# Kunen — Set Theory: An Introduction to Independence Proofs (1980) — Candidate Nodes

## Chapter I — Foundations

ZFC Axioms | axiom | The Zermelo–Fraenkel axioms with Choice: Extensionality, Foundation, Pairing, Union, Power Set, Infinity, Comprehension/Separation schema, Replacement schema, Choice | first-order logic, membership relation
Axiom of Extensionality | axiom | Two sets are equal iff they have the same elements | membership relation
Axiom of Foundation/Regularity | axiom | Every nonempty set has an ∈-minimal element; no infinite descending ∈-chains | membership relation
Axiom of Pairing | axiom | For any two sets there is a set containing exactly them | Extensionality
Axiom of Union | axiom | For any set there is a set that is the union of its members | membership relation
Axiom of Power Set | axiom | For any set there is a set of all its subsets | subset relation
Axiom of Infinity | axiom | There exists an inductive set / a set containing ∅ and closed under successor | empty set, successor
Comprehension/Separation Schema | axiom | For each formula φ, any set has a subset of elements satisfying φ | first-order formulas
Replacement Schema | axiom | The image of a set under a definable class function is a set | first-order formulas, functions
Axiom of Choice | axiom | Every family of nonempty sets has a choice function | functions, families of sets
Ordered Pair (Kuratowski) | state | (a,b) := {{a},{a,b}}, encoding ordered pairs as sets | Pairing
Relation / Function | state | Sets of ordered pairs; function is a single-valued relation | Ordered Pair
Transitive Set | state | A set X with every element a subset: y∈x∈X ⇒ y∈X | membership relation
Ordinal | state | A transitive set well-ordered by ∈ | Transitive Set, well-order
Successor and Limit Ordinals | state | α+1=α∪{α}; a limit ordinal is nonzero and not a successor | Ordinal
Natural Numbers ω | state | The least limit ordinal; finite ordinals modeling ℕ | Ordinal, Axiom of Infinity
Transfinite Induction | technique | Proof method: a property holding below α and at α holds for all ordinals | Ordinal, well-order
Transfinite Recursion | technique | Definition of a class function by recursion along the ordinals | Ordinal, Replacement
Cumulative Hierarchy V_α | state | R(α): V_0=∅, V_{α+1}=P(V_α), V_λ=∪V_β; V=∪V_α | Power Set, Transfinite Recursion, Foundation
Rank of a Set | state | The least α with x∈V_{α+1}, measuring foundation depth | Cumulative Hierarchy
Well-Ordering Theorem | theorem | Every set can be well-ordered (equivalent to AC) | Axiom of Choice
Cardinality / Cardinal | state | A cardinal is an ordinal not in bijection with any smaller ordinal | Ordinal, bijection, Well-Ordering
Aleph Function ℵ_α | state | Enumeration of the infinite cardinals indexed by ordinals | Cardinal, Transfinite Recursion
Cardinal Arithmetic | state | Sum, product, exponentiation of cardinals; κ+λ=κ·λ=max for infinite | Cardinal
Cofinality cf(κ) | state | Least order type of an unbounded subset of κ | Ordinal, unbounded set
Regular / Singular Cardinal | state | κ regular if cf(κ)=κ, singular if cf(κ)<κ | Cofinality
Successor Cardinal is Regular | theorem | Every infinite successor cardinal κ⁺ is regular (under AC) | Cofinality, Axiom of Choice
König's Theorem | theorem | cf(2^κ) > κ; more generally Σκ_i < Πλ_i when κ_i<λ_i | Cofinality, Cardinal Arithmetic, AC
GCH | state | Generalized Continuum Hypothesis: 2^{ℵ_α}=ℵ_{α+1} for all α | Cardinal Arithmetic, Power Set
CH | state | Continuum Hypothesis: 2^{ℵ_0}=ℵ_1 | Cardinal Arithmetic
Club (Closed Unbounded) Set | state | A subset of a regular κ that is closed under limits and unbounded | Cofinality, limit ordinal
Club Filter | state | The filter on κ generated by club sets; closed under <κ intersections | Club Set
Diagonal Intersection | state | △_{α<κ} C_α = {β : β∈C_α for all α<β}; club sets are closed under it | Club Set
Stationary Set | state | A set meeting every club subset of κ | Club Set
Fodor's Lemma (Pressing Down) | theorem | Any regressive function on a stationary set is constant on a stationary subset | Stationary Set, Club Filter
Δ-System (Sunflower) | state | A family of sets with a common pairwise intersection (the root/kernel) | finite sets
Δ-System Lemma | theorem | Every uncountable family of finite sets contains an uncountable Δ-system | Δ-System, Cardinal Arithmetic
Δ-System Method | technique | Use the Δ-system lemma to extract a sunflower for ccc/chain arguments | Δ-System Lemma
Tree (Order-Theoretic) | state | A partial order in which the predecessors of each node are well-ordered | well-order, partial order
Tree Levels and Height | state | Level α = nodes whose predecessor set has order type α; height = sup of levels | Tree
Branch / Chain in a Tree | state | A maximal linearly ordered subset of a tree | Tree
κ-Aronszajn Tree | state | A tree of height κ with all levels <κ and no branch of length κ | Tree, Cardinal
Aronszajn Tree Existence | theorem | There exists an ℵ_1-Aronszajn tree (in ZFC) | Aronszajn Tree
Suslin Tree | state | A tree of height ω_1 with no uncountable chain or antichain | Tree, antichain
Suslin Line / Suslin's Problem | state | A complete dense linear order, ccc but not separable; SH asserts none exists | linear order, ccc
Suslin Hypothesis (SH) | state | Statement that there is no Suslin line (equivalently no Suslin tree) | Suslin Line, Suslin Tree
Kurepa Tree | state | A tree of height ω_1, countable levels, with ≥ℵ_2 branches | Tree, branch
Partition Calculus / κ→(λ)^n_m | state | Arrow notation: every m-coloring of n-subsets of κ has a homogeneous set of size λ | colorings, Cardinal
Ramsey's Theorem | theorem | ℵ_0 → (ℵ_0)^n_k for all finite n,k; infinite homogeneous sets exist | Partition Calculus
Erdős–Rado Theorem | theorem | (2^κ)⁺ → (κ⁺)^2_κ and the general beth-tower partition relation | Partition Calculus, Beth numbers

