Set theory: the terms, and where each one lives¶

Level: reference · for the owner's list of 646 terms of set theory, from "set" to Berkeley cardinals, with the page or glossary entry that covers each one here, and the book that covers the rest

One line: Of the 646 terms, 261 have a home in this library and 385 do not; a term with a home is linked to it, a term without one is named under "not here yet" with the book that treats it, and no stub pages were made, because a page with no argument and no program is what the roadmap rules out, and this list is where such a page would have pointed anyway.

The list is the owner's, sent in October 2026, grouped as they grouped it. A term counts as "here" when a lesson or a glossary entry explains it, not when a page merely mentions it; links to the glossary land on the entry. The book pointers for uncovered sections are from memory and name editions loosely; the reading guide has the verdicts on the books. Topics from this list that had an argument and a program became lessons in the same session: Cantor–Schröder–Bernstein, multisets, Ramsey's theorem, two balls from one, the forms of choice, and the counting of partitions on equivalence relations and partitions. What remains uncovered and is worth a page is on the memory inbox: cardinal arithmetic, order types, filters on a finite set, trees with König's lemma, Cantor's back-and-forth on ℚ.

Contents¶

Foundational concepts — 21 of 21 here
Basic set operations — 18 of 18 here
Set algebra and Boolean structure — 11 of 21 here
Relations — 29 of 30 here
Functions and mappings — 23 of 25 here
Cardinality — 19 of 22 here
Ordinal numbers — 9 of 23 here
Cardinal numbers and cardinal arithmetic — 0 of 24 here
Axiomatic set theory — 26 of 29 here
Well-foundedness and recursion — 13 of 13 here
Constructibility and inner models — 3 of 15 here
Forcing and independence — 6 of 26 here
Large cardinals — 5 of 24 here
Descriptive set theory — 9 of 32 here
Filters, ultrafilters and ideals — 9 of 27 here
Combinatorial set theory — 4 of 27 here
Set-theoretic topology — 4 of 21 here
Trees and orders — 7 of 27 here
Independence and consistency results — 4 of 15 here
Equivalent forms of the axiom of choice — 10 of 14 here
Set theory and logic — 12 of 20 here
Multisets and generalisations — 9 of 13 here
Specific forcing notions and advanced forcing techniques — 0 of 24 here
Cardinal invariants of the continuum (Cichoń’s diagram) — 0 of 21 here
Shelah’s PCF theory — 0 of 8 here
Infinitary combinatorics and walks on ordinals — 0 of 14 here
Determinacy, games and higher descriptive set theory — 0 of 22 here
Inner model theory and fine structure — 0 of 15 here
Extended large cardinal hierarchy — 0 of 12 here
Structural, constructive and alternative set theories — 7 of 12 here
Model-theoretic set theory, admissible sets and logic — 1 of 12 here
Advanced order theory and structure — 2 of 11 here
Non-standard and structural axiomatisations — 0 of 8 here

The terms¶

Foundational concepts¶

Set — what is a set?
Element — what is a set?
Member — what is a set?
Membership — what is a set?
Empty Set — glossary
Universal Set — glossary
Subset — the algebra of sets
Proper Subset — the symbol table: ⊂ versus ⊆
Superset — the algebra of sets
Proper Superset — the symbol table
Equality of Sets — extensionality
Extensionality — glossary
Well-Defined Set — the textbook definition on trial
Naive Set Theory — the reading guide: two subjects with one name
Axiomatic Set Theory — the axioms chapter
Universe of Discourse — predicates and quantifiers
Class — glossary
Proper Class — glossary
Collection — the textbook definition on trial
Family of Sets — unions
Primitive Notion — what is a set?

