We first define the natural numbers, then negative numbers, then fractions(rationals), then the real numbers, and then complex numbers. The reason we start with naturals is probably that we used our fingers as counting objects and hence natural numbers are essentially... natural. This is a related post- Numbers, is there any scope beyond? but it is a little different from what I'm asking.
I was wondering if there was another system where after defining numbers (or whatever they might be), we would end up with another form of consistent mathematics? Or is it that this system is unique? Can we prove it?
Thanks in advance