- Zeno’s Paradoxes
- Hilbert’s Hotel

- Counting to $∞$ : $\aleph_0$
- Ordinals : $𝜔$
- Inaccessible cardinals

Zeno of Elea: In a race, thequickest runnercan`never overtake`

the slowest.

$⇑$

Actually, modern math provide a solution :

$$\sum_{n>0} \frac{1}{2^n}$$ is convergent

- Fully occupied
- Infinitely many rooms

$⟶$ How many additional guests can be housed ?

- For $n$ new guests : a shift of $n$
- How about infintely many new guests ?

**Cardinal** : the number of elements of an *unordered* set

$\downarrow$

If the set of all the integers ($ℕ$) exists, we call $\aleph_0$ its cardinal

NB:Zeno paradox⟹ $\aleph_0$ elements can be "written" within a finite space

**Ordinal** : the first label you’ll have to use in order to append 1 element to an *ordered* set

NB:

For a finite number of elements: ordinal ⟺ cardinal- $\aleph_0 + 1 = \aleph_0$, but $𝜔 + 1 \neq 𝜔$

**Axiom of replacement** : if you take an existing set and replace all elements with something else, you’re left with *an other existing set*.

We’re going to use it to the fullest !

$$𝜔^{2}$$

$𝜔^3$

$𝜔^4$

$𝜔^𝜔$

$𝜔^{5}$

$𝜔^{𝜔^𝜔}$

$𝜔^{𝜔^{𝜔^{\vdots^{𝜔}}}}$

up to

$𝜀_0$

$\omega_1$

$\aleph_1$

$𝜔_2, 𝜔_3, \ldots, 𝜔_{𝜔}, \ldots, 𝜔_{𝜔^{𝜔^{𝜔^{\vdots^{𝜔}}}}}, 𝜔_{𝜀_0}, \ldots$

$\aleph_2, \aleph_3, \ldots$

$|𝒫(E)| = 2^{|E|}$

`+`

Axiom of replacement

$\downarrow$

Inaccessible cardinals