Lecture 3

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 $f(x)=f(y)⟹x=y$.

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 $⟨0,1⟩$ 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 $x\ne \varnothing$.

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.