\mathbb{R}$ of which we can only use an extremely limited subset. Same with how we use functions from $\mathbb{R}$ to $\mathbb{R}$ when we basically only know how to compose continuous ones to make new functions and there's significantly less of those. $\endgroup$ – Luka Horvat Nov 13 '14 at 17:37

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