Basic set operations¶

Union — the algebra of sets
Intersection — the algebra of sets
Difference — the difference law
Complement — glossary
Symmetric Difference — glossary
Cartesian Product — glossary
Power Set — glossary
Disjoint Sets — glossary
Partition — glossary
Cover — equivalence relations and partitions
Refinement — glossary
Closure under Union — glossary
Closure under Intersection — glossary
Finite Union — the algebra of sets
Finite Intersection — the algebra of sets
Arbitrary Union — unions
Arbitrary Intersection — unions
Pairwise Disjoint — equivalence relations and partitions

Set algebra and Boolean structure¶

Boolean Algebra of Sets — glossary
De Morgan's Laws — the de morgan law
Distributive Laws — the distributive law
Associativity — the commutative associative law
Commutativity — the commutative associative law
Idempotency — the idempotent law
Absorption Laws — the absorption law
Double Complement — the double complement law
Lattice of Sets — the absorption law
Sigma-Algebra — the axioms chapter: where the axioms reach other mathematics
Algebra of Sets — the algebra of sets

Not here yet: Sublattice, Boolean Lattice, Complete Lattice, Sigma-Ring, Ring of Sets, Field of Sets, Monotone Class, Pi-System, Lambda-System, Dynkin System. The families of sets that measure theory is built on (σ-algebras, rings, fields, π- and λ-systems, monotone classes, Dynkin systems) are the first chapter of any measure theory book, Halmos’s Measure Theory or Folland’s Real Analysis; 02_Measure_Zero has the sets they are built for. Lattices beyond the absorption law: Davey and Priestley, Introduction to Lattices and Order.

Relations¶

Binary Relation — glossary
Relation on a Set — glossary
Domain — glossary
Range — glossary
Codomain — glossary
Image — relations and functions
Preimage — function katas
Inverse Relation — relations and functions
Composition of Relations — relations and functions
Reflexive Relation — glossary
Irreflexive Relation — glossary
Symmetric Relation — glossary
Antisymmetric Relation — glossary
Asymmetric Relation — glossary
Transitive Relation — glossary
Equivalence Relation — glossary
Equivalence Class — equivalence relations and partitions
Quotient Set — equivalence relations and partitions
Partial Order — glossary
Strict Partial Order — glossary
Total Order — glossary
Linear Order — glossary
Well-Order — glossary
Pre-order — glossary
Compatible Relation — glossary
Congruence Relation — equivalence relations and partitions
Closure of a Relation — glossary
Transitive Closure — glossary
Reflexive Closure — glossary

Not here yet: Partial Equivalence Relation. A partial equivalence relation (symmetric and transitive, not necessarily reflexive) is the one combination the orderings lesson names without a home; it is the setoid of type theory.

Functions and mappings¶

Function — glossary
Partial Function — relations and functions
Total Function — relations and functions
Injective Function — glossary
Surjective Function — glossary
Bijective Function — glossary
Identity Function — orderings
Constant Function — function katas
Restriction of a Function — function katas
Extension of a Function — function katas
Composition of Functions — relations and functions
Inverse Function — function katas
Image of a Set — function katas
Preimage of a Set — function katas
Indexed Family — function katas
Sequence — the symbol table
Choice Function — glossary
Selector — choice
Projection Map — the Cartesian product
Characteristic Function — relations and functions
Indicator Function — relations and functions
Multifunction — relations and functions
Correspondence — relations and functions

Not here yet: Net, Diagonal Map. Nets are a topology notion (Kelley’s General Topology); the diagonal map x ↦ (x, x) is a line in any algebra book.

Cardinality¶

Cardinality — glossary
Finite Set — infinity
Infinite Set — infinity
Countable Set — glossary
Countably Infinite Set — glossary
Uncountable Set — glossary
Equinumerosity — cardinality
One-to-One Correspondence — glossary
Cantor-Bernstein-Schroeder Theorem — glossary
Cardinal Number — cardinality
Continuum — cardinality
Cardinality of the Continuum — cardinality
Countable Union — countable sets
Continuum Hypothesis — the founding papers
Generalized Continuum Hypothesis — the founding papers
Dedekind-Infinite — the reading guide: Dedekind 1888, run
Dedekind-Finite — the reading guide: Dedekind 1888, run
Hartogs Number — glossary
Comparable Cardinals — Cantor–Schröder–Bernstein

Not here yet: Aleph Numbers, Beth Numbers, Amorphous Set. The alephs, beths and Hartogs’s theorem are the cardinal arithmetic on the inbox: Hrbacek and Jech chapters 4, 7 and 9; Jech chapter 3; Komjáth and Totik chapters 1 to 3. Amorphous sets live in models without choice: Jech’s The Axiom of Choice.

