The question
∅∉𝒜 ⇒ ∃f∈∏𝒜or∅∉𝒜 ⇒ ∄f∈∏𝒜?
Which universe are we working in?
A collection of sets without empty sets has a function that chooses an element. Or does it?
One from each · when “obvious” becomes an assumption
A hinge axiom of set theory
Given any family of nonempty sets, may we choose one member from every set—even when no rule tells us how?With Choice every set can be well ordered; Zorn’s lemma (sets have maxima) and Banach–Tarski follow. Reject it and some sets have no choice function, some sets cannot be well ordered, so a different mathematics appears.
Russell’s socks
“Among boots we can distinguish right and left … but with socks no such principle of selection suggests itself.”
A shoe carries a built-in rule: choose every left one. An indistinguishable pair of socks does not. Scale that arbitrary decision to infinitely many pairs and the innocent verb “choose” becomes an axiom.
Bertrand Russell · Introduction to Mathematical Philosophy · 1919 · Read chapter XII ↗Banach–Tarski · the impossible picture
One ball becomes two balls as big as the first.
Assume Choice, and one ball can be partitioned into finitely many non-measurable sets of points and rearranged by rotations and translations into two balls, each the size of the original. The familiar picture is a useful lie: the pieces are not ordinary solids.
See how one becomes two ↗Allan’s point
The great hinge axioms are not bureaucratic preliminaries. They are doors into coherent mathematical worlds. The Parallel Postulate, the Axiom of Determinacy, the Continuum Hypothesis, and the Axiom of Choice are great because mathematics becomes larger when we ask what it's like in each world. But first we must see the door.
Parallel PostulateEuclid built plane geometry from five simple assumptions. The fifth said, essentially, that parallel lines never meet. For 2,000 years mathematicians tried to prove it from the other four. They failed because Euclid was right: it was an assumption. Lobachevsky showed that the opposite assumption yields an equally rich geometry, opening the path to Riemannian geometry—and, ultimately, the geometry Einstein needed for General Relativity.Why the fifth postulate? ↗Axiom of DeterminacyImagine a game where two players alternately choose natural numbers, producing an infinite sequence; a predetermined set of sequences determines who wins. Determinacy says that for every such game, one player has a winning strategy. It conflicts with full Choice—but in return makes sets of real numbers remarkably well-behaved: measurable, regular, and largely free of the pathologies that Choice permits.Play Alice and Bob’s game ↗Continuum HypothesisInfinity comes in different sizes. Cantor showed that the integers are countably infinite, while the real numbers form a larger infinity, the continuum. Is there an infinity in between? The Continuum Hypothesis says no—but neither it nor its opposite follows from the usual axioms of set theory. Cohen's "Forcing" turned such independence into a powerful mathematical tool.About infinity and forcing ↗