Zermelo-Fraenkel set theory with Choice (ZFC), the standard foundation of mainstream mathematics. Provides the container in which other mathematical objects are constructed.
set-theory
Axiom of extensionality
Two sets are equal iff they have the same elements.
Axiom of empty set
There exists a set with no elements.
Axiom of pairing
For any two sets, there exists a set containing exactly those two.
Axiom of union
For any collection of sets, there exists a set containing all their elements.
Axiom of choice
For any collection of nonempty sets, there exists a function selecting one element from each.
Axiom schema of separation
For any set A and any property P, there exists a subset of A containing exactly those elements that satisfy P.
Axiom of power set
For any set A, there exists a set whose elements are exactly the subsets of A.
Axiom of infinity
There exists a set containing the empty set and closed under the successor operation — guaranteeing an infinite set exists.
Axiom schema of replacement
The image of a set under any definable function is also a set.
Axiom of foundation (regularity)
Every non-empty set A contains an element disjoint from A — preventing infinite descending membership chains and sets containing themselves.
Continuum hypothesis
There is no set whose cardinality is strictly between that of the natural numbers and that of the real numbers.
Axiom of choice (AC)
For every family {A_i}_{i∈I} of non-empty sets, there exists a choice function f with f(i) ∈ A_i for each i. Independent of ZF (Gödel…
Zorn's lemma
Every non-empty partially-ordered set in which every chain has an upper bound contains a maximal element. Equivalent to AC over ZF.
Well-ordering theorem
Every set admits a well-ordering. Equivalent to AC; Zermelo's 1904 proof spurred foundational debate and motivated ZFC.
Ordinal number
A transitive set well-ordered by ∈ (von Neumann definition). Canonical representatives of well-ordered types; ordinals extend the natural…
Cardinal number
An ordinal α such that no ordinal <α is in bijection with α. Under AC every set is equinumerous to a unique cardinal; measures 'size'…
Aleph numbers (ℵ_α)
The ordinal-indexed hierarchy of infinite cardinals: ℵ₀ = |ℕ|, ℵ_{α+1} is the least cardinal > ℵ_α, and ℵ_λ = sup_{α<λ} ℵ_α for limits λ.
Cofinality cf(α)
The least order-type of an unbounded subset of α. Regular cardinals (cf(κ)=κ) vs singular cardinals (cf(κ)<κ) is a central dichotomy of…
Forcing
Cohen's 1963 technique for building extensions M[G] of a ground model M that satisfy prescribed new statements. Used to prove independence…
Inaccessible cardinal
An uncountable regular strong-limit cardinal (κ>ℵ₀, cf(κ)=κ, ∀λ<κ: 2^λ<κ). V_κ ⊨ ZFC, so existence is not provable in ZFC (2nd…
Measurable cardinal
An uncountable κ carrying a κ-complete non-principal ultrafilter. Equivalent to existence of a non-trivial elementary embedding j:V→M with…
Gödel's constructible universe L
The minimal inner model of ZFC built by transfinite recursion over definable power sets. V=L implies AC and GCH; Gödel (1938) used L to…
Cardinal arithmetic
κ + λ = max(κ,λ) for infinite; κ · λ = max; but κ^λ subtle. GCH: 2^κ = κ⁺. Hausdorff's formula for cofinality.
Forcing & independence proofs
Cohen 1963: method to prove independence of CH from ZFC by adjoining generic filter to transitive model. Boolean-valued models…
Large cardinals hierarchy
Inaccessible, Mahlo, measurable, supercompact, extendible, Woodin, huge. Form linear-ish consistency hierarchy; resolve V=L status.
Axiom of determinacy (AD)
Every infinite game of perfect information on ω has a winning strategy. Contradicts AC; holds in L(ℝ) assuming Woodin cardinals.
Descriptive set theory
Polish spaces, Borel/projective hierarchy; analytic sets Σ¹_1, co-analytic Π¹_1. Luzin, Sierpiński, Moschovakis.
Inner model L (Gödel)
Constructible universe L = ∪_α L_α defined by restricted power set. L ⊨ GCH + AC + V=L. Core models extend to large-cardinal-compatible V.
Continuum problem (status)
CH: 2^{ℵ_0} = ℵ_1. Gödel (L ⊨ CH), Cohen (¬CH forcing) ⟹ independent from ZFC. Woodin's Ω-logic program, Ultimate-L.
Transfinite induction
Induction principle valid for every ordinal: if P(β) for all β < α implies P(α), then P holds for every ordinal. Derived from the…
Transfinite recursion
Von Neumann's theorem: given a class function G, there exists a unique class function F on ordinals with F(α) = G(F↾α). Extends Dedekind's…
Cantor's theorem
The power set is always strictly larger than the set. Proof by diagonal: for f: X → P(X), the set D = {x : x ∉ f(x)} is not in the image of…
Schröder-Bernstein theorem
Injections in both directions yield a bijection — a result that does not require the axiom of choice. Proof by the Cantor-Bernstein…
Hartogs number
For every set X, the class of ordinals that inject into X is a set, and its union is the least ordinal that does *not* inject into X — the…
Ordinal arithmetic
Addition, multiplication, and exponentiation on ordinals are defined by transfinite recursion and are non-commutative. Lexicographic…
Burali-Forti paradox
If the collection of all ordinals were a set Ω, then Ω would be transitive and well-ordered by ∈, hence itself an ordinal, giving Ω ∈ Ω and…
Limit ordinal
An ordinal that is neither 0 nor a successor. Equivalently, λ = sup{β : β < λ}. The smallest limit ordinal is ω; the next is ω·2. Limit…