Ordinal numbers¶

Ordinal Number — glossary
Successor Ordinal — ordinals
Limit Ordinal — infinity
Zero Ordinal — ordinals
Transfinite Ordinal — ordinals
Well-Ordered Set — glossary
Epsilon Numbers — ordinals
Von Neumann Ordinal — ordinals
Omega (ω) — infinity

Not here yet: Order Type, Ordinal Arithmetic, Ordinal Addition, Ordinal Multiplication, Ordinal Exponentiation, Fixed Point of Ordinal Arithmetic, Omega-1 (ω₁), Countable Ordinal, Uncountable Ordinal, Ordinal Collapse, Club Set, Stationary Set, Regressive Function, Fodor's Lemma. Ordinal arithmetic and order types: Hrbacek and Jech chapter 6, André chapters 34 and 35, Jech chapter 2; a lesson is on the inbox. Clubs, stationary sets and Fodor’s lemma: Jech chapter 8, Kunen chapter II.

Cardinal numbers and cardinal arithmetic¶

Not here yet: Cardinal Arithmetic, Cardinal Addition, Cardinal Multiplication, Cardinal Exponentiation, Regular Cardinal, Singular Cardinal, Strong Limit Cardinal, Weakly Inaccessible Cardinal, Cofinality, König's Theorem, Gimel Function, Power Function, Cardinal Collapse, Successor Cardinal, Limit Cardinal, Cardinal Characteristic of the Continuum, Bounding Number, Dominating Number, Splitting Number, Reaping Number, Ultrafilter Number, Covering Number, Additivity, Uniformity. Cardinal arithmetic and cofinality: Jech chapter 5, Hrbacek and Jech chapter 9, Halbeisen chapter 3. The cardinal characteristics of the continuum: Halbeisen chapter 9, “Thirteen cardinals and their relations”; Blass’s chapter in the Handbook of Set Theory; Bartoszyński and Judah.

Axiomatic set theory¶

Zermelo Set Theory — replacement
Zermelo-Fraenkel Set Theory (ZF) — the axioms chapter
ZFC (ZF with Choice) — the axioms chapter
Axiom of Extensionality — extensionality
Axiom of Empty Set — comprehension
Axiom of Pairing — pairs
Axiom of Union — unions
Axiom of Power Set — power set
Axiom of Infinity — infinity
Axiom of Separation (Schema) — comprehension
Axiom of Replacement (Schema) — replacement
Axiom of Regularity (Foundation) — foundation
Axiom of Choice — choice
Axiom of Dependent Choice — choice: the equivalent forms
Axiom of Countable Choice — choice: the equivalent forms
Axiom of Determinacy — the reading guide, beyond ZFC
Axiom of Projective Determinacy — the reading guide, beyond ZFC
Von Neumann-Bernays-Gödel Set Theory (NBG) — the reading guide, beyond ZFC
Morse-Kelley Set Theory — the reading guide, beyond ZFC
Kripke-Platek Set Theory — the reading guide, beyond ZFC
New Foundations (NF) — the reading guide, beyond ZFC
NFU — the reading guide, beyond ZFC
Anti-Foundation Axiom — foundation
Aczel's Anti-Foundation Axiom — foundation
Scott-Potter Set Theory — the reading guide: Potter
Tarski-Grothendieck Set Theory — the tools table

Not here yet: Global Choice, Axiom of Real Determinacy, Positive Set Theory. The weak forms of choice (countable, dependent, global): Halbeisen chapter 6, Jech’s The Axiom of Choice, Herrlich’s Axiom of Choice. Determinacy: Kanamori, Kechris. Positive set theory is a research niche (Esser, Forti and Hinnion).

Well-foundedness and recursion¶

Well-Founded Relation — foundation
Well-Founded Recursion — ordinals
Well-Founded Induction — foundation
Transfinite Induction — ordinals
Transfinite Recursion — replacement
Rank of a Set — glossary
Von Neumann Hierarchy — foundation
Cumulative Hierarchy — reading a formula
Grounded Set — foundation
Foundation / Regularity — glossary
Non-Well-Founded Set Theory — foundation
Hypersets — foundation
Ill-Founded Set — foundation

Constructibility and inner models¶

Constructible Universe (L) — the founding papers
Gödel's Constructibility — the founding papers
Axiom of Constructibility (V=L) — the founding papers

Not here yet: Relative Constructibility, Core Model, Inner Model, Definable Set, Ordinal-Definable Set, Hereditarily Ordinal-Definable Set (HOD), L[A] (Relativized Constructibility), L[U] (Ultrapower Model), L[E] (Extender Model), Steel's Core Model, Mitchell Order, Fine Structure Theory. Gödel’s L: Jech chapter 13, Kunen chapter VI, Devlin’s Constructibility; HOD and the relativised models: Jech chapter 13; core models and fine structure: Zeman’s Inner Models and Large Cardinals, the Schindler–Zeman and Steel chapters of the Handbook.

