Godel's unprovable statement - Numberfile

  • #1
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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!
 

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  • #3
PeroK
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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.
 
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