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
>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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