Forcing and independence¶

Forcing — the founding papers
Generic Extension — the graduate books
Independence from ZFC — the founding papers
Relative Consistency — the reading guide
Cohen Forcing — the founding papers
Martin's Axiom — the reading guide

Not here yet: Forcing Poset, Forcing Condition, Dense Set (in Forcing), Filter (in Forcing), Generic Filter, Generic Set, Name (in Forcing), Forcing Relation, Boolean-Valued Model, Ground Model, Extension Model, Consistency Strength, Random Forcing, Collapse Forcing, Countable Support Iteration, Finite Support Iteration, Proper Forcing, Semi-Proper Forcing, Proper Forcing Axiom (PFA), Martin's Maximum. Kunen’s Set Theory is the textbook; Halbeisen’s Part III is the gentler route and its Part IV the combinatorics; Chow’s “A beginner’s guide to forcing” says what the construction is for; Bell’s Set Theory: Boolean-Valued Models for that approach; Jech Part III and Shelah’s Proper and Improper Forcing for proper forcing and the forcing axioms.

Large cardinals¶

Weakly Compact Cardinal — glossary
Measurable Cardinal — the founding papers
Supercompact Cardinal — the infographic caution
Woodin Cardinal — the infographic caution
Large Cardinal Hierarchy — the reading guide, beyond ZFC

Not here yet: Inaccessible Cardinal, Mahlo Cardinal, Strongly Compact Cardinal, Extendible Cardinal, Huge Cardinal, Rank-into-Rank Cardinal, Shelah Cardinal, Ramsey Cardinal, Erdős Cardinal, Jónsson Cardinal, Rowbottom Cardinal, Consistency Strength Hierarchy, 0-Sharp (0#), 0-Dagger (0†), Elementary Embedding, Critical Point, Target Model, Ultrapower, Extender. Kanamori’s The Higher Infinite is the whole subject; Jech chapters 9, 17 to 20; Drake’s Set Theory: An Introduction to Large Cardinals for the older, gentler version.

Descriptive set theory¶

Polish Space — the graduate books
Borel Set — the graduate books
Perfect Set — the Cantor set
Baire Category Theorem — glossary
Meager Set — glossary
Cantor-Bendixson Theorem — ordinals
Derived Set — ordinals
Vitali Set — choice
Hamel Basis — choice: the equivalent forms

Not here yet: Borel Hierarchy, Sigma Sets, Pi Sets, Delta Sets, Analytic Set, Co-Analytic Set, Projective Hierarchy, Projective Set, Projective Determinacy, Universally Measurable Set, Perfect Set Property, Baire Property, Comeager Set, Scattered Set, Luzin Set, Sierpiński Set, Bernstein Set, Effective Descriptive Set Theory, Lightface Hierarchy, Boldface Hierarchy, Wadge Degrees, Wadge Hierarchy, Lipschitz Reducibility. Kechris’s Classical Descriptive Set Theory; Moschovakis for the effective theory and Wadge degrees; Oxtoby’s Measure and Category for Baire category, Luzin, Sierpiński and Bernstein sets in a hundred readable pages.

Filters, ultrafilters and ideals¶

Filter — glossary
Ultrafilter — glossary
Principal Filter — glossary
Non-Principal Filter — glossary
Free Ultrafilter — glossary
Fixed Ultrafilter — glossary
Ultrafilter Lemma — glossary
Prime Ideal — choice: the equivalent forms
Measure — measure zero

Not here yet: Ideal, Principal Ideal, Maximal Ideal, Dual Filter, Dual Ideal, Saturated Ideal, Normal Filter, Normal Ultrafilter, Closed Unbounded Filter, Club Filter, Stationary Ideal, Non-Stationary Ideal, Precipitous Ideal, Ultrapower via Ultrafilter, Łoś's Theorem, p-Point Ultrafilter, Selective Ultrafilter, Ramsey Ultrafilter. Jech chapters 7, 8, 10 and 22; Halbeisen chapter 11, “Happy families and their relatives”, for p-points and Ramsey ultrafilters; Comfort and Negrepontis for ultrafilters on their own; Chang and Keisler for Łoś and ultrapowers. A lesson on filters on a finite set, where every ultrafilter is principal, is on the inbox.

