Read-along: Rudin's PoMA: Week 2 (8th April to 14th April)

Goal: The goal for this week is to prove that a mathematical object satisfying the axioms we stated last week exists: i.e. there exists a (unique) complete ordered field, which we will call $ \mathbb{R} $.

Reading: Chapter 1 of Rudin, including the appendix. The chapter is scanned here. An alternative treatment of Dedekind cuts, which includes all detail down to the last $ \varepsilon $, can be found in chapter III of Landau’s Foundations of Analysis; this chapter is scanned here.

Landau’s book is highly recommended if you are interested in foundations: he constructs the complex field, and proves all its fundamental properties, directly from the Peano axioms. The Chicago bibliography has this to say:

This is the book that invented the infamous Landau “Satz-Beweis” (theorem-proof) style. There is nothing in this book except the inexorable progression of theorems and proofs, which is perhaps appropriate for a construction of the real numbers from nothing, but makes horrible bathroom reading. Read for culture.

But from experience the best location to read this book is indeed the bathroom.

In addition, see sections 2.1 to 2.3 of Davidson and Donsig, Real Analysis and Applications (online version via the library). This book is a nice gentle introduction to real analysis, that does everything via the properties of sequences (rather than Rudin’s viewpoint, which is more topological).

Suggested problems: Exercises for Chapter 1, from page 21 onwards: all those not completed last week, including exercise 20.

It would be nice if everyone wrote up their solution to one or two of these problems so that we can collect them all online here and learn from each other ($ \LaTeX $ not required, just scan it if you want to as long as it’s readable!). If we end up having a seminar meeting this week then these solutions are something we can discuss there.

Additional exercises:

Convince yourself philosophically that physically constructing $ \mathbb{R} $ from the rational numbers is a necessary thing to do.
Let $ K $ and $ L $ be ordered fields. An ordered field isomorphism is a bijective function $ \varphi : K \to L $ such that the following are satisfied:
- For all $ x, y \in K $, $ \varphi(x + y) = \varphi(x) + \varphi(y) $ and $ \varphi(xy) = \varphi(x) \varphi(y) $ (note that the operations inside $ \varphi $ are carried out in $ K $, and those outside are carried out in $ L $);
- If $ 1_K $ is the multiplicative identity in $ K $ and $ 1_L $ is the multiplicative identity in $ L $, then $ \varphi(1_K) = 1_L $;
- If $ x \leq y $ in $ K $, then $ \varphi(x) \leq \varphi(y) $.
If such a function exists, $ K $ and $ L $ are called isomorphic. Suppose this is the case; prove that if $ K $ has the least-upper-bound property, then so does $ L $.
An alternative construction for the real numbers proceeds as follows: if $ [9] $ is the set of digits 0 through 9, set $$\mathbb{R}_{10} := \mathbb{Z} \times \{ f : f \text{ a function } \mathbb{N} \to [9]) \} $$
(i.e. the set of all decimal expansions).
- Define the algebraic and ordering operations on this set in the usual way, and show you obtain an ordered field.
- Show that there is a natural ordered field isomorphism $ \varphi : \mathbb{R} \to \mathbb{R}_{10} $, where $ \mathbb{R} $ is the field constructed using Dedekind cuts. Hence conclude that $ \mathbb{R}_{10} $ has the least-upper-bound property.
- Prove the least-upper-bound property for $ \mathbb{R}_{10} $ directly (i.e. without field isomorphisms).
- Generalise the preceeding to arbitrary base $ N \in \mathbb{N} $ (i.e. replace ‘10’ with ‘$ N $’).
- Philosophically speaking, is there any difference between this construction and the Dedekind construction?

Next: Next week the plan is to begin chapter 2; in particular, we will define metric spaces and their associated natural topology.