Can all the truths which can be established under an axiomatic system be discovered by mathematics? — guptanishank
Because true statements have to be proven as true? — guptanishank
In math however if a statement is unproven, we cannot "know", if it was one of these fundamental preexisting truths or not. — guptanishank
Thank you fdrake and others!
Just one last thing:
Where does the law of excluded middle fit into all this?
A statement must be either true or false.
So if it is unprovable, within a formal axiomatic system, and you cannot decide it's truth value even by going outside the system, what value do you assign to that statement?
How does this fit within the context of Godel's theorems? — guptanishank
Well, when they came out they were a massive 'fuck you' to the Hilbert Program — fdrake
Godel's theorem does not say that we cannot prove the statement by going outside the system. Indeed, for the unprovable statements often considered in this context, they can be proved by moving to a meta-system that is larger than the original one and consistent with it. The problem is that there will then be new statements in that meta-system that cannot be proved or disproved (are 'undecidable') in that system. So we need to move to a meta-meta-system to decide those. No matter how many times we do this, the system we end up with will contain new undecidable statements.So if it is unprovable, within a formal axiomatic system, and you cannot decide it's truth value even by going outside the system, what value do you assign to that statement?
How does this fit within the context of Godel's theorems? — guptanishank
Indeed, historians have recently discovered a cache of previously unpublished letters between Gödel and Hilbert. I take the liberty of summarizing them here.
Godel: Fick dich!
Hilbert: Oh no mein freund. Fick DICH!!
Gödel: Fick dich to the n-th power!
Hilbert: Und deine Mutter auch!
etc. — fishfry
Godel's theorem does not say that we cannot prove the statement by going outside the system. Indeed, for the unprovable statements often considered in this context, they can be proved by moving to a meta-system that is larger than the original one and consistent with it. The problem is that there will then be new statements in that meta-system that cannot be proved or disproved (are 'undecidable') in that system. So we need to move to a meta-meta-system to decide those. No matter how many times we do this, the system we end up with will contain new undecidable statements. — andrewk
Godel's theorem does not say that we cannot prove the statement by going outside the system. Indeed, for the unprovable statements often considered in this context, they can be proved by moving to a meta-system that is larger than the original one and consistent with it. The problem is that there will then be new statements in that meta-system that cannot be proved or disproved (are 'undecidable') in that system. So we need to move to a meta-meta-system to decide those. No matter how many times we do this, the system we end up with will contain new undecidable statements. — andrewk
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