Gödel’s theorem is a logical proof about any axiomatic system within which multiplication and division are defined.

By nature, every scientific model that uses basic arithmetic relies on those kinds axioms and is therefore incomplete.

Furthermore, the statement “we live in a simulation” is a logical statement with a truth value. Thus it is within the realm of first order logic, part of mathematics.

The reason you cannot prove the statement is because it itself is standalone. The statement tells you nothing about the universe, so you cannot construct any implication that can be proven directly, or by contradiction, or by proving the converse etc.

As for the latter half of your comment, I don’t think I’m the one who hasn’t thought about this enough.

You are the one repeating the line that “science doesn’t prove things” without realizing that is a generalization not an absolute statement. It also largely depends on what you call science.

Many people say that science doesn’t prove things, it disproves things. Technically both are mathematic proof. In fact, the scientific method is simply proving an implication wrong.

You form a hypothesis to test which is actually an implication “if (assumptions hold true), then (hypothesis holds true).” If your hypothesis is not true then it means your assumptions (your model) are not correct.

However, you can prove things directly in science very easily: Say you have a cat in a box and you think it might be dead. You open the box and it isn’t dead. You now have proven that the cat was not dead. You collected evidence and reached a true conclusion and your limited model of the world with regards to the cat is proven correct. QED.

Say you have two clear crystals in front of you and you know one is quartz and one is calcite but you don’t remember which. But you have vinegar with you and you remember that it should cause a reaction with only the calcite. You place a drop of vinegar on the rocks and one starts fizzing slightly. Viola, you have just directly proven that rock is the calcite.

Now you can only do this kind of proof when your axioms (that one rock is calcite, one rock is quartz, and only the calcite will react with the vinegar) hold true.

The quest of science, of philosophy, is to find axioms that hold true enough we can do these proofs to predict and manipulate the world around us.

Just like in mathematics, there are often multiple different sets of axioms that can explain the same things. It doesn’t matter if you have “the right ones” You only need ones that are not wrong in your use case, and that are useful for whatever you want to prove things with.

The laws of thermodynamics have not been proven. They have been proven statistically but I get the feeling that you wouldn’t count statistics as a valid form of proof.

Fortunately, engineers don’t care what you think, and with those laws as axioms, engineers have proven that there cannot be any perpetual motion machines. Furthermore, Carnot was able to prove that there is a maximum efficiency heat engine and he was able to derive the processes needed to create one.

All inventions typically start as proof based on axioms found by science. And often times, science proves a model wrong by trying to do something, assuming the model was right, and then failing.

The point is that if our scientific axioms weren’t true, we would not be able to build things with them. We would not predict the world accurately. (Notice that statement is an implication) When this happens, (when that implication is proven false) science finds the assumption/axiom in our model that was proven wrong and replaces it with one or more assumptions that are more correct.

Science is a single massive logical proof by process of elimination.

The only arguments I’ve ever seen that it isn’t real proof are in the same vein as the “you can’t prove the world isn’t a simulation.” Yep, it’s impossible to be 100% certain that all of science is correct. However, that doesn’t matter.

It is absolutely possible to know/prove if science dealing with a limited scope is a valid model because if it isn’t, you’ll be able to prove it wrong. “Oh but there could be multiple explanations” yep, the same thing happens in mathematics.

You can usually find multiple sets of axioms that prove the same things. Some of them might allow you to prove more than the others. Maybe they even disagree on certain kinds of statements. But if you are dealing with statements in that zone of disagreement, you can prove which set of axioms is wrong, and if you don’t deal with those statements at all, then both are equally valid models.

Science can never prove that only a single model is correct… because it is certain that you can construct multiple models that will be equally correct. The perfect model doesn’t matter because it doesn’t exist. What matters is what models/axioms are true enough that they can be useful, and science is proving what that is and isn’t.

Sorry, the point I was trying to make is that we will be able to know if any statement that is testable is correct.

I just wanted to clarify that your initial comment is only true when you are counting things that don’t actually matter in science. Anything that actually matters can be tested/proven which means that science can be 100% correct for anything that’s actually relevant.