Godel's unprovable statement - Numberfile

[jamalkoiyess]

Hello PF,

I was watching this video

and around 8:00 the speaker says that the statement that cannot be proven by the axioms is supposed first as false which would make it provable by the axioms, "which would make it true since it was proven". However, the proof can be a proof that the statement is false so that would not make the contradiction.

I think this is some kind of mind twister that I cannot get over.

Thanks!

 

[jedishrfu]

What Godel's Incompleteness Theorems say is that in any given system of logic, there exist statements whose truth are undecided ie there are statements which cannot be proven true or false given the axioms of the system. The mind twister is an undecided statement.

https://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems

 

[PeroK]

I think this is some kind of mind twister that I cannot get over.

Thanks!

The idea is a version of Russell's paradox:

https://en.wikipedia.org/wiki/Russell's_paradox

What Godel did was to construct a statement (by ingeniously getting number theory to talk about itself) that effectively said:

"This statement cannot be proved from the given axioms."

Now, if you prove that statement, then you have a contradiction.

And, if you cannot prove it, then it is a true statement that cannot be proved. This is the key point. The statement itself is true, in the sense that it cannot be false. But, this cannot be proved using the axioms. Hence, your axiomatic system cannot be used to prove all true statements.

This leads to the concept of an undecidable proposition.