The Axiom of Choice, Uncertainty, and the Limits of Mathematical Knowledge

Introduction:
The axiom of choice is one of the most fundamental, powerful, and controversial principles in mathematics. It states that given any collection of non-empty sets, we can construct a function that picks an element from each set. This simple-sounding statement has far-reaching consequences across mathematics, but it also raises deep questions about the nature of mathematical existence and the limits of what we can know.

The Scenario:
Consider the following situation. We have two sets, A and B. We define a third set, C, as the set of elements that are in either A or B, but not in both. In mathematical notation:

C = (A ∪ B) \ (A ∩ B)

Now, suppose we don’t know whether A and B are equal or not. This means we don’t know whether C is empty or not.

The Question:
What can we say about the choice functions for C? A choice function is a function that picks an element from each non-empty subset of C. The axiom of choice guarantees that such a function exists for any non-empty set. But in this case, we don’t know if C is non-empty. So can we still apply the axiom of choice?

The Uncertainty Principle:
This scenario highlights a fundamental limitation of the axiom of choice, and of mathematical knowledge more broadly. The axiom of choice is an existential principle — it tells us that certain functions exist, but it doesn’t tell us how to construct them or what they look like.

In this case, the existence of a choice function for C depends on whether C is empty or not, which in turn depends on whether A and B are equal or not. But we don’t know whether A and B are equal. So we’re in a state of irreducible uncertainty about the choice functions for C.

The Constructive Perspective:
One way to look at this is through the lens of constructive mathematics. In constructive mathematics, to prove that something exists, you have to be able to construct it. From this perspective, we might say that the axiom of choice doesn’t apply to C, because we can’t construct a choice function for C without knowing whether it’s empty or not.

This suggests a possible reformulation of the axiom of choice: instead of just asserting the existence of a choice function, we could require the set of all choice functions to be non-empty. Under this reformulation, the axiom of choice would indeed fail for C in our scenario.

The Limits of Knowledge:
But this reformulation also highlights the limits of what we can know in mathematics. Even with a principle as powerful as the axiom of choice, there are situations where we bump up against the boundaries of mathematical certainty.

In this case, our uncertainty about A and B translates directly into an uncertainty about the choice functions for C. No amount of mathematical sophistication can eliminate this uncertainty, because it’s inherent in the setup of the problem.

Conclusion:
This simple scenario illuminates the complex interplay between existence, construction, and knowledge in mathematics. It shows how even the most fundamental principles, like the axiom of choice, can lead to situations of irreducible uncertainty, like for instance the Banach-Tarski paradox.

But it also shows the power of mathematical thinking to probe the limits of its own certainty. By carefully analyzing the assumptions and implications of our mathematical statements, we can gain a deeper understanding of what we know, what we can know, and what we can’t know.

In the end, mathematics is not just about proving theorems and solving problems. It’s also about asking questions, exploring boundaries, and grappling with the deep mysteries of abstract thought. And it’s in this questioning and grappling that the true beauty and profundity of mathematics reveals itself.