Specifically, he proved that there is a true statement in a formal system that cannot be proven by the formal system. There is an additional option: that of a formal system which is incapable of self-reference or self-description.
That would be a system in which “There does not exist a valid proof that “There does not exist a valid proof that the statement “There does not exist a valid proof that the statement “”There does not exist a valid proof that the statement… …is true” is true” is true” cannot even be constructed.
I should have been more careful when I said “arithmetic” too, because if I recall correctly a system that has only addition and not multiplication is insufficient to construct a self referencing statement.
I guess I am making the assumption that if the rules of the universe exist, they at least contain addition and multiplication since we can construct both of them.
Specifically, he proved that there is a true statement in a formal system that cannot be proven by the formal system. There is an additional option: that of a formal system which is incapable of self-reference or self-description.
That would be a system in which “There does not exist a valid proof that “There does not exist a valid proof that the statement “There does not exist a valid proof that the statement “”There does not exist a valid proof that the statement… …is true” is true” is true” cannot even be constructed.
I should have been more careful when I said “arithmetic” too, because if I recall correctly a system that has only addition and not multiplication is insufficient to construct a self referencing statement.
I guess I am making the assumption that if the rules of the universe exist, they at least contain addition and multiplication since we can construct both of them.