Combinatorial set theory¶

Ramsey Theory — Ramsey
Ramsey's Theorem — Ramsey
Partition Calculus — glossary
Arrow Notation — glossary

Not here yet: Erdős-Rado Theorem, Delta-System, Sunflower Lemma, Almost Disjoint Family, Independent Family, Mad Family (Maximal Almost Disjoint), Luzin Family, Kurepa Tree, Kurepa Hypothesis, Aronszajn Tree, Suslin Tree, Suslin Line, Suslin Hypothesis, Special Aronszajn Tree, Tree Property, Branch, Cofinal Branch, Square Principle, Diamond Principle, Club Guessing, Winding Number, Combinatorial Principle, Ostaszewski Space. Halbeisen’s Combinatorial Set Theory (3rd ed.), whose contents the owner sent, is the book: Ramsey’s theorem is its overture, chapter 4; the Δ-system lemma, trees and the Suslin problem are chapters 14 and 20; Kunen chapter II; Komjáth and Totik for problems; Erdős, Hajnal, Máté and Rado for the partition calculus. “Winding number” is topology and does not belong in this list.

Set-theoretic topology¶

Dense Set — glossary
Nowhere Dense Set — glossary
First Category — glossary
Cantor Space — the Cantor set

Not here yet: Topological Space (Set-Theoretic View), Open Set, Closed Set, Clopen Set, Second Category, Baire Space, Baire Space (ωω), Stone Space, Stone-Čech Compactification, Extremally Disconnected Space, βω (Stone-Čech of ω), P-Space, F-Space, Zero-Dimensional Space, Regular Open Set, Topology Generated by a Set, Čech-Complete Space. Engelking’s General Topology; the Kunen–Vaughan Handbook of Set-Theoretic Topology; Walker’s The Stone–Čech Compactification for βω. This library’s measure zero chapter has the Cantor set and nowhere-dense sets.

Trees and orders¶

Finitely Branching Tree — glossary
König's Infinity Lemma — glossary
Partial Order — glossary
Linear Extension — orderings
Chain — glossary
Antichain — glossary
Zorn's Lemma — glossary

Not here yet: Tree, Tree Ordering, Branch of a Tree, Level of a Tree, Height of a Tree, Root of a Tree, Body of a Tree, Well-Pruned Tree, ω-Tree, κ-Tree, Closed Tree, Ill-Founded Tree, Cofinal Branch, Binary Tree, Perfect Binary Tree, Dilworth's Theorem, Mirsky's Theorem, Width, Height, Hausdorff's Maximal Principle. Trees and König’s lemma: Kunen chapter II, Jech chapter 9, Halbeisen chapter 20; Dilworth and Mirsky, the finite theorems about chains and antichains, in any combinatorics book (van Lint and Wilson); a tree lesson with König’s lemma run on a finitely branching tree is on the inbox.

Independence and consistency results¶

Continuum Hypothesis — the founding papers
Generalized Continuum Hypothesis — the founding papers
Suslin's Problem — the graduate books
Martin's Axiom — the reading guide

Not here yet: Whitehead Problem, Borel Conjecture, Luzin's Problem, Souslin Hypothesis, Kurepa Hypothesis, Proper Forcing Axiom, Bounded Proper Forcing Axiom, Open Coloring Axiom, Projective Uniformization, Projective Measurability, Large Cardinal Equiconsistency. Kunen for the method, Jech Parts II and III for the results; Halbeisen chapters 16 and 19 to 20 for CH, Martin’s axiom and Suslin’s problem; Todorcevic’s Partition Problems in Topology for the open colouring axiom.

