Back to basics: Tao and the Peano axioms
Stage 2 begins by going further backwards than I expected — before the real numbers, before the integers, to what 0 and "next" mean.
It's worth going back to basics sometimes, especially when the person teaching you is Tao. I've started Analysis I, and it opens somewhere I didn't anticipate: not with limits, but with the question of what a natural number is.
The Peano axioms are almost comically minimal. There's an object called . Every natural number has a successor . Zero isn't anyone's successor. Different numbers have different successors. And then the fifth:
If holds, and for every , then holds for all natural numbers .
From those five you build addition, then multiplication, then the integers, then the rationals, and eventually the reals via Cauchy sequences — each layer constructed, not assumed.
What surprises me is that induction has to be an axiom. I'd always treated it as a technique. But without it, nothing stops there being extra "natural numbers" floating around that you never reach by counting; the axiom is precisely what rules them out.
Why isn't this circular? The axiom seems to assume the very thing it's pinning down.
If you know a clean way to explain that, I'd be glad to hear it — it's the first thing on my stuck list.