$∞$ 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.

Achilles and the Tortoise

Achilles tries hard

$⇑$

Achilles : the explanation

Zeno_Dichotomy_Paradox

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 ?

Hilbert hotel

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

aleph_0 elements

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 $𝜔 + 1 \neq 𝜔$

omega elements

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 !

$$𝜔^{2}$$

omega_2

$𝜔^3$

omega_3

$𝜔^4$

omega_4

$𝜔^𝜔$

omega_4

$𝜔^{5}$

omega_4

$𝜔^{𝜔^𝜔}$

omega_4

$𝜔^{𝜔^{𝜔^{\vdots^{𝜔}}}}$

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