When trying to learn analysis from bottom up, what numbers should I first construct?

Solution 1:

My advice is that, if your goal is to study analysis, choose any construction of the real numbers that makes sense to you and move on to actually studying analysis.

Solution 2:

As littleO pointed out in the comments, I think its a great idea to build real analysis starting from an axiomatic description of the real numbers. The reals are decribed uniquely by a surprisingly short list of properties: every complete ordered field is functionally equivalent to $\mathbb{R}$.

Two textbooks that follow this approach are Anbar Sengupta's Notes in Introductory Real Analysis and William Trench's Introduction to Real Analysis.

The reason I think it's a good idea to start from an axiomatic description of the real numbers is that, as you've discovered, there are tons of very different-looking ways to construct the real numbers. Each one has its own flavor and advantages, but they all give you $\mathbb{R}$ in the end. Thinking of the reals axiomatically means that you focus on the properties of ther reals that actually matter, rather than the weird little quirks of a particular construction.

If you're a programmer, you're already familiar with this idea. When you write a Scheme program, you usually don't care whether it's going to be interpreted with Guile, compiled into machine code with Scheme 48, or compiled into bytecode with Kawa. Everything that matters about Scheme is described by the language specification, so you don't have to worry about implementation details, except under some exceptional circumstances.

An axiom system is like a language specification, or an API, for a mathematical object. It lets you focus on playing with the object, instead of getting bogged down in the details of its implementation.

Solution 3:

Do not focus on constructing the real numbers. Just learn their properties, and move into doing analysis. Once you are comfortable with analysis: convergence, integration, derivatives, ect. Go back to the real numbers and see if you can construct them. You appreciate what is being done with Dedekind's construction more if you already done some analysis.