Set theory forms the foundation of modern mathematics, providing a framework for defining and working with collections of objects. It introduces fundamental concepts like sets, subsets, and operations that are crucial for understanding mathematical structures.

In this section, we explore the basics of set theory, including definitions, operations, and key axioms. We'll see how these concepts apply to various mathematical structures and discuss some limitations and paradoxes that arise in naive set theory.

Set Theory Fundamentals

Basic Concepts and Definitions

• A set is a well-defined collection of distinct objects
  • Objects in a set are called elements or members
  • Sets are typically denoted using capital letters (A, B, C)
  • Elements are listed between curly braces, separated by commas (e.g., {1, 2, 3})
• Two sets are equal if and only if they have exactly the same elements
  • Order and repetition of elements do not matter ({1, 2, 3} = {3, 2, 1})
• The empty set, denoted by ∅ or {}, contains no elements
• A set A is a subset of a set B if every element of A is also an element of B
  • Denoted as A ⊆ B
  • Every set is a subset of itself (A ⊆ A)
  • The empty set is a subset of every set (∅ ⊆ A)

Power Sets and Axioms

• The power set of a set A, denoted by P(A) or 2^A, is the set of all subsets of A
  • If A has n elements, P(A) has 2^n elements
  • Example: If A = {1, 2}, then P(A) = {∅, {1}, {2}, {1, 2}}
• The axiom of extensionality states that two sets are equal if and only if they have the same elements
  • This axiom defines set equality based on membership, not on how sets are described
• The axiom of specification allows the creation of a new set by selecting elements from an existing set that satisfy a given property
  • Also known as the axiom of separation or subset axiom
  • Allows for the construction of subsets based on a predicate

Applying Set Theory

Set Operations

• The union of two sets A and B, denoted by A ∪ B, contains all elements that belong to either A or B, or both
  • Example: If A = {1, 2, 3} and B = {3, 4, 5}, then A ∪ B = {1, 2, 3, 4, 5}
• The intersection of two sets A and B, denoted by A ∩ B, contains all elements that belong to both A and B
  • Example: If A = {1, 2, 3} and B = {3, 4, 5}, then A ∩ B = {3}
• The difference of two sets A and B, denoted by A - B or A \ B, contains all elements of A that are not elements of B
  • Example: If A = {1, 2, 3} and B = {3, 4, 5}, then A - B = {1, 2}
• The complement of a set A, denoted by A^c or A', is the set of all elements in the universal set that are not in A
  • The universal set, often denoted by U, is the set of all elements under consideration
  • Example: If U = {1, 2, 3, 4, 5} and A = {1, 2, 3}, then A^c = {4, 5}

Laws and Principles

• De Morgan's laws state that (A ∪ B)^c = A^c ∩ B^c and (A ∩ B)^c = A^c ∪ B^c
  • These laws relate the complement of a union to the intersection of complements, and vice versa
• The cartesian product of two sets A and B, denoted by A × B, is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B
  • Example: If A = {1, 2} and B = {x, y}, then A × B = {(1, x), (1, y), (2, x), (2, y)}
  • Cartesian products are used to define relations and functions between sets

Set Theory in Mathematics

Foundations and Definitions

• Set theory provides a rigorous foundation for mathematics by defining basic concepts in terms of sets
  • Numbers, functions, and mathematical structures can be constructed using sets
• Natural numbers can be defined as the finite ordinals
  • Each natural number is the set of all preceding natural numbers (e.g., 3 = {0, 1, 2})
• Real numbers can be defined as Dedekind cuts or equivalence classes of Cauchy sequences
  • Dedekind cuts partition the rational numbers into two non-empty sets (lower and upper cuts)
  • Cauchy sequences are sequences of rational numbers that converge to a unique real number
• Functions can be defined as sets of ordered pairs, with the domain and codomain being sets
  • A function f: A → B is a subset of A × B such that for each a ∈ A, there is exactly one b ∈ B with (a, b) ∈ f

Mathematical Structures and Axioms

• Mathematical structures, such as groups, rings, and fields, can be defined as sets with specific properties and operations
  • A group is a set with a binary operation that satisfies the group axioms (closure, associativity, identity, inverses)
  • A ring is a set with two binary operations (addition and multiplication) that satisfy the ring axioms
  • A field is a ring in which the non-zero elements form a group under multiplication
• The axioms of set theory, such as the Zermelo-Fraenkel axioms (ZF) or the axiom of choice (AC), provide a consistent framework for mathematical reasoning
  • ZF axioms include the axioms of extensionality, pairing, union, power set, infinity, separation, and replacement
  • The axiom of choice states that for any set of non-empty sets, there exists a function that chooses an element from each set
  • AC is independent of ZF and has important consequences in mathematics (e.g., the well-ordering theorem)

Naive Set Theory Limitations vs Paradoxes

Paradoxes in Naive Set Theory

• Naive set theory, which allows unrestricted comprehension, leads to paradoxes such as Russell's paradox
  • Russell's paradox considers the set R of all sets that do not contain themselves as elements
    • If R ∈ R, then R ∉ R, and if R ∉ R, then R ∈ R, leading to a contradiction
• The Burali-Forti paradox arises when considering the set of all ordinal numbers
  • The set of all ordinals would have to be its own successor, which is impossible
• Cantor's paradox involves the power set of the universal set
  • The power set of the universal set would have to be strictly larger than the universal set, which is contradictory

Axiomatic Set Theory and Independence Results

• These paradoxes led to the development of axiomatic set theory, such as Zermelo-Fraenkel set theory (ZF)
  • ZF imposes restrictions on set formation to avoid inconsistencies
  • The axiom schema of separation replaces unrestricted comprehension, allowing only subsets of existing sets to be formed
• The independence of the continuum hypothesis (CH) and the axiom of choice (AC) from ZF demonstrates the limitations of set theory
  • CH states that there is no set with cardinality strictly between that of the natural numbers and the real numbers
  • Both CH and its negation are consistent with ZF, as shown by Gödel and Cohen
  • Similarly, AC and its negation are consistent with ZF
  • These independence results show that certain mathematical questions cannot be resolved using the axioms of ZF alone