Equivalent forms of the axiom of choice¶

Well-Ordering Theorem — glossary
Zorn's Lemma — glossary
Tukey's Lemma — glossary
Tychonoff's Theorem — choice: the equivalent forms
Ultrafilter Lemma — glossary
Boolean Prime Ideal Theorem — glossary
Banach-Tarski Paradox — two balls from one
Existence of Hamel Basis — choice: the equivalent forms
Comparability of Cardinals — glossary
Krull's Theorem — choice: the equivalent forms

Not here yet: Hausdorff Maximal Principle, Antichain Principle, Partition Principle, Nielsen-Schreier Theorem. Rubin and Rubin’s Equivalents of the Axiom of Choice is the catalogue; Jech’s The Axiom of Choice and Halbeisen chapter 6 the courses; Howard and Rubin’s Consequences of the Axiom of Choice the database. Banach–Tarski is Halbeisen chapter 7, “How to make two balls from one”, and Wagon’s book. The ultrafilter lemma and the Boolean prime ideal theorem are strictly weaker than choice; the partition principle is an open problem.

Set theory and logic¶

First-Order Logic — reading a formula
Model — reading a formula
Structure — reading a formula
Interpretation — reading a formula
Satisfaction — reading a formula
Consistency — the reading guide, on consistency
Gödel's Incompleteness Theorems — the reading guide
Relative Consistency Proof — the founding papers
Downward Löwenheim-Skolem Theorem — power set
Skolem Paradox — power set
Definability — comprehension
Formal Language — reading a formula

Not here yet: Second-Order Logic, Completeness, Soundness, Absoluteness, Upward Löwenheim-Skolem Theorem, Reflection Principle, Set-Theoretic Definability, Ordinal-Definability. Enderton’s A Mathematical Introduction to Logic, Cori and Lascar’s Part I, Kunen’s Foundations of Mathematics, Halbeisen chapter 2 (“First-order logic in a nutshell”); absoluteness and reflection are Kunen chapter IV and Halbeisen chapter 16. A reading guide for logic is on the inbox.

Multisets and generalisations¶

Multiset — multisets
Multiplicity — glossary
Multiset Union — multisets
Multiset Intersection — multisets
Multiset Sum — multisets
Fuzzy Set ↗ — day or night
Hyperset — foundation
Non-Well-Founded Set — foundation
Quine Atom — foundation

Not here yet: Rough Set, Soft Set, Hyperclass, Conglomerate. Rough sets are Pawlak’s, soft sets Molodtsov’s, both in data-mining texts; hyperclasses and conglomerates are the category theorists’ layers above classes (Adámek, Herrlich and Strecker).

Specific forcing notions and advanced forcing techniques¶

Not here yet: Sacks Forcing, Laver Forcing, Mathias Forcing, Prikry Forcing, Silver Forcing, Magidor Forcing, Radin Forcing, Namba Forcing, Hechler Forcing, Tree Forcing, Side Conditions in Forcing, Class Forcing, Subcomplete Forcing, Subproper Forcing, Two-Step Iteration, Finite Support Iteration, Countable Support Iteration, Matrix Iteration, Ground Model Definability, Laver’s Theorem on Ground Model Definability, Symmetric Submodels, Symmetric Intermediate Models, Boolean-Valued Ultrapowers, Quotient Algebras and Forcing. Halbeisen’s Part IV walks through Sacks, Silver-like, Miller, Mathias and Laver forcing one chapter each, with symmetric submodels in chapter 17 and iterations in chapter 18; Jech chapters 15, 16, 21 and 23; Prikry, Magidor and Radin forcing in Gitik’s chapter of the Handbook; Laver’s ground-model definability theorem is a 2007 paper.

Cardinal invariants of the continuum (Cichoń’s diagram)¶

Not here yet: Unbounding Number (b), Dominating Number (d), Pseudo-Intersection Number (p), Tower Number (t), Splitting Number (s), Reaping / Unreaping Number (r), Almost Disjointness Number (a), Ultrafilter Number (u), Independence Number (i), Groupwise Density Number (g), Evasion Number, Aronszajn Tree Number, Additivity of Null Sets, Coverage of Null Sets, Uniformity of Null Sets, Cofinality of Null Sets, Additivity of Meager Sets, Coverage of Meager Sets, Uniformity of Meager Sets, Cofinality of Meager Sets, Cichoń's Diagram Relationships. Halbeisen chapter 9 defines thirteen of these and Part IV separates them by forcing; Bartoszyński and Judah’s Set Theory: On the Structure of the Real Line is Cichoń’s diagram at book length; Blass’s Handbook chapter is the survey. That p = t, Malliaris and Shelah 2016, is the recent theorem.

