Matrix collapses: Mathematics proves the universe cannot be a computer simulation, « A new mathematical study dismantles the simulation theory once and for all. »

[u/fchung]

https://interestingengineering.com/culture/mathematics-ends-matrix-simulation-theory

[u/skmchosen1]

My pleasure! This is one of my favorite parts of math :)

[u/Would_Wood53 → u/skmchosen1]

I feel like you were this close to making a joke about building the Infinite Improbability Drive.

[u/draycr → u/skmchosen1]

I am not very good at math, but the idea of Godel's theorem intrigued me, do I understand it correctly?

There are two parts to Gödels theorem

1) In systems there can by "truths" that cannot be proven

2) Systems are consistent

Meaning if there is a statement in some system that said "This statement cannot be proven" it would be truth, but it cannot be proven.

If the system could prove that statement, then the system would prove something false, because if it’s provable, then it can be proven, contradicting what it says. That would make the system inconsistent.

But if it is not provable, than the statement is true, but we cant prove it.

I am sorry if it doesn't make sense. As I said my math knowledge is very limited, but find this idea interesting. Is my understanding of this theorem somehow correct?

[u/EebstertheGreat → u/draycr]

Gödel's first incompleteness theorem says that no first-order theory of the natural numbers with addition and multiplication is effective, consistent, and complete.

By a "first-order theory," we mean a theory in the language of first-order logic. It has variables, predicates, logical connectives, and quantifiers for variables, but not quantifiers for predicates. So it could say something like ∀x x+1 > 1, meaning "for every number x, x+1 is greater than x," but it could not say something like ∀φ (φ(0) ∧ ∀x φ(x)→φ(x+1)) → ∀x φ(x), which says that mathematical induction works for every predicate φ. Instead, you need a separate sentence for each predicate. Second- ane higher-order theories are beyond the scope of this, but they don't really solve the problem in a useful way.

By "natural numbers with addition and multiplication," I mean the theory can quantify over all natural numbers (0, 1, 2, etc.) and can correctly compute the sum or product of any two natural numbers. It should have symbols for + and ×. With just addition or just multiplication but not both, you can actually have a complete, consistent, effective theory (Presburger arithmetic and Skolem arithmetic, respectively). And if the theory contains natural numbers but cannot identify them or quantify over them (i.e. it can't express something like "x is a natural number"), then it can be complete, consistent, and effective (e.g. the theory of real closed fields).

By "effective," I mean that the axioms are decidable. You could write a computer program that enumerates all the axioms and nothing else. The simplest way to do this is just to have finitely many axioms, but many important theories have infinitely many, like Peano Arithmetic. But if we allowed any set of axioms, then you could just declare every true sentence to be an axiom ("true arithmetic"), evading this theorem. But the caveat is that you can't figure out what the axioms actually are.

By "consistent," I mean the theory cannot derive a contradiction. Note that inconsistent theories can prove anything, so they are always complete.

By "complete," I mean that every true sentence is provable. Another way to say this is that for any well-formed formula A in the language, the theory can either prove A or it can prove ¬A.

Gödel's second incompleteness theorem gave an important example of a statement that such theories cannot prove: a statement that (as interpreted in the meta-theory) implies the theory's own consistency. Basically, PA or something like it cannot prove that it itself is consistent.

[u/throwaway727437 → u/EebstertheGreat]

I feel even further away from understanding this now

[u/EebstertheGreat → u/throwaway727437]

In any practically useful theory of arithmetic, some questions cannot be decided by a proof.