# $∞$ Infinity $∞$ ###### Younesse Kaddar

I. What is it that we call infinity ?

1) Zeno’s Paradoxes 2) Hilbert’s Hotel

II. How big is it ?

1) Counting to $∞$ : $\aleph_0$ 2) Ordinals : $𝜔$ 3) Inaccessible cardinals

Zeno’s paradoxes

Zeno of Elea : In a race, the quickest runner can never overtake the slowest.

 $⇑$ 

Actually, modern math provide a solution :

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

Hilbert's Hotel

- Fully occupied - Infinitely many rooms $⟶$ How many additional guests can be housed ?

1) For $n$ new guests : a shift of $n$ 2) How about infintely many new guests ?

II. How big is infinity ?

Cardinals : $\aleph_0$

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


Ordinals : $𝜔$

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 : to infinity and beyond !

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 $𝜀_0$

And all that is smaller than …

$\omega_1$

corresponding to the cardinal …

$\aleph_1$

Then …

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

and the corresponding cardinals

$\aleph_2, \aleph_3, \ldots$

Power sets : becoming exponentially big

$

𝒫(E)

= 2^{

E

}$

+

Axiom of replacement

$\downarrow$

We iterate over and over both operations !

But all of that has an order type … so : it remains smaller than …

Inaccessible cardinals

Conclusion