Shelah’s PCF theory¶

Not here yet: Possible Cofinalities Set (pcf(A)), PCF Topology, Pseudopower, Scale in PCF Theory, Generators of PCF, Exact Upper Bound Principle, Bound on 2^(aleph_omega) (Shelah's Bound), Reduced Products and Ideals. Shelah’s Cardinal Arithmetic; the Abraham–Magidor chapter of the Handbook; Holz, Steffens and Weitz’s Introduction to Cardinal Arithmetic for the readable version of the bound ℵ_ω^ℵ₀ < ℵ_{ω₄} when ℵ_ω is a strong limit.

Infinitary combinatorics and walks on ordinals¶

Not here yet: Todorcevic's Walks on Ordinals, Minimal Walk Steps, Coherence Principles, Gaps and Gap Hierarchy, Hausdorff Gaps, (ω₁, ω₁)-Gaps, Incoherent Sequences, Jensen’s Diamond Variations, Jensen’s Square Variations, Weak Diamond, Matrix Principles, Coloring Principles, S-Spaces and L-Spaces, Happy Families. Todorcevic’s Walks on Ordinals and Their Characteristics; Jensen’s diamond and square in Jech chapter 27 and Devlin’s Constructibility*; Hausdorff gaps in Scheepers’s survey; happy families are Mathias’s and Halbeisen’s chapter 11.

Determinacy, games and higher descriptive set theory¶

Not here yet: Infinite Games of Perfect Information, Gale-Stewart Theorem, Open Determinacy, Borel Determinacy, Axiom of Real Determinacy, Banach-Mazur Game, Wadge Lemma, Wadge Reducibility, Wadge Hierarchy, Scale Property, Prewellordering Property, Uniformization Property, Generalized Baire Spaces, Generalized Cantor Spaces, κ-Borel Sets, κ-Analytic Sets, Equivalence Relation DST, Borel Equivalence Relations, Orbit Equivalence Relations, Polish Group Actions, Hyperfinite Equivalence Relations, Reducibility of Equivalence Relations (E₀, E₁, E₂). Kechris for Gale–Stewart, Borel determinacy (Martin 1975) and the Banach–Mazur game; Moschovakis for scales and prewellorderings; Gao’s Invariant Descriptive Set Theory and Kanovei’s Borel Equivalence Relations for the equivalence-relation theory, which is where equivalence relations go when the sets are uncountable; generalised Baire spaces are a current research programme (Friedman, Hyttinen and Kulikov).

Inner model theory and fine structure¶

Not here yet: Mice and Premice, Extenders and Extender Sequences, Core Model (K), Dodd-Jensen Core Model, Steel Core Model, Covering Lemma, Jensen's Fine Structure Theory, Projective Determinacy in Inner Models, Mouse Order, Iterability and Iteration Trees, Comparison Strategy, Sharp Operator, Indiscernibles, Silver Indiscernibles, HOD Hypothesis (Woodin). Zeman; the Steel, Schindler–Zeman and Mitchell chapters of the Handbook; Jech chapters 18 and 35 for 0# and Silver indiscernibles; Woodin’s HOD hypothesis is in his recent papers only.

Extended large cardinal hierarchy¶

Not here yet: Strongly Unfoldable Cardinal, Subcompact Cardinal, Superstrong Cardinal, Woodin for Strongness, Super-Woodin Cardinal, Almost Huge Cardinal, Superhuge Cardinal, Wholeness Axiom (WA), Berkeley Cardinals, Vopěnka's Principle, Vopěnka Scheme, Extended Large Cardinal Inclusions. Kanamori for everything up to huge; Bagaria’s papers for Vopěnka’s principle; Bagaria, Koellner and Woodin (2019) for Berkeley cardinals, which contradict choice.