## Chapter II — Infinitary Combinatorics

Martin's Axiom MA(κ) | axiom | For every ccc poset and ≤κ dense sets there is a filter meeting them all | ccc, dense set, filter
Martin's Axiom MA | state | MA(κ) holds for all κ<2^{ℵ_0} | MA(κ), Cardinal Arithmetic
MA ⇒ ¬(Suslin Tree) | theorem | MA(ℵ_1) implies there are no Suslin trees, hence SH | Martin's Axiom, Suslin Tree
MA(κ) ⇒ 2^{ℵ_0}>κ | theorem | Under MA, the continuum is large; MA+¬CH is consistent | Martin's Axiom, CH
MA and Almost Disjoint Families | theorem | MA implies properties of almost disjoint and dominating families (e.g. p=c) | Martin's Axiom, almost disjoint family
Almost Disjoint Family | state | A family of infinite subsets of ω with pairwise finite intersections | subsets of ω
MA ⇒ Union of <c Meager/Null is Meager/Null | theorem | MA gives additivity of measure and category up to the continuum | Martin's Axiom, measure, category
Diamond Principle ◊ | axiom | A sequence ⟨A_α : α<ω_1⟩ with A_α⊆α guessing every X⊆ω_1 stationarily | Stationary Set, ω_1
◊ ⇒ CH | theorem | The diamond principle implies the Continuum Hypothesis | Diamond Principle, CH
◊ ⇒ Suslin Tree | theorem | Diamond yields a Suslin tree, so ◊ ⇒ ¬SH | Diamond Principle, Suslin Tree
Club Principle ♣ | axiom | A sequence guessing cofinal subsets at limit stages; weaker than ◊ | Stationary Set, club
◊⁺ | axiom | A strengthening of diamond guessing countable subsets; yields Kurepa trees | Diamond Principle, Kurepa Tree
◊⁺ ⇒ Kurepa Tree | theorem | Diamond-plus implies the existence of a Kurepa tree | Diamond Principle, Kurepa Tree

