An Introduction to Set Theory#

Course Description#

This course provides a rigorous introduction to the foundations of modern mathematics through the lens of axiomatic set theory. Starting with the intuitive ideas of Georg Cantor and the paradoxes they engendered, we will build the formal language of mathematics from the ground up using the Zermelo-Fraenkel axioms with the Axiom of Choice (ZFC). Students will learn how fundamental mathematical objects such as numbers, relations, and functions are constructed and will gain a deep understanding of the concepts of cardinality, ordinals, and the profound role of the Axiom of Choice.

Course Objectives#

Upon successful completion of this course, students will be able to:

• Understand the historical context and motivation for axiomatic set theory.
• Identify and explain the axioms of ZFC and their purpose.
• Use the language of set theory to define and prove properties of relations, functions, and orderings.
• Distinguish between finite, countable, and uncountable sets and formally prove the cardinality of given sets.
• Understand the construction of ordinal and cardinal numbers.
• Analyze the role and consequences of the Axiom of Choice and its logical equivalents.

Module 1: The Genesis of Set Theory (Naive Sets & The Foundational Crisis)#

Cantor's Paradise#

Basic Concepts: Sets, elements, subsets, power sets, union, intersection, complement.

Georg Cantor's revolutionary ideas: one-to-one correspondence and the "size" of infinite sets.

Countable sets: The integers ($\mathbb{Z}$) and rational numbers ($\mathbb{Q}$).

The Hierarchy of Infinities & The Crisis#

Uncountable sets: Cantor's diagonal argument and the uncountability of the real numbers ($\mathbb{R}$).

The discovery of paradoxes: Russell's Paradox and its implications.

The need for a formal system: From naive set theory to the axiomatic approach.

Module 2: The ZFC Axioms - A Formal Foundation#

The Language of Set Theory and Basic Axioms#

Introduction to first-order logic for set theory.

Axiom of Extensionality: A set is determined by its members.

Axiom of the Empty Set.

Axiom of Pairing.

Axiom of Union.

Axioms for Building and Restricting Sets#

Axiom Schema of Specification (or Separation): Resolving Russell's Paradox.

Axiom of the Power Set.

Axiom of Infinity: Guaranteeing the existence of an infinite set.

Axiom of Regularity (or Foundation): Prohibiting infinite descending membership chains.

The Axiom of Choice and Replacement#

Axiom Schema of Replacement: A powerful tool for constructing sets.

The Axiom of Choice (AC): Statement and intuitive meaning. Initial discussion of its controversial nature.

Module 3: Relations, Functions, and Orderings#

Ordered Pairs and Relations#

Constructing the ordered pair.

Cartesian products.

Relations, domains, and ranges.

Equivalence relations, partitions, and quotient sets.

Functions and Orderings#

Formal definition of a function.

Injections, surjections, and bijections.

Partial and total orderings.

Well-orderings and the principle of transfinite induction.

Module 4: Cardinality - The Measure of Sets#

Cardinal Numbers#

Equinumerosity and the formal definition of cardinal numbers.

Finite vs. Infinite sets.

The cardinal of the natural numbers: $\aleph_0$ (Aleph-Naught).

Cardinal Arithmetic#

Addition, multiplication, and exponentiation of cardinal numbers.

Cantor's Theorem: $|A| < |P(A)|$.

The Continuum Hypothesis (CH): Statement and historical significance.

Module 5: Ordinal Numbers - The Shape of Well-Ordered Sets#

Well-Ordered Sets and Ordinals#

The definition of ordinal numbers (von Neumann ordinals).

Successor and limit ordinals.

Transfinite Induction and Recursion#

The Principle of Transfinite Induction revisited.

Defining operations on ordinals via Transfinite Recursion.

Ordinal arithmetic (addition, multiplication, exponentiation) and its non-commutative nature.

Module 6: Deeper into the Axiom of Choice#

Equivalents and Consequences of AC#

The Well-Ordering Theorem: Every set can be well-ordered.

Zorn's Lemma: A powerful tool in algebra and topology.

Proving the equivalence of AC, the Well-Ordering Theorem, and Zorn's Lemma.

Applications of Zorn's Lemma (e.g., existence of a basis for every vector space).

Module 7: Advanced Topics and The Limits of Set Theory#

The Cumulative Hierarchy and Models of Set Theory#

The von Neumann universe ($V$) as the universe of all sets.

Brief introduction to the idea of models of ZFC.

Independence and Large Cardinals#

The work of Gödel and Cohen: The independence of AC and CH from ZF.

A glimpse beyond ZFC: The search for new axioms (e.g., large cardinal axioms).