In reference to your comments on arithmetic, I’m pretty sure Godel’s Incompleteness Theorem states that you -can’t- have a fully self-describing mathematical system. But I may be misunderstanding what you’re saying there.
Godel created a numbering scheme for statements in a formal system that was strong enough to contain arithmetic. The syntactic rules of the logic could be represented as statements in that logic using only arithmetic operations on the numeric representation of statements. Each statement could also be numbered according to the same scheme, and the system was now self-describing because its definition was in the same language that it operated on. From there it was a matter of encoding the statement Y=”There does not exist a sequence of valid derivations in X that results in Y” using the numbering scheme and replacing X and Y where they appear in the statement with the numerical representations of X and Y.
In short, Godel used a self-describing formal system to prove that all formal systems capable of arithmetic were either inconsistent or incomplete. Self-description is not a problem in general, but Godel’s construction works at a meta-level above the system itself. From inside the universe we will only have evidence that our natural laws match or do not match the actual rules.
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.
Godel created a numbering scheme for statements in a formal system that was strong enough to contain arithmetic. The syntactic rules of the logic could be represented as statements in that logic using only arithmetic operations on the numeric representation of statements. Each statement could also be numbered according to the same scheme, and the system was now self-describing because its definition was in the same language that it operated on. From there it was a matter of encoding the statement Y=”There does not exist a sequence of valid derivations in X that results in Y” using the numbering scheme and replacing X and Y where they appear in the statement with the numerical representations of X and Y.
In short, Godel used a self-describing formal system to prove that all formal systems capable of arithmetic were either inconsistent or incomplete. Self-description is not a problem in general, but Godel’s construction works at a meta-level above the system itself. From inside the universe we will only have evidence that our natural laws match or do not match the actual rules.
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.