Comment by swagmoney1606
3 days ago
In my mind this is literally what math is. We start with axioms, and derive conclusions. There's probably more to it than that, but that's the understanding I'm at now.
3 days ago
In my mind this is literally what math is. We start with axioms, and derive conclusions. There's probably more to it than that, but that's the understanding I'm at now.
But shouldn't it also be part of the axioms what are the rules that allow you to derive new theorems from them?
So then you could self-apply it and start ... deriving new rules of how you can derive new theorems and thus also new rules, from axioms?
I'm jusr confused a bit about "axioms" and "rules". What's the difference?
The rules that you use to compose axioms and propositions are a different set of axioms defined by the Logic system you're using. e.g., can a proof consist of infinitely many steps? Can I use the law of excluded middle? Some logic systems won't let you re-use the same proposition more than once, etc,...
They're usually considered separate, because they're orthogonal to the foundational axioms you're using to build up your mathematical systems. With the exact same system of axioms, you might be able to prove or disprove certain things using some logic systems, but not others.
Choosing the axioms is difficult.
Presumably made easier by something like Lean where you can have a very minimal set of axioms, because things you might use as axioms already have proved versions, in Lean.