Structural, constructive and alternative set theories¶

Constructive Zermelo-Fraenkel (CZF) — the reading guide, beyond ZFC
Intuitionistic ZFC (IZF) — the reading guide, beyond ZFC
Elementary Theory of the Category of Sets (ETCS) — set theory among the branches
Permutation Models — the reading guide: Jech's The Axiom of Choice
Fraenkel-Mostowski Models — the reading guide: Jech's The Axiom of Choice
Quine's New Foundations with Atoms (NFU) — the reading guide, beyond ZFC
Univalent Foundations / Set-Truncation in HoTT — the tools table

Not here yet: Sets, Elements, And Relations (SEAR), Algebraic Set Theory (Joyal-Moerdijk), Class Categories, Small Maps in Category-Theoretic Set Theory, Topos-Theoretic Set Models (Set Objects). Aczel and Rathjen’s notes on CZF; Lawvere and Rosebrugh for ETCS; SEAR is Shulman’s nLab proposal; Joyal and Moerdijk’s Algebraic Set Theory; the HoTT book; permutation models are Halbeisen chapter 8 and Jech’s The Axiom of Choice.

Model-theoretic set theory, admissible sets and logic¶

Kripke-Platek Set Theory with Ordinals — the reading guide, beyond ZFC

Not here yet: Admissible Sets, Admissible Ordinals, Hyperarithmetical Sets (HYP), Barwise Compactness Theorem, Infinitary Logics, Hanf Numbers, Chang's Conjecture, End Extensions, Ill-Founded Models of Set Theory, Standard System of a Model, Automorphism Groups of Models of Set Theory. Barwise’s Admissible Sets and Structures; Sacks’s Higher Recursion Theory for HYP; Chang and Keisler for Hanf numbers and Chang’s conjecture; Kaye and Hamkins for nonstandard models.

Advanced order theory and structure¶

Dense Linear Orders (DLO) — glossary
Antichain Structure in Partial Orders — glossary

Not here yet: Scattered Linear Orders, Hausdorff Classification of Scattered Linear Orders, Special Aronszajn Trees, Rigid Trees, Well-Quasi-Orderings (WQO), Better-Quasi-Orderings (BQO), Kruskal's Tree Theorem, Higman's Lemma, Fraïssé Limits. Rosenstein’s Linear Orderings for DLO, scattered orders and Hausdorff’s classification; Fraïssé’s Theory of Relations; Kruskal and Higman in Diestel’s Graph Theory, chapter 12; Cantor’s back-and-forth on ℚ, the first fact about dense linear orders, is on the inbox as a lesson.

Non-standard and structural axiomatisations¶

Not here yet: Freiling's Axiom of Symmetry, Axiom of Freese-Nation, Singular Cardinals Hypothesis (SCH), Failure of SCH, Subproper and Subcomplete Forcing Axioms, Strong Open Coloring Axiom (OCA), Bounded Proper Forcing Axiom (BPFA), Bounded Martin's Maximum (BMM). Freiling’s 1986 paper; SCH in Jech chapters 8 and 24; the forcing axioms in Jech chapter 31 and Todorcevic’s work; the Freese–Nation property is Fuchino, Koppelberg and Shelah.

Po polsku, w skrócie¶

Właściciel biblioteki przysłał listę 646 terminów teorii mnogości, od „zbioru" po kardynały Berkeley, i spytał, które mają tu swoje strony. Ta strona jest odpowiedzią: 261 terminów ma lekcję albo hasło w słowniku i jest do nich podlinkowanych, 385 nie ma i są wymienione w każdym dziale pod „jeszcze nie tutaj" razem z książką, która je omawia. Pustych stron-zalążków nie zrobiono, bo reguła biblioteki (ROADMAP) mówi, że strona bez argumentu i bez programu nie powstaje; ta lista jest tym, co taka strona by i tak zawierała. Cztery tematy z listy, które mają argument i program, dostały w tej samej sesji lekcje: twierdzenie Cantora–Schrödera–Bernsteina, multizbiory, twierdzenie Ramseya i liczenie podziałów. Reszta wartych strony tematów czeka w skrzynce pomysłów.

Auf Deutsch: Stichwörter¶

Die Liste von rund 650 Begriffen der Mengenlehre, jeder mit der Seite oder dem Glossareintrag, der ihn hier erklärt, der Rest mit dem Buch, das es tut.

Stichwörter: Begriffsliste, Mengenlehre, Glossareintrag, Kardinalzahl, Ordinalzahl, Forcing, große Kardinalzahlen, deskriptive Mengenlehre, Filter und Ultrafilter, Bäume, nicht abgedeckt.