How to prove induction principle by real number axioms?

Question

In axiomatic approach to real numbers, that is by defining them to be the complete ordered field, one is expected to prove every theorem and solve every problem by using ultimately only the axioms. I was trying to solve a Spivak's calculus problem that asked to show that the sum of any number of real numbers is still meaningful without using any parentheses. I think we have to use induction principle to solve this problem. But I don't know how to prove the induction principle using axioms of the reals.

Answer 1

The fundamental principle behind induction is that if $S$ is some subset of $\mathbb{N}$ with the property that
1. $S$ contains $1$
2. whenever $S$ contains a natural number $n$, it also contains $n+1$, then it must be that $S = \mathbb{N}$.(Abbot, Understanding Analysis)

Axioms of the reals in Spivak's Calculus and induction principle are irrelevant. So it can't be derived from these axioms

Answer 2

The Principle of Induction is a property of the natural number system; see, for example, Peano's axioms. Hence, there isn't really any way to "prove" the Principle of Induction from the ordered field axioms, i.e., $(\text{P}1)$-$(\text{P}12)$ in Spivak. As Spivak himself says in the text of Chapter 2:

It is also possible to prove the principle of induction from the well-ordering principle (Problem 9). Either principle may be considered a basic assumption about the natural numbers.

I have added the emphasis, here.

However, as I recall, there's a neat exercise towards the end of Chapter 2 (Problem 2-23 in mine) in which, starting with the reals $\Bbb{R}$, you define the natural numbers $\Bbb{N}$, which you do, basically, "by induction" (or rather, you define the notion of an "inductive set" of real numbers, and then the natural numbers are those numbers in every inductive set). In this sense, you can start with the reals, and construct the natural numbers in such a way that you are guaranteed that the Principle of Induction applies; probably this is as close as you're going to get.