5.1 Zermelo-Fraenkel axioms

The Zermelo-Fraenkel axioms form the backbone of modern set theory. These rules define how sets can be created, combined, and manipulated, providing a solid foundation for mathematical reasoning and avoiding paradoxes that plagued earlier set theories.

From basic operations like pairing and union to more complex concepts like power sets and infinite sets, these axioms cover it all. They ensure that set theory remains consistent and powerful, allowing mathematicians to build complex structures on a firm logical base.

Axioms of Set Existence

Creating New Sets from Existing Sets

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• Axiom of Pairing states that for any two sets $A$ and $B$, there exists a set $C$ that contains exactly $A$ and $B$ as elements
  • Allows for the creation of unordered pairs and singletons (sets with only one element)
  • Example: If $A = \{1, 2\}$ and $B = \{3, 4\}$, then $C = \{\{1, 2\}, \{3, 4\}\}$
• Axiom of Union states that for any set $A$, there exists a set $B$ that contains all the elements of the elements of $A$
  • Allows for the creation of a set that combines all elements from a collection of sets
  • Example: If $A = \{\{1, 2\}, \{3, 4\}\}$, then $B = \{1, 2, 3, 4\}$
• Axiom of Power Set states that for any set $A$, there exists a set $B$ that contains all the subsets of $A$
  • The power set of $A$ is denoted as $\mathcal{P}(A)$ or $2^A$
  • Example: If $A = \{1, 2\}$, then $\mathcal{P}(A) = \{\emptyset, \{1\}, \{2\}, \{1, 2\}\}$

Existence of an Infinite Set

• Axiom of Infinity states that there exists a set $A$ such that $\emptyset \in A$ and for every $x \in A$, $x \cup \{x\} \in A$
  • Guarantees the existence of an infinite set, which is crucial for the construction of natural numbers and other infinite sets
  • The smallest set satisfying the Axiom of Infinity is the set of natural numbers $\mathbb{N} = \{0, 1, 2, 3, \ldots\}$

Axioms of Set Properties

Equality and Uniqueness of Sets

• Axiom of Extensionality states that two sets are equal if and only if they have the same elements
  • Establishes the criterion for set equality based on their contents, rather than their descriptions or representations
  • Example: If $A = \{1, 2, 3\}$ and $B = \{3, 2, 1\}$, then $A = B$ because they contain the same elements
• Axiom of Foundation (also known as the Axiom of Regularity) states that every non-empty set $A$ contains an element $B$ such that $A$ and $B$ are disjoint sets
  • Prevents the existence of sets that are elements of themselves or contain infinite descending chains of membership
  • Ensures that set membership is well-founded and avoids paradoxes like Russell's paradox

Axiom Schemas

Constructing Subsets and Replacing Elements

• Axiom Schema of Separation states that for any set $A$ and any predicate $P(x)$, there exists a set $B$ that contains all the elements $x$ of $A$ for which $P(x)$ is true
  • Allows for the construction of subsets of a given set based on a specific property or condition
  • Example: If $A = \{1, 2, 3, 4\}$ and $P(x)$ is "$x$ is even," then $B = \{2, 4\}$
• Axiom Schema of Replacement states that for any set $A$ and any function $f$, there exists a set $B$ that contains the elements $f(x)$ for each $x \in A$
  • Allows for the creation of a new set by replacing each element of a given set with its image under a function
  • Example: If $A = \{1, 2, 3\}$ and $f(x) = x^2$, then $B = \{1, 4, 9\}$

ZF Set Theory

Zermelo-Fraenkel Set Theory

• ZF Set Theory is an axiomatic system that forms the foundation of modern set theory
  • Consists of the axioms mentioned above (Pairing, Union, Power Set, Infinity, Extensionality, Foundation, Separation, and Replacement)
  • Provides a rigorous framework for the construction and manipulation of sets, avoiding paradoxes and inconsistencies found in naive set theory
• ZF Set Theory is widely accepted as a consistent and powerful system for studying sets and their properties
  • Serves as the basis for many branches of mathematics, including topology, analysis, and algebra
  • Can be extended with additional axioms, such as the Axiom of Choice (ZFC), to prove more advanced results and theorems