Set Theory

Origin. Georg Cantor created set theory (1874-1897), introducing infinite cardinals and ordinals. Paradoxes (Russell 1901, Burali-Forti 1897) required axiomatization. Ernst Zermelo (1908) and Abraham Fraenkel (1922) developed ZF. John von Neumann added the axiom of regularity. ZFC (ZF + Axiom of Choice) became the standard foundation for mathematics.

Models. Collections and membership. The fundamental relation ∈ (is an element of) underlies all mathematical structures. Numbers, functions, spaces — all encoded as sets. Set theory provides a universal language and foundation: any mathematical object can be constructed from sets.

Formalism.

ZFC axioms:

1. Extensionality: ∀x∀y(∀z(z ∈ x ↔ z ∈ y) → x = y) — sets with same members are equal
2. Empty set: ∃x∀y(y ∉ x) — the empty set ∅ exists
3. Pairing: ∀x∀y∃z(x ∈ z ∧ y ∈ z) — for any x, y, the pair {x, y} exists
4. Union: ∀x∃y∀z(z ∈ y ↔ ∃w(w ∈ x ∧ z ∈ w)) — ∪x exists
5. Power set: ∀x∃y∀z(z ∈ y ↔ z ⊆ x) — P(x) exists
6. Infinity: ∃x(∅ ∈ x ∧ ∀y(y ∈ x → y ∪ {y} ∈ x)) — an infinite set exists
7. Separation (schema): ∀x∃y∀z(z ∈ y ↔ (z ∈ x ∧ φ(z))) — {z ∈ x : φ(z)} exists
8. Replacement (schema): if φ defines a function on x, its range exists
9. Regularity (Foundation): ∀x(x ≠ ∅ → ∃y(y ∈ x ∧ y ∩ x = ∅)) — no infinite descending ∈-chains
10. Choice (AC): every family of non-empty sets has a choice function

Key constructions:

• Ordered pair: (a, b) = {{a}, {a, b}} (Kuratowski)
• Natural numbers: 0 = ∅, n+1 = n ∪ {n} (von Neumann)
• Functions: sets of ordered pairs with functional property
• Cardinals and ordinals: equivalence classes / well-ordered representatives

Symbols.

Metatheory. Gödel (1938–40) constructed L, the constructible universe, an inner model of ZF in which AC and the generalized continuum hypothesis both hold; this gives Con(ZF) → Con(ZFC + GCH). Cohen (1963) invented forcing and obtained the complementary results, Con(ZF) → Con(ZFC + ¬CH) and Con(ZF) → Con(ZF + ¬AC), so CH and AC are independent of ZF. Easton (1970) showed the continuum function on regular cardinals is otherwise unconstrained by ZFC — any monotone assignment respecting König's theorem cf(2^κ) > κ is consistent — while Silver (1975) proved GCH cannot first fail at a singular cardinal of uncountable cofinality, and Shelah's PCF theory bounds 2^ℵ_ω below ℵ_ω₄ when ℵ_ω is a strong limit, so the singular case is not free. ZFC is not finitely axiomatizable (Montague 1961): by the Montague–Lévy reflection theorem ZFC proves the consistency of each of its finite subtheories, so by Gödel's second theorem no finite subtheory suffices. Kunen (1971) proved there is no non-trivial elementary embedding j : V → V, which caps the large cardinal hierarchy from above. Martin (1975) proved Borel determinacy in ZFC, and Friedman (1971) showed the proof needs ω₁ iterations of the power set, so it is unavailable in Zermelo set theory.

Applies to. Foundations of mathematics. Encoding all mathematical structures. Infinite combinatorics. Model theory. Descriptive set theory. Proof that various mathematical objects exist.

Limitations. ZFC does not determine its own subject matter. It cannot prove Con(ZFC), so it cannot prove it has a model; by Löwenheim–Skolem any model has a countable elementary submodel, in which the set the theory calls uncountable is countable from outside, and ∈ can be interpreted by relations that are not membership. Independence is pervasive rather than exceptional: CH, Suslin's hypothesis, the Whitehead problem (Shelah 1974), the Borel conjecture and the value of 2^ℵ₀ on every regular cardinal are all undecided, and Easton's theorem shows the continuum function is not merely unknown but unconstrained. ZFC decides very little about the projective sets — Lebesgue measurability at Σ¹₂ and projective determinacy both exceed it and require large cardinals whose consistency it cannot establish. AC is non-constructive: it yields non-measurable sets and the Banach–Tarski decomposition while exhibiting no well-ordering of ℝ. Proper classes are not objects of the theory, so quantification over them must be simulated. And the encoding is arbitrary in a way the axioms cannot repair: ZFC assigns a truth value to 3 ∈ π that depends only on the chosen coding of numbers as sets.

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