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:
- Extensionality: ∀x∀y(∀z(z ∈ x ↔ z ∈ y) → x = y) — sets with same members are equal
- Empty set: ∃x∀y(y ∉ x) — the empty set ∅ exists
- Pairing: ∀x∀y∃z(x ∈ z ∧ y ∈ z) — for any x, y, the pair {x, y} exists
- Union: ∀x∃y∀z(z ∈ y ↔ ∃w(w ∈ x ∧ z ∈ w)) — ∪x exists
- Power set: ∀x∃y∀z(z ∈ y ↔ z ⊆ x) — P(x) exists
- Infinity: ∃x(∅ ∈ x ∧ ∀y(y ∈ x → y ∪ {y} ∈ x)) — an infinite set exists
- Separation (schema): ∀x∃y∀z(z ∈ y ↔ (z ∈ x ∧ φ(z))) — {z ∈ x : φ(z)} exists
- Replacement (schema): if φ defines a function on x, its range exists
- Regularity (Foundation): ∀x(x ≠ ∅ → ∃y(y ∈ x ∧ y ∩ x = ∅)) — no infinite descending ∈-chains
- 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.
| Symbol | Unicode | Name | Meaning |
|---|---|---|---|
| ∈ | U+2208 | Element of | Set membership |
| ∉ | U+2209 | Not element of | Non-membership |
| ⊆ | U+2286 | Subset | Subset or equal |
| ⊂ | U+2282 | Proper subset | Strict subset |
| ∪ | U+222A | Union | Set union |
| ∩ | U+2229 | Intersection | Set intersection |
| ∅ | U+2205 | Empty set | Set with no elements |
| P(x) | — | Power set | Set of all subsets |
| {x : φ} | — | Set-builder | Set of x satisfying φ |
| ω | U+03C9 | Omega | First infinite ordinal |
| ℵ₀ | U+2135 + ₀ | Aleph-null | First infinite cardinal |
| ℶ | U+2136 | Beth | Beth numbers |
Metatheory. ZFC is incomplete (Gödel): some statements (like CH, the Continuum Hypothesis) are independent. ZFC is not finitely axiomatizable. Consistency of ZFC cannot be proved within ZFC (Gödel). Relative consistency: if ZFC is consistent, so is ZFC + CH and ZFC + ¬CH (Gödel, Cohen). Large cardinal axioms (inaccessibles, measurables, etc.) extend ZFC but their consistency is not provable in ZFC.
Applies to. Foundations of mathematics. Encoding all mathematical structures. Infinite combinatorics. Model theory. Descriptive set theory. Proof that various mathematical objects exist.
Limitations. First-order ZFC cannot distinguish "intended" models (where ∈ means real membership) from nonstandard models. Set theory is one foundation among several (type theory, category theory). The axiom of choice is non-constructive; some mathematicians reject it. Proper classes (like the class of all sets) are not sets, requiring awkward workarounds. The encoding of mathematics in sets is unnatural (is 3 ∈ π meaningful?). Independence results mean many natural questions have no answer in ZFC alone.
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