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
>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.
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