TL;DR: Within axiomatic systems, TRUE
statements can be provable
or unprovable
.
Before this, mathematicians thought that anything TRUE
had to be provable
. Ever since the ancient greeks, we believed that any true statement in math had a proof.
BUT there might be conjectures that are true (Goldbachs Conjecture) in mathematics, even if there is no proof.
Liars Paradox
This statement is false
This is an example of the Liar Paradox. It’s self-referential and leads to a contradiction.
If the statement is assumed to be true, then it must be false. But if it’s assumed to be false, then it must be true as it claims.
This leads to a paradox, or an infinite loop.Verbal paradoxes are fine since, you don’t expect sentences to have inherit truth in them. BUT in mathematics, all statements are known to be either
TRUE
orFALSE
Godel created the Godel Coding System which could turn any mathematical statement into a number. So any axioms could be turned into a code number.
And every code number could be worked back into a statement. Just like how computers turn text into ASCII numbers.
With his system, he could start to talk about mathematics using mathematics itself. So now you can use this system to figure out if a particular statement is provable from mathematical axioms
If Godel coded a statement that expresses itself to be unprovable
mathematically, then is it either TRUE
or FALSE
in his Coding System?
First let’s assume it’s FALSE
→ This means this statement is actually provable
. But in mathematics, this is impossible since a provable
statement must be TRUE
. So we started with something we assumed was FALSE
, and came to the conclusion that it is TRUE
which is inconsistent.
SO this means the statement can’t be false, since that would be a mathematical contradiction.
This means the statement has to be TRUE
→ A mathematical equation must be either true or false. Which means this TRUE
statement is claiming it is unprovable
.
SO Godel showed that you can have a TRUE
statement which cannot be provable
within the system of mathematics.
This basically leads to an infinite regress problem, since you can continue to add axioms that are TRUE
but unprovable
to mathematics, thus you can never really prove mathematics as a completely provable
and consistent system.
From Wikipedia:
The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an effective procedure (i.e. an algorithm) is capable of proving all truths about the arithmetic of natural numbers. For any such consistent formal system, there will always be statements about natural numbers that are true, but that are unprovable within the system.
The second incompleteness theorem, an extension of the first, shows that the system cannot demonstrate its own consistency.