---
lecture: 3
date: 2025-01-13
---
# Proposition 1.1.4
Each real number is a limit of a sequence of rational numbers.
$$\mathbb{Q} \subset \mathbb{R}$$
As an ordered [[Number Field]] $\mathbb{R}$ is **complete**, meaning that every [[Cauchy Sequence]] in $\mathbb{R}$ converges to a real number.
Equivalently, the real numbers have the **[[Least Upper Bound Property]]**; $\forall X \subset \mathbb{R}$ bounded above has a least upper bound denoted by $sup(X) \in \mathbb{R}$. Eq. an $inf(Y) \in \mathbb{R}$ if $Y$ bounded below.
# Functions and Cardinality
A function $f : X \to Y$ is **[[Injective]]** if $f(x) = f(y) \implies x = y$.

[[Surjective]] if $f(x) = y$.

[[Bijective]] if it is both injective and surjective.

Then we write $X \simeq Y$.
$$|X| = |Y| \; \text{(cardinality)}$$
Say $X$ is [[Countable]] if $|X| = |\mathbb{N}|$; this means that the members of $X$ can be listed as a sequence with $x_{n} = f(n)$, where $f: \text{IN} \to X$ is some bijection.
## Cantor's Diagonal Argument
The real numbers cannot be listed, or they are **uncountable**.

Indeed, present a list of the real numbers in $\langle 0, 1 \rangle$ written as binary expansions. Then the number that has as its $n$-th digit, the opposite value to the $n$-th digit of the $n$-th number of the list, will never be in the list.


| **\\** |       |       |       |       |     |     |     |     |     |     |
| ------ | ----- | ----- | ----- | ----- | --- | --- | --- | --- | --- | --- |
| 0      | **1** | 0     | 1     | 1     | 0   | 0   | 1   | 1   | 1   | ... |
| 0      | 1     | **1** | 1     | 1     | 1   | 1   | 1   | 1   | 1   | ... |
| 0      | 0     | 0     | **0** | 0     | 1   | 0   | 0   | 0   | 0   | ... |
| 0      | 1     | 0     | 1     | **0** | 1   | 0   | 0   | 0   | 0   | ... |
| ...    |       |       |       |       |     |     |     |     |     |     |
So here the bold numbers going diagonally from the **\\** shows that they cannot be countable as they are not the same number.
Cantor: 0.0011...
# Axiom of Choice
Any **[[Direct Product]]**
$$\Pi_{i \in I} \, X_{i} \equiv \{ x : I \to \cup_{i \in I} X_{i} | x_{i} \equiv x(i) \in X_{i} \}$$
is non-empty when all $x \neq \emptyset$.

Any $x \in \Pi_{i \in I} \, X_{i}$ is called a choice function.

The [[Power Set]] $\wp(X)$ of $X$ consists of all the subsets of $X$.

$\exists$ bijection $\wp(X) \to \Pi_{x} \{ 0, \, 1 \} = \{ f : X \to \{ 0, \, 1 \} \}$ that sends $Y \subset X$ to its [[Characteristic Function]].