Set Theory

Prerequisite path Preparation & skip guidance · Proofs & Mathematical Writing

Studying Set Theory

Counting elements and comparing order types answer different questions. Two sets may be equinumerous while their chosen well-orders behave differently under addition. Button's text makes that transition within an introductory axiomatic route; Halmos offers a shorter account of sets, order and cardinality. Either can support the early language, but the ordinal calculations deserve their own written examples rather than being absorbed as unfamiliar notation.

What to work on

Prove countability by constructing an enumeration or injection.
Apply Cantor's diagonal argument to a power set.
Distinguish ordinal addition from cardinal addition.
Explain the role of replacement, transfinite recursion and choice in basic constructions.

A starting text

Set Theory: An Open Introduction — Tim Button, based on the Open Logic Project. Use sections 5.2 and 5.6-5.9 for cardinality; 10.3-10.7 and 10.9-10.10 for well-orders, ordinals and replacement; 11.2 for transfinite recursion; 13.2-13.3 for ordinal addition; 15.1-15.2 for cardinal arithmetic; and 16.4 for choice and well-ordering.

Choosing what to read

Construct actual injections and bijections before using cardinal equations. Draw the order types of omega, omega plus one and one plus omega, then explain the difference without appealing only to symbols. When a construction iterates through all earlier stages, identify the recursion theorem and the relevant existence assumptions. Keep an axiom ledger for replacement and choice: record what each permits and which argument would be incomplete without it.

Set Theory: An Open Introduction

Tim Button, based on the Open Logic Project

Separate cardinality, well-orders and transfinite recursion in a single route.

How it reads: Introductory axiomatic route with explicit ordinal and cardinal constructions.
What it expects: Proof-writing and elementary logic; ability to work with functions and relations.
What to keep in mind: The chosen sections are preparation for, rather than a course in, forcing or large cardinals.
Practice: Exercises provided; public solutions not verified. Chapter 5 Problems, printed page 70; chapter 10 Problems, page 148; chapter 11 Problems, page 160; chapter 13 Problems, page 186; chapter 16 Problems, page 229.

Complete public PDF.

Edition, reading range and source

Use sections 5.2 and 5.6-5.9 for cardinality; 10.3-10.7 and 10.9-10.10 for well-orders, ordinals and replacement; 11.2 for transfinite recursion; 13.2-13.3 for ordinal addition; 15.1-15.2 for cardinal arithmetic; and 16.4 for choice and well-ordering.

Source · Checked 2026-10-16

Naive Set Theory

Paul R. Halmos

Obtain a second account of maps and order before the more explicit ordinal route.

How it reads: Short exposition of elementary sets, order and cardinality.
What it expects: Comfort with mathematical proofs and basic logical notation.
What to keep in mind: Its compact scope leaves advanced axiomatic and independence methods for later study.
Practice: Use the acquired edition's exercises where present; public solutions and exercise coverage were not verified.

Purchase or eligible institutional access; publisher samples are not the complete book.

Edition, reading range and source

Select the set, order, cardinality and ordinal discussions using the linked reprint's contents; no edition-specific page range is asserted.

Source · Checked 2026-10-11

Practice

Cardinality, ordinal and choice problems. Chapter 5 Problems, printed page 70; chapter 10 Problems, page 148; chapter 11 Problems, page 160; chapter 13 Problems, page 186; chapter 16 Problems, page 229.. Exercises provided; public solutions not verified.

Try these before moving on

Try a proof or calculation before opening the hint. These solutions address the checks below, not the exercises in the linked textbooks.

Construct an explicit injection from the integers into the natural numbers, then prove no map A -> P(A) can be surjective.
A hint
Send nonnegative and negative integers to disjoint parity classes. For the power set, form a subset whose membership disagrees with its proposed preimage.
Work through the solution
Using $\mathbb N=\{0,1,2,\ldots\}$ , define $j(n)=2n$ for $n\ge0$ and $j(n)=-2n-1$ for $n<0$ . The first branch has even values and the second has odd values, so equality of outputs forces the inputs into the same branch. Each branch is injective, hence so is $j$ .

For any set $A$ and map $f:A\to\mathcal P(A)$ , the subset $D=\{a\in A:a\notin f(a)\}$ belongs to $\mathcal P(A)$ . If $f$ were onto, some $b\in A$ would satisfy $f(b)=D$ . Then $b\in D$ if and only if $b\notin f(b)=D$ , a contradiction. Thus no such map is surjective; the argument also covers $A=\varnothing$ .

If this was the sticking point:
- Proofs & Mathematical Writing — Practice piecewise definitions, injectivity and proof by contradiction before cardinal comparisons.

