# EURO PROJECT
## PROPOSITIONS
### SET THEORY, A HOLE IN MATH: ADONIS
#### Introduction
Set theory is the study of objects called *sets*.
To put it simply a set is something that contains other things (a set can not contain itself). It was formalized by Georg Cantor in the 1870's. His initial axioms lead to many paradoxes such as [Russell's paradox](https://www.scientificamerican.com/article/what-is-russells-paradox/) which lead to the discovery of the now standard [ZFC Set theory](https://plato.stanford.edu/entries/set-theory/ZF.html). All modern maths can be formulated using ZFC
but due to [Gödel’s Incompleteness Theorems](https://plato.stanford.edu/entries/goedel-incompleteness/),
ZFC as well as all any formal axiomatic theory can either be
inconsistent or incomplete. Jhon could talk about the historical implecations of these theories (Turing machines, computability) as well as the life of some of the mathmaticians. Timon and I could talk about one of the following:
- How does set theory encapsulate all of maths and reasoning for example how does it encapsulate numbers?
- What are the axioms of ZFC?
- Gödel’s Incompleteness Theorems
#### The Peano Axioms and Zermelo's Construction
##### The Peano Axioms
The peano axioms define what the natural numbers are it goes like this:
0 ∈ ℕ (zero is a member of the set ℕ of natural numbers).
Now that we have zero in our set we can define all natural numbers as successors of 0 (for example the successor of 0 is 1, the successor of 1 is 2, etc). All we have to now is to define what successor means. To do that we can use an object from set theory called a *function* a function is an object that takes an element from a set and produces another object (in reality a subset of the Cartesian product between to sets). successor is a function called S. S(n) should satisfy the following axioms:
1. n ∈ ℕ → S(n) ∈ n (if n is a natural number than S(n) is also a natural number)
2. ∀n ∈ ℕ. S(n) != 0 (no number has zero as a successor)
3. ∀n,m ∈ ℕ. S(n) = S(m) → m = n (if 2 numbers have the same successor, they are equal)
4. [T ⊂ ℕ ∧ 0 ∈ ℕ ∧ (n ∈ T → S(n) ∈ T)] → T = ℕ (ℕ is the minimal set that satisfies axioms 1 to 3)
###### Equality under the Peano axioms
Under the Peano Axioms, equality is a very simple concept.
It should satisfy the following:
- reflexivity: x = x
- symmetry: x = y ↔ y = x
- transitivity: (a = b ∧ b = c) → a = c
- the natural numbers are closed under equality: ∀a ∈ ℕ. a = b -> b ∈ ℕ
##### Zermelo's Construction
TODO: lol I have a history homework todo can't spend too much time on this.
#### ZFC axioms
TODO: lol I have a history homework todo can't spend too much time on this.
#### Gödel’s Incompleteness Theorems
TODO: lol I have a history homework todo can't spend too much time on this.

Note to Jhon: I never cared to lookup the historical context but I know that it is pretty juicy so if we were to chose this subject, you need to do your own research.
#### Criticism
##### Jhon
[insert criticism here]
##### Timon
[insert criticism here]