Choosing Elements

Set Theory · Axiom Academy

How hard can it be to pick one item from each basket? Discover why infinity changes everything! You have three baskets, each containing different items. Click one item from each basket to make your selection! Now imagine you need to describe your choice to someone else. Click the "Apply Rule" button to see how a systematic rule makes choosing easy! Scroll through this infinite collection of baskets. Try clicking "Choose Randomly" to make arbitrary choices. Can you describe a rule for what you chose? Compare these two scenarios. When do we know a choice is possible? Choosing one element from each set is easy when you can give an explicit rule. But for infinitely many arbitrary sets, the Axiom of Choice guarantees that such a choice is always possible—even when you can't write down the rule! Given any collection of non-empty sets A i , there exists a function f (called a choice function ) such that f(A i ) ∈ A i for every set in the collection. The Axiom of Choice seems obvious at first ("just pick one!"), but it has profound consequences: it allows us to prove the existence of mathematical objects without constructing them explicitly. This power—and controversy—makes AC one of the most fascinating axioms in mathematics!

This is the written version of the interactive lesson above. See the full Set Theory course.