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
It isn’t trivial because in almost all cases, from some well chosen axioms we can either prove or disprove D.
Some D being true but unprovable is something that until Godel we didn’t know would always happen.
In your example. Gödel’s theorem basically says that no matter how many of the A, B, C etc you take, there will always be D that isn’t provable by any of their combinations. A, B and C aren’t just separate entities, they allow for other theorems to be derived from them. All Euclidean geometry was initially built on 5 axioms (and modern definitions include up to 20 of them) but look how many provable theorems there are in geometry.
>Gödel proves that there are (true) theorems in math that cannot be proven formally from their axioms.
What’s “their” axioms here? There are no ownership relationship between a true statement and axioms. What’s the axioms of “there are no biggest natural number”?
That statement about Godel theorem is too imprecise to even say where your misunderstanding happen. My guess here is that you think it said that there exist some true claims, and there exist some axioms (called “their” axioms), such that the true claims cannot be proved from these axioms. If that’s what you thought it said, then yeah the theorem sound trivial.
The actual Godel theorem said that *any* consistent, enumerable set of axioms that interpret Peano’s arithmetic will never be sufficient to *prove* or *disprove* every claims.
The theorem talk about *any* set of axioms with the above property. It’s easy to intentionally invent a set of axioms that is just bad; but the point is that this is a fundamental limitation that you cannot fix even if you try.
It’s also important to notice the order of quantifier: for any set of axioms then there exist some statements. The set of true claims that cannot be proved is not a fixed set independent of the axioms. Each set of axioms will have different set of unprovable true claims.
Not the most important misunderstanding, but still, you need to consider everything the axioms can prove or disprove. Sure, D might not be listed among A,B,C but depends on what A,B,C are, you might still be able to prove D from it.
All of the conditions in the theorems (consistent, enumerable, interpret Peano arithmetic) are important. You *can* have a set of axioms that let you prove everything if your axioms include contradictions (not consistent). You can have a set of axiom that let you prove all true claims if your axioms are not enumerable. You can have a set of axioms that let you prove or disprove every claims if you just give up on interpreting Peano’s arithmetic; in fact we do actually use these. This makes a vague summary like “there are (true) theorems in math that cannot be proven formally from their axioms” very misleading.
In science, there is a quest to find an equation for everything. The idea with the Grand Unified Theory is that we could come up with some way to completely explain everything in the physical universe.
Gödel tells us that for maths, that’s impossible. We are always trying to find more things out, but some of them might simply be impossible to prove. In maths, everything is built of proof. We can say that we *think* something is true, but that’s not enough. We need a rigorous proof. There are plenty of things that we can prove are true if this one other thing is also true. But the proof of that has escaped us for years. Fermat’s Last Theorem was one example for hundreds of years. So maybe there’s a group of mathematicians out there working tirelessly on a problem that they simply cannot solve because it’s actually impossible. The thing is, we can’t know if something that’s unprovable is actually unprovable!
If D is not true in your system, then it isn’t a theorem. Theorems are only things that can be proven true in a given system.
Prior to Godel it was not obvious that there are true theorems that couldn’t be derived from a set of axioms. In fact, one of the things mathematicians were doing at the time was just that: trying to demonstrate that you *could* prove everything from a given set of axioms.
Almost everything in maths is an axiom. We’re basically unable to prove that 1+1 is always 2 and that there isn’t a way to get 3 out of it.
Also if we take the universe we aren’t able to prove anything because we are inside the system and so anything we figure out is just an axiom that works for itself but can never be truly proven.
But none of that is a problem, because things work on that scale that we can observe and in which we can operate….
Kurt Gödel in the other hand feared his colleagues wraith and starved himself to dear because he thought they would poison him. 😅
*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.”
>While the theorem might be true, it obviously can’t be proven from the axioms.
What ? Don’t you know how proofs work ?
Example :
Axioms are :
* A : All humans are primates
* B : No primate can lay eggs
* C : I am human
Theorem : D = I cannot lay eggs.
Proof : from axioms A and C, we can conclude that I am a primate (statement E). Assume that theorem D is false, which means I can lay eggs (statement F). From statements E and F, we can conclude that some primates can lay eggs (statement G). Axiom B and statement G contradict each other, therefore we made an invalid assumption. Since the only assumption made is that theorem D is false, we can conclude that theorem D is not false. Thus, theorem D is true, QED.
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