I’m not a mathematician. Gödel proves that there are (true) theorems in math that cannot be proven formally from their axioms.
How is this significant? Isn’t it trivial?
Example:
“Axioms are: A,B,C are each true. Theorem is: D is true.”
While the theorem might be true, it obviously can’t be proven from the axioms.
I mean the example is utterly trivial, what is the catch? Does the example fall into those that Gödel addresses?
In: Mathematics
*For any* set of axioms [1], there is a true statement for which *a proof does not exist*.
So to prove Godel’s theorem, it doesn’t suffice to say “Okay I picked this set of axioms that are too weak to prove much of anything. Here’s an example of a thing I can’t prove.” Instead you have to say “Okay, Bob picked a set of axioms that Bob thinks are strong enough to prove all of math as we know it. Here’s a fact that’s true in every world where Bob’s axioms are true, and here’s why Bob can’t ever prove that fact.” You have no control over what axioms Bob picks [2]. You have to show an unprovable fact exists *regardless of how Bob chooses his axioms*.
You’re also missing some details involving your unprovable statement D [3]. In particular, the whole “is true in every world where the axioms are true, but a proof does not exist” part is *very* tricky. You have to figure out how to show that D is true without proving it from the axioms, which is already mind-bendingly hard. You have to show this not for some specific set of axioms, but for *any possible* set of axioms that Bob could pick — which makes it even more difficult. Then you have to show that a proof *cannot exist* — no matter how clever someone is, how much computer power they have, how many trillions of lines of tricky math tricks they throw at the problem, *no possible mathematical proof* proves D. We’ve entered the realm of an *extremely challenging* problem.
[1] Technically, for any set of axioms that can describe both addition and multiplication of integers.
[2] Again, there are some technicalities I’m leaving out. The axioms can’t contradict each other. The worlds they describe have to allow integers that can be both added and multiplied. And the axioms have to be specified precisely enough that a computer program can figure out in a finite amount of time whether something’s an axiom, which prevents Bob from cheating by simply saying “I declare every true statement to be an axiom.”
[3] You say D is a “theorem that cannot be proved,” but mathematicians usually reserve the term “theorem” for something that can be proved, and would call D an “unprovable statement,” or “a statement that is true but cannot be proved.”
Latest Answers