Boolean Algebras of Sets

Core Idea

The power set P(X) of any set X forms a Boolean algebra under union (join), intersection (meet), and complementation (negation), with ∅ as 0 and X as 1. This algebraic structure — capturing the laws of union, intersection, and complement — generalizes to abstract Boolean algebras, which need not consist of sets at all. A filter on a Boolean algebra is an upward-closed collection closed under finite meets (modeling 'large' sets); an ultrafilter is a maximal filter (for every element, it contains either the element or its complement). Stone's representation theorem establishes that every Boolean algebra is isomorphic to a field of sets — the clopen subsets of some compact totally disconnected space (its Stone space). Boolean algebras appear throughout set theory: in the algebraic formulation of forcing, the forcing poset is embedded in a complete Boolean algebra, and generic ultrafilters on this algebra correspond to forcing extensions.

How It's Best Learned

Start with P({1,2,3}): draw its Hasse diagram under ⊆, identify unions as joins and intersections as meets, and verify the Boolean algebra axioms (complementation, distributivity, De Morgan's laws). Then define a filter (e.g., the Fréchet filter of cofinite subsets of ℕ) and an ultrafilter (e.g., a principal ultrafilter generated by a single point). State Stone's theorem and see that P(X) corresponds to the discrete space on X. The connection to forcing becomes natural: a generic filter is simply an ultrafilter that intersects every dense subset of the Boolean algebra.

Common Misconceptions

• Not every Boolean algebra is a power set — there are countable atomless Boolean algebras (like the clopen sets of the Cantor space) that are not isomorphic to any P(X).
• The existence of non-principal ultrafilters on P(ℕ) requires the axiom of choice (specifically, the Boolean prime ideal theorem or ultrafilter lemma), which is strictly weaker than full AC.

Explainer

You already know about the power set P(X) from the axiom of power set, and you know the basic set operations — union, intersection, and complement — from naive set theory. A Boolean algebra is the algebraic abstraction of exactly this structure. The power set P(X) with ∪ (join), ∩ (meet), complement (negation), ∅ (bottom, or 0), and X (top, or 1) satisfies a list of equational laws — commutativity, associativity, distributivity of ∩ over ∪ and vice versa, identity laws, and complement laws. These are the Boolean algebra axioms. Any algebraic structure satisfying these laws is a Boolean algebra, whether or not its elements are sets.

A filter on a Boolean algebra B is a subset F ⊆ B that is (1) upward-closed: if a ∈ F and a ≤ b then b ∈ F, and (2) closed under finite meets: if a, b ∈ F then a ∧ b ∈ F. In P(ℕ), the Fréchet filter — the collection of all cofinite subsets (subsets with finite complement) — is a canonical example. It models the intuition of "almost all": a set is "large" (in the filter) if it contains all but finitely many natural numbers. Filters formalize a notion of largeness or genericity that does not specify which particular large set you have, only that it satisfies the closure conditions.

An ultrafilter is a filter U that is maximal — you cannot add any more elements to U without violating the filter axioms. The equivalent characterization: U is an ultrafilter if and only if for every element a, either a ∈ U or its complement ¬a ∈ U (but not both). An ultrafilter is decisive — it assigns every element of the algebra to one side or the other. Principal ultrafilters are easy to construct: given a point x ∈ X, the collection of all sets containing x is an ultrafilter on P(X). Non-principal ultrafilters — ones not generated by a single point — require the axiom of choice to prove they exist; they cannot be described explicitly.

Stone's representation theorem reveals that abstract Boolean algebras are not really more general than power sets — every Boolean algebra is isomorphic to a field of sets (the clopen subsets of its Stone space). The Stone space of B is a compact, totally disconnected topological space whose points are the ultrafilters of B. This connects Boolean algebra to topology: ultrafilters become points, Boolean operations become set operations on clopen sets. The connection to forcing is the culmination: Cohen's forcing poset embeds into a complete Boolean algebra (one where every subset has a supremum), and a generic filter — the new object adjoined to the ground model — is exactly an ultrafilter that meets every dense subset of the algebra. The abstract algebraic machinery of Boolean algebras is not an end in itself; it is the scaffolding on which forcing is built.