Peano Axioms and Arithmetic: How we define how we count
Caution It's another expository post (*I mean it's better than just ranting, but Samyak please stop playing an expert*). The same cautions ahead(I'm no expert; I may be wrong). With that out of the way, let's begin: Foreplay Introduction So! How do you define natural numbers? As is the case with other topics in intro analysis, you'll think (I definitely thought) that this is so boringly obvious . I mean, it's just how we count right? But many things about $\mathbb{N}$ aren't so obvious. For example, how would you prove that $a+b=b+a$ for $a,b\in \mathbb{N} $? You'd say it's blindingly obvious , but commutativity is not obvious for many other things(try, for example, rotating a Rubik's cube and you'll get it). Note that you can't justify with examples (you'd have to give infinite examples to prove this property) Now, this was a problem that presented itself in the mid-to-late $19^{th}$ century. People always assumed that numbers, in g