## Chapter III — Models, Absoluteness, Reflection

Relativization φ^M | technique | Restricting quantifiers of a formula to a class/set M | first-order formulas, class
Transitive Model | state | A transitive set/class M satisfying given axioms | Transitive Set, satisfaction
Absoluteness | state | A formula has the same truth value in M as in V | Relativization, Transitive Model
Δ_0 / Σ_0 Formula | state | A formula with only bounded quantifiers; absolute for transitive models | bounded quantifiers
Lévy Hierarchy (Σ_n / Π_n) | state | Classification of formulas by quantifier alternation over Σ_0 | Δ_0 Formula
Σ_1 / Π_1 Absoluteness | theorem | Σ_1 formulas are upward absolute, Π_1 downward, for transitive models | Lévy Hierarchy, Transitive Model
Absoluteness of Basic Notions | theorem | ∈, ⊆, ordinal, ω, ordered pair, function are absolute for transitive models | Δ_0 Formula, Transitive Set
Reflection Theorem | theorem | For each finite list of formulas there are arbitrarily large V_α reflecting them | Cumulative Hierarchy, Relativization
Reflection Method | technique | Use reflection to get set models of finite fragments for consistency proofs | Reflection Theorem
Mostowski Collapse | technique | The transitive collapse of an extensional well-founded relation, unique and ∈-isomorphic | Extensionality, well-founded relation, Transfinite Recursion
Mostowski Collapsing Lemma | theorem | Every well-founded extensional relation is isomorphic to a unique transitive set | Mostowski Collapse
Well-Founded Relation | state | A relation in which every nonempty subset has a minimal element | Foundation
ZF Without AC | state | The theory ZF and the study of choiceless models and consequences | ZFC Axioms

## Chapter IV — Easy Consistency Proofs

Consistency / Con(T) | state | The metamathematical statement that theory T proves no contradiction | first-order logic
Countable Transitive Model (CTM) | state | A countable transitive set model of a finite fragment of ZFC | Transitive Model, Löwenheim–Skolem
Finite ZFC Fragments | technique | Working with finitely many axioms to sidestep Gödel's second incompleteness | ZFC Axioms, Reflection Theorem
Relative Consistency Proof | technique | Build a model of T' inside a model of T to show Con(T)→Con(T') | Consistency, Relativization
Löwenheim–Skolem / Countable Elementary Submodel | theorem | Every model has a countable elementary submodel; collapse gives a CTM | Mostowski Collapse, elementary submodel

## Chapter V — The Constructible Universe L

Gödel Operations / Definable Power Set | state | Def(M): the set of subsets of M definable with parameters over (M,∈) | first-order formulas, satisfaction
Constructible Hierarchy L_α | state | L_0=∅, L_{α+1}=Def(L_α), L_λ=∪L_β | Definable Power Set, Transfinite Recursion
Constructible Universe L | state | L=∪_α L_α, the class of constructible sets | Constructible Hierarchy
Axiom of Constructibility V=L | axiom | The assertion that every set is constructible | Constructible Universe
L is an Inner Model of ZF | theorem | L is a transitive class model of ZF containing all ordinals | Constructible Universe, Transitive Model
Condensation Lemma | theorem | An elementary submodel of L_α collapses to some L_β (β≤α) | Mostowski Collapse, Constructible Hierarchy, elementary submodel
L ⊨ AC | theorem | There is a definable well-ordering <_L of L, so L satisfies Choice | Constructible Universe, Well-Ordering
L ⊨ GCH | theorem | Condensation implies the GCH holds in L | Condensation Lemma, GCH
L ⊨ ◊ | theorem | The diamond principle holds in L | Constructible Universe, Diamond Principle
Con(ZFC) → Con(ZFC+GCH) | theorem | Gödel: relative consistency of GCH (and of V=L) via L | L is an Inner Model, L ⊨ GCH
L ⊨ □ (Square) | theorem | The square / global square principle holds in L (advanced) | Constructible Universe
Relative Constructibility L[A] | state | The class constructible relative to a predicate/set A | Constructible Universe, Definable Power Set
Absoluteness of L | theorem | The notion "x∈L" and L_α are absolute for transitive models of ZF | Absoluteness, Constructible Hierarchy

