Axiom

• Axioms

Axioms

Def: An axiom is a proposition that is assumed to be true.

The key in math is to identify what your assumptions are. so people can see them. And the idea is that when you do a proof, anybody who agrees with your assumptions or your axioms can follow you proof. And they have to agree with your conclusion. Now, they might disagree with your axioms, in which case, they’re not going to buy your proof. Now, there are lots of axioms used in math. For example, if an equals b and b equals c, then an equals c. There is no proof of that. But it seems pretty good. And so we just throw it in the bucket of axioms and use it. Now axioms can be contradictory in different contexts. Here’s a good example. In Euclidean geometry, there’s a central axiom that says given a line L and a point p not on L, there is exactly one line through p parallel to L. You all saw this in geometry in middle school, Right? You’ve got a point in a line. There’s exactly another line through the point that’s parallel to the line. Now, there’s also a field called spherical geometry. And there, you have an axiom that contradicts this. It says, given a line L and a point p not on L, there is no line through p parallel to L on the sphere. There’s a field called hyperbolic geometry. And there’s an axiom that says, given a line L and a point p not on L, there are infinitely many lines through p parallel to L.

So how can this be?

Does that mean one of these fields is totally bogus, or two of them are? Because they’ve got contradictory axes. That’s OK. Just whatever field you’re in, state
you’re axioms. And they do make sense in their various fields. This is planar geometry. This is on the sphere. And this is on hyperbolic geometry. They make sense in those contexts. So you can have more or less whatever axioms you want. There are sort of two guiding principles to axioms. Axioms should be – it’s called consistent-- and complete.

Axiom should be:

1. consistant
2. complete

Def: A set of axioms is consistent, if no proposition can be proved to be both true and false.

Def: A set of axioms is said to be complete, if it can be used to prove every proposition is either true or false.

In fact, many logicians spent their careers–famous logicians–trying to find a set of axioms, just one set, that was consistent and complete. In fact, Russell and Whitehead are probably the two most famous. They spent their entire careers doing this, and they never got there. Then one day, this guy named Kurt Godel showed up. And in 1930s, he proved it’s not possible that there exists any set of axioms that are both consistent and complete. Now, this discovery devastated the field. It was a huge discovery. Imagine poor Russell and Whitehead. They spent their entire careers going after this holy grail. Then Kurt shows up and said, hey, guys. There’s no grail. It doesn’t exist.

And that’s a little depressing-- pretty bad day when that heppened. Now, it’s an amazing result, because it says if you want consistency-- and that’s a must-- there will be true facts that you will never be able to prove. It’s proved in a logic course.

For example, maybe Goldbach’s conjecture is true and it is impossible be prove. You can state a problem that you can’t prove is true or false.