∞Intro to the Theory of Sets Unit 5 – Axioms of Set Theory: ZFC System

Key Concepts and Definitions

• Set theory provides a rigorous foundation for mathematics by defining basic concepts and operations on sets
• A set is a collection of distinct objects, called elements or members, without regard to order
• The empty set, denoted by $\emptyset$ or {}, contains no elements
• Set-builder notation, {x | P(x)}, defines a set by a property P(x) that its elements x must satisfy
• Membership, denoted by $\in$, determines whether an object belongs to a set
  • Example: If A = {1, 2, 3}, then 2 $\in$ A, but 4 $\notin$ A
• Subset, denoted by $\subseteq$, indicates that every element of one set is also an element of another set
  • Example: If A = {1, 2} and B = {1, 2, 3}, then A $\subseteq$ B
• Power set, denoted by $\mathcal{P}(A)$, is the set of all subsets of A
• Cardinality, denoted by |A|, is the number of elements in a set A

Historical Context and Development

• Set theory emerged in the late 19th century to address foundational issues in mathematics
• Georg Cantor introduced the concept of sets and developed the theory of transfinite numbers
  • Cantor's work led to the discovery of different sizes of infinity and the continuum hypothesis
• Bertrand Russell discovered Russell's paradox, which exposed inconsistencies in naive set theory
  • Russell's paradox considers the set of all sets that do not contain themselves, leading to a contradiction
• Ernst Zermelo proposed the first axiomatization of set theory in 1908 to avoid paradoxes and inconsistencies
• Abraham Fraenkel and Thoralf Skolem independently improved Zermelo's axioms, resulting in the Zermelo-Fraenkel (ZF) axioms
• The axiom of choice (AC) was added to ZF, forming the widely-accepted ZFC axiom system

Axioms of ZFC: Overview

• ZFC consists of nine axioms that define the properties and behavior of sets
• The axioms are designed to avoid paradoxes and provide a consistent foundation for mathematics
• The axioms of ZFC are:
  1. Axiom of Extensionality
  2. Axiom of Empty Set
  3. Axiom of Pairing
  4. Axiom of Union
  5. Axiom of Power Set
  6. Axiom of Infinity
  7. Axiom Schema of Separation
  8. Axiom Schema of Replacement
  9. Axiom of Choice
• The axioms are independent, meaning no axiom can be derived from the others
• The consistency of ZFC is an open question, but it is widely believed to be consistent

Detailed Breakdown of ZFC Axioms

1. Axiom of Extensionality: Two sets are equal if and only if they have the same elements
   • Formally: $\forall A \forall B (\forall x (x \in A \leftrightarrow x \in B) \rightarrow A = B)$
2. Axiom of Empty Set: There exists a set with no elements, called the empty set
   • Formally: $\exists B \forall x (x \notin B)$
3. Axiom of Pairing: For any two sets A and B, there exists a set containing exactly A and B
   • Formally: $\forall A \forall B \exists C \forall x (x \in C \leftrightarrow (x = A \lor x = B))$
4. Axiom of Union: For any set A, there exists a set containing all elements of elements of A
   • Formally: $\forall A \exists B \forall x (x \in B \leftrightarrow \exists C (C \in A \land x \in C))$
5. Axiom of Power Set: For any set A, there exists a set containing all subsets of A
   • Formally: $\forall A \exists B \forall x (x \in B \leftrightarrow x \subseteq A)$
6. Axiom of Infinity: There exists a set containing the empty set and closed under the successor operation
   • Formally: $\exists A (\emptyset \in A \land \forall x (x \in A \rightarrow x \cup {x} \in A))$
7. Axiom Schema of Separation: For any set A and any property P(x), there exists a set containing all elements of A that satisfy P(x)
   • Formally: $\forall A \exists B \forall x (x \in B \leftrightarrow (x \in A \land P(x)))$
8. Axiom Schema of Replacement: For any set A and any function f, there exists a set containing f(x) for all x in A
   • Formally: $\forall A \forall f \exists B \forall y (y \in B \leftrightarrow \exists x (x \in A \land y = f(x)))$
9. Axiom of Choice: For any set A of non-empty sets, there exists a function f that chooses an element from each set in A
   • Formally: $\forall A (\emptyset \notin A \rightarrow \exists f: A \rightarrow \bigcup A \forall B (B \in A \rightarrow f(B) \in B))$

Applications and Examples

• Set theory serves as a foundation for various branches of mathematics, including analysis, topology, and algebra
• The natural numbers can be constructed using sets, starting with the empty set and applying the successor operation
  • Example: 0 = $\emptyset$, 1 = {$\emptyset$}, 2 = {$\emptyset$, {$\emptyset$}}, etc.
• Functions can be defined as sets of ordered pairs, where each element of the domain is paired with a unique element of the codomain
• The Cartesian product of two sets A and B, denoted by A × B, is the set of all ordered pairs (a, b) with a $\in$ A and b $\in$ B
  • Example: If A = {1, 2} and B = {x, y}, then A × B = {(1, x), (1, y), (2, x), (2, y)}
• Set operations, such as union ($\cup$), intersection ($\cap$), and difference (), can be used to manipulate and combine sets
• The axiom of choice is equivalent to several important mathematical statements, such as the well-ordering theorem and Zorn's lemma

Common Misconceptions and Pitfalls

• Confusing the empty set ($\emptyset$) with the set containing the empty set ({$\emptyset$})
• Misunderstanding the difference between a set and a class, which is a collection too large to be a set
  • Example: The class of all sets is not a set itself, as it would lead to Russell's paradox
• Incorrectly applying the axiom schema of separation or replacement, leading to inconsistencies or paradoxes
• Assuming that the axiom of choice is always necessary or that it has no consequences
  • The axiom of choice is independent of the other ZFC axioms and can lead to counterintuitive results, such as the Banach-Tarski paradox
• Neglecting the importance of proper set-theoretic notation and formalism in proofs and definitions

Related Theories and Extensions

• Von Neumann-Bernays-Gödel (NBG) set theory is an extension of ZFC that introduces classes to avoid paradoxes
• Morse-Kelley (MK) set theory is another extension that allows classes to be members of other classes
• Constructive set theories, such as Intuitionistic Zermelo-Fraenkel (IZF), reject the law of excluded middle and the axiom of choice
• Topos theory is a generalization of set theory that uses category theory to study mathematical structures
• Large cardinal axioms extend ZFC by postulating the existence of sets with certain properties, such as inaccessible, measurable, or supercompact cardinals
  • Large cardinals are used to study the consistency and independence of mathematical statements

Practice Problems and Exercises

1. Prove that the empty set is a subset of every set.
2. Show that the power set of the empty set is not empty.
3. Prove that the union of a set A with the empty set is equal to A.
4. Demonstrate that the intersection of a set A with the empty set is always empty.
5. Prove that the power set of a set A is always larger than A, except when A is empty.
6. Show that the axiom of pairing can be derived from the axioms of empty set, power set, and separation.
7. Prove that the Cartesian product of a set A with the empty set is always empty.
8. Demonstrate that the axiom of choice implies the existence of a choice function for any family of non-empty sets.
9. Show that the well-ordering theorem (every set can be well-ordered) is equivalent to the axiom of choice.
10. Prove that the union of a countable collection of countable sets is countable, using the axiom of choice.