## Chapter VI — HOD / Inner Models

HOD | state | The class of hereditarily ordinal-definable sets; an inner model of ZFC | ordinal-definable, Transitive Set
OD / Ordinal-Definable | state | Sets definable from ordinal parameters | first-order formulas, Ordinal
HOD ⊨ ZFC | theorem | HOD is a transitive class model of ZFC | HOD, Well-Ordering

## Chapter VII — Forcing

Forcing Poset (P,≤) | axiom | A partial order with a top element 1; conditions, p≤q means p extends q | partial order
Dense Set (in P) | state | A subset D such that every condition has an extension in D | Forcing Poset
Compatible / Incompatible | state | p,q compatible if they have a common extension; antichain = pairwise incompatible | Forcing Poset
Filter on P | state | An upward-closed, downward-directed subset of P | Forcing Poset
Generic Filter G | state | A filter meeting every dense set of M; exists for countable M | Filter, Dense Set, CTM
Generic Filter Existence | theorem | For a CTM M and poset P∈M, an M-generic filter exists (Rasiowa–Sikorski) | Generic Filter, Dense Set
P-Name | state | A set of pairs (name, condition) defined by ∈-recursion; elements of M^P | Transfinite Recursion, Forcing Poset
Valuation / Interpretation val(τ,G) | state | The G-interpretation of a P-name τ_G | P-Name, Generic Filter
Generic Extension M[G] | state | The model {τ_G : τ a P-name in M}; M⊆M[G], G∈M[G] | Valuation, Generic Filter
Canonical Names x̌ and Ġ | state | The check-name for ground elements and the canonical name Ġ for G | P-Name, Generic Filter
Forcing Relation p ⊩ φ | state | Condition p forces formula φ; definable inside M | Forcing Poset, P-Name, satisfaction
Definability Lemma | theorem | The relation p⊩φ is definable in the ground model M | Forcing Relation, Relativization
Truth Lemma | theorem | M[G]⊨φ iff some p∈G forces φ | Forcing Relation, Generic Extension
Forcing Theorem | theorem | Combination of Definability+Truth: forcing captures truth in M[G] | Definability Lemma, Truth Lemma
M[G] ⊨ ZFC | theorem | The generic extension satisfies all of ZFC | Generic Extension, Forcing Theorem
Forcing Method | technique | Construct generic extensions M[G] to prove independence/relative consistency | Generic Extension, Forcing Theorem
Countable Chain Condition (ccc) | state | Every antichain in P is countable | antichain, Forcing Poset
ccc Preserves Cardinals/Cofinalities | theorem | ccc forcing preserves all cardinals and cofinalities | ccc, Generic Extension
κ-Closed Forcing | state | A poset where every descending <κ sequence of conditions has a lower bound | Forcing Poset
κ-Closed Adds No New <κ-Sequences | theorem | κ-closed forcing adds no new sequences of length <κ; preserves cardinals ≤κ | κ-Closed Forcing, Generic Extension
Cohen Forcing Fn(I,2) | state | Finite partial functions from I to 2 ordered by reverse inclusion | Forcing Poset, finite functions
Cohen Real | state | The generic real/subset of ω added by Cohen forcing | Cohen Forcing, Generic Filter
Cohen Forcing is ccc | theorem | Fn(I,2) is ccc, proven via the Δ-system lemma | Cohen Forcing, Δ-System Lemma, ccc
Consistency of ¬CH | theorem | Adding ℵ_2 Cohen reals gives a model of 2^{ℵ_0}=ℵ_2, so Con(ZFC+¬CH) | Cohen Forcing, ccc Preserves Cardinals
Consistency of CH via Forcing | theorem | Countably closed collapse/Add forcing yields models of CH | κ-Closed Forcing
Product Forcing | state | The product of posets P×Q; conditions are pairs | Forcing Poset
Product Lemma | theorem | G generic for P×Q iff G=G_1×G_2 with mutual genericity; M[G]=M[G_1][G_2] | Product Forcing, Generic Filter
Collapsing Cardinals Coll(κ,λ) | state | Forcing with partial functions that collapses λ to κ | Forcing Poset, Cardinal
Lévy Collapse | technique | Coll(ω,<κ) collapsing all cardinals below κ to ω; used for symmetric/Solovay models | Collapsing Cardinals, Forcing Method
Separative Poset | state | A poset where p⊀q implies some r≤p incompatible with q | Forcing Poset
Separative Quotient | state | The separative poset canonically associated to any poset, same forcing | Separative Poset
Regular Open Algebra RO(P) | state | The complete Boolean algebra of regular open sets of P; dense embedding of P | Separative Poset, Boolean algebra
Boolean-Valued Model V^B | state | The B-valued universe over a complete Boolean algebra B | Regular Open Algebra, P-Name
Boolean-Valued Method | technique | Develop forcing via [[φ]]∈B truth values in a Boolean-valued model | Boolean-Valued Model
Equivalence of Forcing and Boolean-Valued Approaches | theorem | Forcing with P matches V^{RO(P)}; quotient by an ultrafilter gives M[G] | Boolean-Valued Model, Regular Open Algebra
Easton Forcing | state | A class product (Easton support) controlling 2^κ on regular cardinals | Product Forcing, regular Cardinal
Easton's Theorem | theorem | Any reasonable (monotone, König-respecting) function on regulars is the continuum function in some extension | Easton Forcing, König's Theorem
Random Real Forcing | state | Forcing with positive-measure Borel sets / the measure algebra adding a random real | measure algebra, ccc
Nice Name | state | A P-name for a subset of a ground set using one antichain per element; used for counting | P-Name, antichain
Counting Nice Names | technique | Bound 2^κ in M[G] by counting nice names for subsets | Nice Name, ccc
Preservation of GCH / Cardinal Computation in Extensions | technique | Use ccc/closure + nice-name counting to compute the continuum in M[G] | Counting Nice Names, ccc Preserves Cardinals