Already comfortable with this? Skip the naive-set review if you can construct countability proofs and prove Cantor's theorem. Skip the ordinal selection only if you can work with order types and explain why transfinite recursion requires more than ordinary induction.

Review the prerequisites · Compare the full subject collection

Where this can lead

Mathematical Logic & Computability — Study formal languages when you want to distinguish an axiom system from the structures satisfying it.
Model Theory — Apply cardinal and compactness arguments to elementary extensions and types.

Reading-source dates

Set Theory: An Open Introduction · 2026-10-16

Reading choices revised 2026-10-17.

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Set Theory: An Open Introduction

Online text

Tim Button · Textbook · Introductory

Explains the set-theoretic foundations of mathematics and the motivation for ZFC. Its emphasis is conceptual and philosophical, not graduate forcing theory.

Preparation: Basic logic and high-school mathematics.

Full book & PDF

Additional reading

Search the complete catalogue.

Introductory reading

Naive Set Theory

Paul R. Halmos · Textbook · Introductory · English · Purchase / institutional access

Purchase or institutional access required; previews are not the complete book.

A short introduction to sets, order, cardinality and ordinal ideas. Useful preparation before axiomatic and forcing-based study.

Preparation: Proof-writing; basic logic and sets.

Official page / access details

Edition and sources

Version: Linked Springer reprint, 1974; original 1960 · Checked 2026-10-11

Foundations of Set Theory

OperatorP · Course notes · Introductory · Chinese · Online text

Online lecture text; chapters can be read separately.

An introduction to logic and set theory, developing the language of sets and basic constructions used throughout undergraduate mathematics.

Preparation: School mathematics; no previous axiomatic set theory assumed.

Use: Read the logical language and basic constructions before the more advanced axiomatic-set-theory sequel.

Open online text / archive

Edition and sources

Original title: 集合论基础

Version: Online lecture manuscript · Checked 2026-11-03

Intermediate reading

A Course on Set Theory

Ernest Schimmerling · Textbook · Intermediate · English · Purchase / institutional access

Purchase or institutional access required; previews are not the complete book.

An undergraduate treatment of ZFC, order, cardinality, trees, filters and ideals. A bridge from elementary sets toward axiomatic set theory.

Preparation: Proof-writing; basic logic and sets.

Official page / access details

Edition and sources

Version: Print 2011; electronic publication 2012 · Checked 2026-10-11

Set Theory: An Introduction

Robert L. Vaught · Textbook · Intermediate · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

A compact progression from sets, ordinals and cardinals toward axiomatic considerations. Good for preparing to understand consistency questions.

Preparation: Proof-writing; elementary logic.

Official page / access details

Edition and sources

Version: 2001 reprint · Checked 2026-10-11

Graduate reading

Classical Descriptive Set Theory

Alexander S. Kechris · Textbook · Graduate · English · Purchase / institutional access

The complete published book is subscription or purchase content. A preview or sample is not the complete book; institutional access depends on your library licence.

Studies definable sets in Polish spaces and their interaction with analysis.

Preparation: Topology; measure theory; elementary set theory.

Use: Enter descriptive set theory and its analytic applications.

Publisher / access page

Edition and sources

Version: 1995 · Checked 2026-10-11

Thomas Jech · Textbook · Graduate · English · Purchase / institutional access

Purchase or institutional access required; previews are not the complete book.

A large reference progressing from set-theoretic foundations to forcing and advanced results. Select chapters according to the intended topic.

Preparation: Proof-writing; basic logic and sets.

Official page / access details

Edition and sources

Version: Third Millennium edition, revised and expanded, 2003 · Checked 2026-10-11

Set Theory and the Continuum Hypothesis

Paul J. Cohen · Textbook · Graduate · English · Purchase / institutional access

Purchase or institutional access required; previews are not the complete book.

A foundational exposition of independence and forcing. Read after basic logic and axiomatic set theory, not as a first introduction to mathematical proof.

Preparation: Proof-writing; basic logic and sets.

Official page / access details

Edition and sources

Version: Dover reprint; consult publisher for current printing · Checked 2026-10-11

Combinatorial Set Theory: With a Gentle Introduction to Forcing

Lorenz J. Halbeisen · Textbook · Graduate · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

Develops combinatorial set theory, choice principles and forcing. Follow the foundational chapters before selecting a specialized continuum problem.

Preparation: Axiomatic set theory and first-order logic.

