How someone got the idea for the completeness axiom?

Question

I don't understand what was the Motivation for the completeness Axiom and why Analysis and calculus would not work without it:

Answer 1

I don't have any good historical information, but I do have some intuition for you.

Analysis, and therefore calculus, is centered around the idea of limits. What happens as we approach a certain point? What is the result of repeating a process indefinitely? The Greeks used this idea in their method of exhaustion to calculate the volume of solids. Newton and Leibniz used this idea with their "infinitesimals" and "fluxions." None of these were rigorous methods, but they got the job done.

When we finally started to introduce rigor into analysis, we needed to really understand what we meant by a limit. This gave rise to the $\epsilon$-$\delta$ definition of the limit and various other formalizations of ideas that were only intuitive up to that point.

The notion of limit and supremum (and infimum) are intimately connected. Roughly, we want to be able to say that, in as many cases as we reasonably can, a process that we carry out ought to have a limit. The completeness axiom (though it can be proven with some set theory) tells us that this always works in the context of bounded sets. Given a set bounded from above, we can inch our way closer and closer to the upper bound, and are guaranteed to approach some limit, namely the supremum.

This idea turns out to be crucial in defining and proving various things that calculus students often take for granted in analysis. As a basic example, how do you define $$x^p$$ for arbitrary real $p$? Calculus students are well aware that $$\frac{dx^p}{dx} = p x^{p - 1},$$ but how do we prove that? It turns out that one way to define this quantity is as $$x^p = \sup_{\substack{a/b \in \mathbb{Q} \\ a/b \leq p}} x^{a/b},$$ where $$x^{a/b} = \left(\sqrt[b]{x}\right)^a.$$ To even make this definition we must assume that such an upper bound exists for every suitable $x$.

So really, the completeness axiom is about just what is says: completeness. We want to be justified in taking "limits" - or things that look like limits - as often as possible; that is, we want our system to be as complete as possible.

Answer 2

Core theorems in analysis rely on this. Examples:

• Heine-Borel
• Continuous functions on closed intervals attain maxima/minima
• intermediate value theorem.

These theorems are essential in developping calculus (differentiation and integration).

Answer 3

Consider sequence or rational numbers $a_n = (1+\frac 1 n)^n$. Does it converge? If we're working with real numbers, then yes, $\lim_{x\to\infty}a_n = e$. If we were working with rational numbers (which don't satisfy the completness axiom) then no, it doesn't. Completness axiom ensures that sequences which "ought" to converge, actually do.