Constraining mathematics to a subset of $\mathbb{R}$ 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.


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.