## Chapter VIII — Iterated Forcing

Two-Step Iteration P*Q̇ | state | Forcing with P then a P-name Q̇ for a poset; conditions are pairs (p, q̇) | Forcing Poset, P-Name
Two-Step Iteration Theorem | theorem | M[G][H] for P*Q̇ equals a single generic extension by P*Q̇ | Two-Step Iteration, Generic Extension
Finite Support Iteration | state | An iteration ⟨P_α, Q̇_α⟩ where conditions have finite support | Two-Step Iteration, direct limit
Direct/Inverse Limit | state | Limit stages of an iteration taken with finite (direct) or full (inverse) support | Finite Support Iteration
ccc Preserved Under Finite Support Iteration | theorem | A finite-support iteration of ccc posets is ccc (Δ-system argument) | Finite Support Iteration, ccc, Δ-System Lemma
Solovay–Tennenbaum Theorem | theorem | Con(ZFC) → Con(ZFC + MA + ¬CH) via a finite-support ccc iteration | Finite Support Iteration, Martin's Axiom, ccc Preserved
Iterated Forcing Method | technique | Iterate posets to handle ℵ_1-many tasks (e.g. all ccc posets for MA) | Finite Support Iteration, Forcing Method
Bookkeeping / Anticipating All Posets | technique | Enumerate all relevant names along the iteration to handle every dense requirement | Iterated Forcing Method

## Appendices / Metamathematics

Con(ZFC) Cannot Be Proved in ZFC | theorem | Gödel's Second Incompleteness: ZFC ⊬ Con(ZFC) if ZFC is consistent | Consistency, first-order logic
Relative Consistency Results Framework | technique | All independence results are relative: Con(ZFC)→Con(ZFC+φ) | Consistency, Forcing Method, Reflection Method
Formalizing the Metatheory | state | Treating provability and models within a weak metatheory / arithmetic | first-order logic, Consistency
Independence of CH | theorem | CH is independent of ZFC: Con(ZFC+CH) (Gödel) and Con(ZFC+¬CH) (Cohen) | Con(ZFC)→Con(ZFC+GCH), Consistency of ¬CH
