Theorem in applied mathematics that relies on the axiom of choice

[Berrius]

Hi there,

Im looking for a theorem that relies on the axiom of choice, but is used in applied mathematics (economics, physics, biology, whatever). In other words a mathematical theory we use to say something about the real world.
This is because I'm wondering if discarding the axiom of choice has any consequence for the mathematics we use to tell us something about the real world. In other words, are we worst of if we discard it, or is it only used in theorems relevant in pure math.

 

[Number Nine]

You know that all of the mathematics that we use to describe the "real world" has theoretical underpinnings, right? Eliminating the AC would change pure mathematics, which would alter the theory underlying all of applied mathematics. The fact that \mathbb{R}^+ and \mathbb{C}^+ are isomorphic is somewhat important, and requires the axiom of choice.

 

[Berrius]

Yes, but I'm writing a text non-mathematicians so I'm searching for for a very concrete example.

 

[ImaLooser]

Hi there,

Im looking for a theorem that relies on the axiom of choice, but is used in applied mathematics (economics, physics, biology, whatever). In other words a mathematical theory we use to say something about the real world.
This is because I'm wondering if discarding the axiom of choice has any consequence for the mathematics we use to tell us something about the real world. In other words, are we worst of if we discard it, or is it only used in theorems relevant in pure math.

You could have math that doesn't use the axiom of choice. You could use math that doesn't have any infinities at all. It's not hard, just messy, and would be fine for real-world use.

The axiom of choice is used in probability where they are talking about things like choosing a random number from the real interval [0,1]. It isn't possibible to really do that, so it is necessary to assume the axiom of choice to allow this. But like I said, you don't really need this. It's just an unrewarding mess to work around not using it.

 

[micromass]

The axiom of choice is used in probability where they are talking about things like choosing a random number from the real interval [0,1]. It isn't possibible to really do that, so it is necessary to assume the axiom of choice to allow this.

How does this relate to the axiom of choice??