I don't know a lot of mathematics but I have noticed that every branch of Mathematics has the same structure: some axioms (For example in Geometry might be Euclid's Axioms, in Probability might be Kolmogorov Axioms), and some theorems that are originated from them. So, as the way I understand it, mathematics is laid out in something called a "Formal System". I also have heard that this concept has created a lot of debate starting from 1921 when David Hilbert proposed to use it as the foundation for the knowledge in mathematics. Then Kurt Godel came and the rest is famous history that I don't understand. But before trying to grasp at: What are "Godel's Incompleteness Theorems"?, I would like to understand first of all, what is a Formal System?
Formal systems have these things:
1) A Language
2) A Logical Formal System
3) Indefinite Terms
4) Definitions
5) Propositions (Axioms)
6) Propositions (Theorems)
For example, in the case of Real Numbers $ \mathbb{R} $ we have:
1) A Language
...
2) A Logical Formal System
...
3) Indefinite Terms
...
4) Definitions
...
5) Axioms
There are three axioms:
Field Axioms
F1: If $ a, b \in \mathbb{R} $, so $ a + b $, $ ab \in \mathbb{R} $ ... (Closeness Axioms)
F2: If $ a, b \in \mathbb{R} $, so $ a + b = b + a $ and $ ab = ba $ ... (Commutativity Axioms)
F3: If $ a, b, c \in \mathbb{R} $, so $ a + (b+c) = (a+b) + c $ and $ a(bc) = (ab)c $ ... (Associativity Axioms)
F4: If $ a, b, c \in \mathbb{R} $, so $ a(b+c) = ab +ac $ ... (Distributivity Axiom)
F5: There are $0,1 \in \mathbb{R} $, with $ 0 \neq 1 $, so that if $ a \in \mathbb{R}$, so $ a +0 =a $ and $ (a)(1) = a $ ... (Identities Axioms)
F6: If $ a \in \mathbb{R} $, there is $ a_1 \in \mathbb{R} $, so that $ a + a_1 = 0 $, and if a $ a \in \mathbb{R} $ with $ a \neq 0 $, so there is $ a_2 \in \mathbb{R} $ so that $ (a)(a_2) =1 $ ... (Inverses Axioms)
Order Axioms
O1: If $ a,b,c \in \mathbb{R}$, then, only and just only of the next propositions is true:
i) $ a= b $
ii) $ a<b $
iii) $ b<a $ ... (Trichotomy Axiom)
O2: If $a,b,c \in \mathbb{R} $ and $ a<b $, $b<c$, so $ a<c$ ... (Transitivity Axiom)
O3: If $ a,b,c \in \mathbb{R}, c> 0 $ and $ a<b $, so $ ab < bc $ ... (Product Consistency)
O4: if $a,b,c \in \mathbb{R} $ and $ a<b $, so $ a+c <b+c $ ... (Addition Consistency)
Completeness Axiom
C1: If $A \subseteq \mathbb{R} $, so that:
1) $ a \neq \emptyset $
2) $ A $ is bounded from above
so, $ A $ has a supremum in $ \mathbb{R} $
6) Theorems
There are many theorems, for example, "The Addition of Fractions":
$ \frac{a}{b} + \frac{c}{d} = \frac{ad+bc}{bd} $
and the proof is:
$ \frac{a}{b} + \frac{c}{d} = ab^{-1} + cd^{-1} $ (Definition)
$ = (ab^{-1})1 + (cd^{-1})1 $ (F5)
$ = (ab^{-1})(dd^{-1}) + (cd^{-1})(b b^{-1}) $ (F6)
$ = a (b^{-1} d d^{-1} ) + c(d^{-1}bb^{-1}) $ (F3)
$ = a (db^{-1} d^{-1}) + c(bd^{-1}b^{-1}) $ (F2)
$ = (ad) (b^{-1} d^{-1}) + (cb) ( b^{-1} d^{-1} ) $ (F3)
$ = (ad) (b^{-1} d^{-1}) + (bc)(b^{-1} d^{-1}) $ (F2)
$ = (ad+bc)(b^{-1} d^{-1}) $ (F4)
$ = (ad + bc)(bd)^{-1} $ (it can be proved with help of another theorem)
$ = \frac{ad + bc}{bd} $ ... QED
I don't understand very well, in the case of Real Numbers, what are: "The Language", "The Logical Formal System", "The Indefinite Terms" and "The Definitions"?, are they really a "Formal System"? an special case of a Formal system called: An Axiomatic System? What is really a Formal System? Who was the first one that created these axioms for Real Numbers? if formal systems are not the foundation knowledge for mathematics, then, what do we continue learning Axioms of Real Numbers this way? What would be the real foundation of Real Numbers?
