This doubt might be a mistake on my part, but its confused me for a while now so I decided to ask here.
In Terrence Tao’s “Analysis 1”, at the beginning of the chapter about Natural Numbers (where he uses the Peano Axioms to define them) he asks the reader to forget everything about the standard rules of algebra and to start from scratch.
However, when he starts to define incrementation as counting forward, or an increase in number, I hit a block. How can I think of increase in numbers without the order axioms? The increment operation itself makes no sense to me without a previously established notion of what is greater and lesser, i.e., some form of the order axioms, and to set the order axioms we need the addition axioms, and this seems to oppose the notion that the Peano Axioms are the most fundamental axioms needed to define natural numbers.
Maybe I’m just interpreting what Tao meant by “counting forward” wrongly, in which case, I would appreciate if someone gave me some pointers about the right way to think about this.