Are the addition and multiplication of real numbers, as we know them, unique?

The short answer is no: the operation defined by $a+_3 b=(a^3+b^3)^{1/3}$ also has all the properties 1 through 6 over the reals. This distributes over cannonical multiplication: for any $a,b,z\in \mathbb{R}$,

$z(a+_3b)=z(a^3+b^3)^{1/3}=(z^3)^{1/3}(a^3+b^3)^{1/3}=((za)^3+(zb)^3)^{1/3}=(za)+_3(zb).$

It might, however, be the case (and this is entirely speculation, not necessarily true) that only operations of the form $a+_f b = f^{-1}(f(a)+f(b))$ (where $f:\mathbb{R}\to \mathbb{R}$ is bijective and fixes the origin; or stated differently, $f$ is a permutation of the real numbers and $f(0)=0$) have all the properties 1 through 6. That would mean that addition is unique up to automorphism on the real numbers.

I would agree that this is not a trivial question.

The proof goes in four steps:

• Addition on the natural numbers is uniquely determined by the successor operation (proved using induction)

• Addition on the integers is uniquely determined by addition on the natural numbers

• Addition on the rational numbers is uniquely determined by addition of integers

• Addition of real numbers is uniquely determined by addition of rationals, by continuity, using the fact that the reals are a complete ordered field containing the rationals as its prime subfield.

The same four steps can be used to show that multiplication on the reals is ultimately uniquely determined by the successor operation on the natural numbers.