In mathematics we use:
For example, we can't prove both 2+2 = 4 and 2+2 != 4 in the system. Only one can be proven.
Now we know we can't prove both. Do we know we can prove the true one in all cases?
Godel theorem says no. In a consistent system, there will always be statements which are true but not provable! Sounds odd, but that's exactly the reason the theorem is puzzling and popular!
THIS STATEMENT IS NOT PROVABLE
The implication is although mathematicians may be working on things like Fermat's last theorem they may not provable even if they are true. So they may be just wasting time [in some sense] trying to prove it. Of course Fermat's last theorem itself was provable and proved by Andrew Wiles.
- a set of assumptions (axioms) and
- use a set of rules for deriving conclusions (inference)
- Proofs are statements which are derivable from axioms using these rules.
For example, we can't prove both 2+2 = 4 and 2+2 != 4 in the system. Only one can be proven.
Now we know we can't prove both. Do we know we can prove the true one in all cases?
Godel theorem says no. In a consistent system, there will always be statements which are true but not provable! Sounds odd, but that's exactly the reason the theorem is puzzling and popular!
THIS STATEMENT IS NOT PROVABLE
The implication is although mathematicians may be working on things like Fermat's last theorem they may not provable even if they are true. So they may be just wasting time [in some sense] trying to prove it. Of course Fermat's last theorem itself was provable and proved by Andrew Wiles.
