[alipang]

I've heard the objection about complex number not being real many times. I think the sensible answer is to argue that the natural numbers don't "actually exist" either. They're an abstraction just like the complex numbers.

Arguably we might one day find out that the universe is discrete at which point we could begin to try to define the naturals as something that "exists", at least up to some maximum large number. But even then the numbers are probably still best thought of as just a helpful abstraction.

[soVeryTired]

I have a much harder time beliving in the full set of real numbers than I do believing in the basic construction of complex numbers. The full set of real numbers requires me to accept things like the axiom of choice, and to believe that non-computable numbers 'exist' on the same level as computable ones. That doesn't sit right with everyone.

Basic complex numbers, on the other hand, just require me to expand what I accept as the solution of an equation. Note that I've already done this with fractions.

Fractions: Given integers a and b, ax + b = 0 has a meaningful solution

Complex numbers: The equation x^2 + 1 = 0 has a meaningful solution

[robinzfc]

The construction of real numbers does not need the axiom of choice in the sense that there are constructions of models of real numbers that do not need it. One example of such construction is described on the Wikipedia page [1], look for "Edudoxus reals" there.

[1] https://en.wikipedia.org/wiki/Construction_of_the_real_numbe...

[red_trumpet]

At which point does the construction of the reals require the axiom of choice (AC)? I'm not familiar what exactly happens without AC, but defining R as the set of rational Cauchy-sequences modulo zero-sequences does not seem to use it?

[heinrich5991]

The moment where you want a Cauchy sequence of real numbers to converge.

You have a Cauchy sequence of real numbers (a_i)_i. You pick a representative (b_ij)_j (a Cauchy sequence of rational numbers) for each sequence member a_i (axiom of choice!) and then produce the diagonal sequence (b_ii)_i which is the constructed limit of the Cauchy sequence (a_i)_i.