I understand that in mathematics and logic we can continue to reduce things to simpler axioms, principles, and so on, and we have to "stop" at some point otherwise we're just going on forever. We eventually say that some axiom or principle is good enough, so that we accept it to be valid, true, useful, sensible, and so on.
That being said, my question is whether mathematical induction is one of these "fundamental" concepts we just accept, or if it follows from some even deeper or simpler concept.
Sometimes I see people say that it works because of the well-ordering principle of the natural numbers, but this doesn't satisfy me. In Tao's Analysis Vol I, we say $m \leq n$ (for natural numbers $m$ and $n$) iff $m = n + a$ for some natural number $a$. But then if I wanted to prove that any arbitrary set of natural numbers has a "least element" (the definition of the well-ordering principle), I'd be resorting to induction, the very thing I'm trying to "prove."
Does this mean the concept of induction is just something we all accept as one of those sufficiently simple, intuitive things that require no further proof, that comes from no simpler means?
Best Answer
My viewpoint is the same as in Noah's first comment: for me, induction is part of the essence of what I mean when I talk about the natural numbers, so the thing I take on faith is not that induction is true but that the natural numbers "exist," whatever that means. Axioms don't tell you what to take on faith: they're a way for two people to agree that they're talking about the same thing.
Some people called ultrafinitists would in some sense deny that the natural numbers exist.
Here is a blog post which describes in detail the sense in which the Peano axioms are a way for two people to agree that they mean the same thing by "the natural numbers."