Set Theory

Goal

To gain a real understanding on real numbers.

Analytical construction

I “swallowed” the Compleness Axiom, then I worked on exercises on $\sup$ and $\inf$, and then the $\epsilon$-$\delta$ criterion for limits, before completing $\Q$ with Cauchy sequences.

I’ve also heard about the completion of a metric space in a more general setting. My professor once said that it suffices to view this proof once throughout lifetime: the proof itself wasn’t very useful.

The basic arithmetic properties of $\R$, as an equivalence class of Cauchy sequences sharing the same limits, didn’t arouse our interests. That’s just an extension of its rational counterpart due to some arithmetic properties of limits.

Purpose

This post aims at recapturing the main ideas of the formal proofs that I’ve read. It never tries to replace them. You may consult the references if you need any of them.

Some notations

Unless otherwise specified, all cardinalities here are infinite. Denote $\mathfrak{a} = \card{A}$, $\mathfrak{b} = \card{B}$ and $\mathfrak{i} = \card{I}$.

Sum

$\mathfrak{a} + \mathfrak{b} = \card{A \cup B}$ provided that $A \cap B =\varnothing$.

Product

$\mathfrak{a} \, \mathfrak{b} = \card{A \times B}$

Power

$\mathfrak{a}^\mathfrak{i} = \card{A^I}$, where $A^I = \lbrace f \mid f: I \to A \rbrace$ denotes the set of functions from $I$ to $A$.

I've chosen $I$ instead of $B$ to express the index set because this reminds me of an array of $(a_i)_i$ indexed by $I$.

This isn’t a substitute for books .

Reminder

• $A \preceq B$: $A$ can be “injected” into $B$.
• $A \sim B$: $A$ and $B$ share the same cardinality.
• $A \prec B$: $A$ can be “injected” into $B$, but it’s “smaller” than $B$.
• A finite set can be “counted” from one to some nonnegative integer.
• Infinite is the “antonym” of finite.

Wolf’s proof

When I first saw this proof in Robert S. Wolf’s Proof, Logic and Conjecture: The Mathematician’s Toolbox, I gave it up since it wasn’t as intuitive as the statement.