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
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.
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