Collatz_problem [comrade/them]

Kind Vladimir Ilyich would have shot the political leadership of every post-Soviet country.

This means that you could write a syntactically valid statement which cannot be proven from the axioms of that system even if you were to add more axioms.

You can actually add the statement itself as an axiom. The point of the theorem is that no finite number of additional axioms will completely eliminate all unprovable true statements from the theory.

Also, it relies on consistency of the formal system, because inconsistent system can prove anything. In fact, you can prove consistency of a formal system if and only if it is inconsistent.

Information-theoretic incompleteness is new to me, but seems to be similar to Gödel’s theorem but with a focus on computation saying that if you have a complex enough system there are functions that won’t be recursively definable. As in you can’t just break it down into smaller parts that can be computed and work upwards to it.

In fact, any function, growing fast enough, will be non-recursive. And the same applies to various similar definitions, resulting in fast-growing hierarchy.

All in all this means that no algorithmic theory could actually describe everything. This means you cannot break all of physics down into a finite set of rules that can be used to compute reality. Ergo, we can’t be in a simulation because there are physical phenomena that exist which are impossible to compute.

It should be noted that it doesn’t rule out analog simulations.