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

Question

I am interested in studying more analysis and related topics. However, I want to make sure I do so well and without making too many broad jumps in my learning.

In some books I have seen, the author will for example construct $\mathbb{R}$ from $\mathbb{Q}$ with assuming the properties of $\mathbb{Q}$, in other texts, I have seen first construction of $\mathbb{Q}$ assuming properties of $\mathbb{Z}$. I have also seen that $\mathbb{Z}$ and $\mathbb{N}$ can be constructed themselves.

So, I am wondering, what order should I start with, what is the very bottom in a sense? Should I first try to understand construction of $\mathbb{N}$ via set theory, and then use that to construct $\mathbb{Z}$, then $\mathbb{Q}$ and then lastly $\mathbb{R}$? ( And I assume $\mathbb{C}$ usually will come last as you must have $\mathbb{R}$ first? Or should this just a basic contraction following $\mathbb{R}$)?

Anyone have insight, or recommendations about this?

Thanks!

$\mathbf{Update:}$ Thanks for all the comments so far, maybe I should provide some more context also about what I am looking to achieve. Basically, I am looking for a solid understanding of this, all in the context of say an honours undergraduate program in math. Ie, as much as I am interested in all the grit and details, my main focus is a practical understanding, and one that will be sufficient in succeeding at this level of maths!

Also, it seems as if it may be a good idea to simply just learn the rules of the real numbers, so can anyone provide me with a way to learn this/pdf of these? Pr is it simply the rules of fields that are ordered?

Thanks for any and all advice, I will be starting soon

Update: After reading all the responses I decided I would not worry about constructing any numbers just yet, and have begun studying analysis just using a set of axioms for the real numbers. I have not had any issues with this approach so far.

Answer 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.

Answer 2

If you want to go from first principles, you can read Principia Mathematica in which it takes hundreds of pages to get to the point where they prove $1+1=2$. However for most people this is going too far. :-)

From a practical standpoint, you can understand analysis fine if you start with $\mathbb{Q}$, as many textbooks do.

Answer 3

Landau's beautiful "Foundations of Analysis" gives an excellent account of the standard way of progressing from the natural numbers to the real and complex numbers. There are lots of alternative routes, but you should understand the standard account first.

Answer 4

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.

Answer 5

To some extent, the "proper" answer depends on why you wish to do this. Starting with the axioms of ZF set theory is, in a sense, "minimalist"; however, this theory was developed with an aim to formalize a logical foundation of mathematics, and isn't the most intuitive way to proceed.

A more intuitive approach might start with the Peano axioms for the natural numbers, which "make sense" given a typical understanding of basic arithmetic. Indeed, since many analytic proofs make use of recursion and induction, examining the natural numbers first puts these methods on firm rigorous ground.

That said, nothing truly extraordinary happens (and certainly nothing with an "analytic", that is to say topological, flavor) until one investigates the "gap" between the rationals and the reals. The practical methods of the algebra of a field of fractions are covered (in a perfunctory way) fairly early in most educational systems (in my life, it was around 3rd grade or 4th grade, maybe?).

While it is possible to start with an axiomatization of a complete archimedean ordered field, the so-called "13th axiom" (the completeness axiom, or it's more-often used equivalent, the least-upper-bound property) is clearly something different than the other field axioms. It is insightful to see why it's needed (note there are other ways to extend the reals-such as the algebraic extensions used in Galois theory, so the mere existence of irrationals isn't "quite" the problem we're seeking to solve).

I suppose at the very bottom we have the notion of distinction-something represented by "set membership" (the basic predicate), either "in", or "out". It is no accident that binary coding (a mechanical way of modelling logic) needs just 1's and 0's. Verifying that the things we've always suspected are true (like $2+2 = 4$) can be proven is somewhat of a tedious task, the details of which may take you further afield from the analysis you say you wish to study.

Personally, I would recommend Landau's account, as another poster has already suggested.

Answer 6

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.

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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.

Answer 7

Conway, in On Numbers and Game recommended starting with $\mathbb N = \mathbb Z_{\ge 0}$, then going to $\mathbb Q_{\ge 0}$, then to $\mathbb R_{\ge 0}$, then to $\mathbb R$. The reason is that it reduces the number of cases, as opposed to going doing $\mathbb N \to \mathbb Z \to \mathbb Q \to \mathbb R$, which requires many special cases.

You construct $\mathbb N$ using Peano Axioms.

Then you construct $\mathbb Q_{\ge 0}$ by taking formal quotients of natural numbers, and setting equivalence classes and such.

Now you construct $\mathbb R_{\ge 0}$ by take Dedekind cuts of members of $\mathbb Q_{\ge 0}$, note, that since you only doing positive reals, you may just need the right cut, but I'm not sure.

Now you construct $\mathbb R$ by taking formal differences of $\mathbb R_{\ge 0}$. Do not introduce subtraction at all until this point.

This path ensures that you don't have to taking Dedekind cuts of $\mathbb Q$, which is different for positive, negative, and zero (which is up to nine cases for each operation!)

Starting analysis from the bottom up usually isn't done, but it does give you a unique perspective on the reals. I personally believe in formalism, since then you always know what it is that you are talking about, instead of many things about the thing you are talking about.

Hint for Division in $\mathbb R$: For $r, s \in \mathbb R_{\ge 0}$:

$$\frac 1{r-s}=\frac{r-s}{(r-s)(r-s)}=\frac r{(r-s)^2} - \frac s{(r-s)^2}$$

where $(r-s)^2 \in \mathbb R_{\ge 0}$.

Answer 8

Analysis text of Terence Tao - Analysis I takes the axiomatic approach of first constructing natural numbers using Peano Axioms in a very intuitive manner. Maybe, you can try looking at that to see if you are comfortable with his approach.