Hi DrummerDick,
The question was about knowledge. Can knowledge exist without proof? Goedel's Theorem states that in a sufficiently complex formal system (SCFS), there may exist propositions which cannot be proved to be consistent within the system, and if no such statements exist, it cannot be proven so within the system.
This gives us three cases:
1. In a SCFS, an inconsistent proposition can be proved to be true. In this case, we would declare that the axioms of the system themselves are inconsistent, and the system is not covered by Goedel's theorem.
2. In a SCFS, an inconsistent proposition exists but there is no proof within the system that the proposition is either true or false. In this case, the system is inconsistent, but we cannot know it.
3. In a SCFS, a consistent proposition exists but there is no proof within the system that the proposition is either true or false. In this case, the system is incomplete, but again, we cannot know it.
So, what we know from Goedel's theorem is that every SCFS is either incomplete or inconsistent, but we cannot know which.
My original question was whether or not we can know something without proof. In your example of Goedel's Theorem, the only thing we know is what I said in the previous paragraph, and that has been proven. I'm looking for something we know which has not been proven.
***There are also examples of computer results in math that are true bit have not yet been proven from first principles.***
Such as?
Warm regards,
Paul |