Official page / access details

Edition and sources

Version: Third edition, 2025 · Checked 2026-10-11

Axiomatic Set Theory

Gaisi Takeuti and Wilson M. Zaring · Textbook · Graduate · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

Studies model-construction methods including constructibility, forcing and Boolean-valued models. Historical notation is best compared with a current course.

Preparation: Axiomatic set theory; first-order logic.

Official page / access details

Edition and sources

Version: 1973 · Checked 2026-10-11

Lectures in Set Theory: With Particular Emphasis on the Method of Forcing

Thomas J. Jech · Textbook · Graduate · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

A short, concentrated course on forcing and consistency methods. Distinct from Jech’s later comprehensive Set Theory reference.

Preparation: ZFC, ordinals and cardinals; models of first-order theories.

Official page / access details

Edition and sources

Version: Lecture Notes in Mathematics 217, 1971 · Checked 2026-10-11

A Course on Borel Sets

Shashi Mohan Srivastava · Textbook · Graduate · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

Develops Borel sets and measurable selections as a bridge to descriptive set theory. A different branch from forcing and large cardinals.

Preparation: Topology; measure theory; set-theoretic language.

Official page / access details

Edition and sources

Version: 1998 · Checked 2026-10-11

Descriptive Set Theory

David Marker · Course notes · Graduate · English · PDF

Covers Polish spaces, Borel and analytic sets, effective hierarchies, determinacy and Borel equivalence relations.

Preparation: Topology, mathematical logic and elementary set theory.

Use: Bridge classical descriptive set theory and classification by equivalence relations.

Open full PDF Source / terms

Edition and sources

Version: Fall 2002 course notes · Checked 2026-10-13

Axiomatic Set Theory

Asaf Karagila · Course notes · Graduate · English · PDF

Develops axioms, ordinals, cardinal arithmetic, absoluteness, reflection and constructibility, followed by combinatorial examples.

Preparation: Elementary set theory and first-order logic.

Use: Bridge basic sets and transfinite methods with inner-model arguments.

Open full PDF Source / terms

Edition and sources

Version: Updated November 18, 2025; course taught in 2016–2017 · Checked 2026-10-13

Further Set Theory A: Axiomatic Foundations

OperatorP · Course notes · Graduate · Chinese · Online text

Online lecture text; chapters can be read separately.

A developing sequence on formal set-theoretic theories and construction principles. Read the existing chapters selectively; a listed language-extension section is still unfinished.

Preparation: First-course set theory and first-order logic.

Use: Compare the axioms and the resulting theories after a first course in sets.

Open online text / archive

Edition and sources

Original title: 集合论进阶 A：公理集合论基础

Version: Online manuscript; unfinished · Checked 2026-11-03

Research reading

The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings

Akihiro Kanamori · Textbook · Research · English · Purchase / institutional access

Purchase or library subscription required; previews are not the complete book.

A detailed development of large-cardinal principles and their relationships. Use after axiomatic set theory and introductory forcing.

Preparation: ZFC, ordinals, cardinals, elementary embeddings and forcing.

Official page / access details

Edition and sources

Version: Second edition, 2003 · Checked 2026-10-11

Descriptive Set Theory and Forcing: How to Prove Theorems about Borel Sets the Hard Way

Arnold W. Miller · Lecture notes · Research · English · PDF

Complete author-deposited PDF opens on arXiv.

Uses forcing methods to study Borel hierarchies, analytic sets, separation theorems and Gandy forcing.

Preparation: Elementary mathematical logic and prior knowledge of forcing.

Use: Study descriptive set theory from a forcing-oriented perspective.

Open full PDF Source / terms

Edition and sources

Version: arXiv v1, January 6, 1994; Spring 1993 course · Checked 2026-10-13

Forcing & Symmetric Extensions

Asaf Karagila · Course notes · Research · English · PDF

Develops forcing, iterations and forcing axioms before automorphisms, symmetric extensions and choiceless model constructions.

Preparation: Axiomatic set theory, ordinals, cardinals and elementary model theory.

Use: Learn forcing and continue into symmetric-extension techniques.

Open full PDF Source / terms

Edition and sources

Version: Updated August 24, 2026; Leeds 2022–2023 course · Checked 2026-10-13

Large Cardinals

Asaf Karagila · Advanced course notes · Research · English · PDF

Surveys large-cardinal hierarchies through ultrapowers, extenders, compactness, inconsistency bounds and choiceless principles.

Preparation: Axiomatic set theory, elementary embeddings and model-theoretic methods.

Use: Study large-cardinal techniques and their relationships before specialist papers.

Open full PDF Source / terms

Edition and sources

Version: Updated April 13, 2026; Spring 2025 Leeds course · Checked 2026-10-13