∞Intro to the Theory of Sets Unit 8 – Cardinal Numbers and Arithmetic

What Are Cardinal Numbers?

• Cardinal numbers represent the size or cardinality of a set, indicating how many elements are in the set
• Finite cardinal numbers correspond to the natural numbers (0, 1, 2, 3, ...) and represent the number of elements in a finite set
• Infinite cardinal numbers, such as $\aleph_0$ (aleph-null), represent the cardinality of infinite sets (the set of natural numbers)
• Two sets have the same cardinality if there exists a bijection (one-to-one correspondence) between them
• The cardinality of a set A is denoted as $|A|$
• The empty set, denoted as $\emptyset$, has a cardinality of 0
• Cardinal numbers are used to compare the sizes of sets and perform arithmetic operations on them

Properties of Cardinal Numbers

• Cardinal numbers are well-ordered, meaning that for any two cardinal numbers, one is always less than or equal to the other
• The smallest infinite cardinal number is $\aleph_0$, which represents the cardinality of the set of natural numbers
• For any cardinal number $\kappa$, there exists a next larger cardinal number, denoted as $\kappa^+$
  • For example, the next larger cardinal after $\aleph_0$ is $\aleph_1$
• The continuum hypothesis states that there is no cardinal number between $\aleph_0$ and $2^{\aleph_0}$ (the cardinality of the real numbers)
  • The continuum hypothesis is independent of the standard axioms of set theory (ZFC)
• The generalized continuum hypothesis states that for any infinite cardinal $\kappa$, there is no cardinal between $\kappa$ and $2^\kappa$
• The axiom of choice implies that every set can be well-ordered, and thus, every set has a cardinal number

Basic Arithmetic with Cardinals

• Addition of cardinal numbers: For two sets A and B, $|A| + |B| = |A \cup B|$ if A and B are disjoint
  • If A and B are not disjoint, $|A| + |B| = |A \cup B| + |A \cap B|$
• Multiplication of cardinal numbers: For two sets A and B, $|A| \cdot |B| = |A \times B|$, where $A \times B$ is the Cartesian product of A and B
• Exponentiation of cardinal numbers: For two sets A and B, $|A|^{|B|} = |B^A|$, where $B^A$ is the set of all functions from A to B
• For infinite cardinal numbers, addition and multiplication are idempotent, meaning $\kappa + \kappa = \kappa$ and $\kappa \cdot \kappa = \kappa$ for any infinite cardinal $\kappa$
• The distributive law holds for cardinal arithmetic: $\kappa \cdot (\lambda + \mu) = \kappa \cdot \lambda + \kappa \cdot \mu$

Comparing Cardinal Numbers

• Two sets A and B have the same cardinality ($|A| = |B|$) if there exists a bijection between them
• A set A has a smaller cardinality than a set B ($|A| < |B|$) if there exists an injection from A to B, but no bijection between them
• Cantor's theorem states that for any set A, the power set of A ($\mathcal{P}(A)$) has a strictly larger cardinality than A
  • This implies that there is no "largest" cardinal number
• The Cantor-Bernstein-Schroeder theorem states that if there exist injections from A to B and from B to A, then $|A| = |B|$
• The cardinality of the natural numbers ($\aleph_0$) is less than the cardinality of the real numbers ($2^{\aleph_0}$)
  • This is demonstrated by Cantor's diagonalization argument

Infinite Cardinals and Aleph Numbers

• Aleph numbers ($\aleph_0, \aleph_1, \aleph_2, ...$ are used to represent the cardinalities of infinite sets
• $\aleph_0$ represents the cardinality of the set of natural numbers, which is the smallest infinite cardinal
• The next larger cardinal after $\aleph_0$ is $\aleph_1$, followed by $\aleph_2$, and so on
• The continuum hypothesis states that $2^{\aleph_0} = \aleph_1$, but this is independent of the standard axioms of set theory (ZFC)
• The generalized continuum hypothesis states that for any infinite cardinal $\kappa$, $2^\kappa = \kappa^+$, where $\kappa^+$ is the next larger cardinal after $\kappa$
• The cofinality of an infinite cardinal $\kappa$ is the smallest cardinal $\lambda$ such that $\kappa$ can be expressed as the union of $\lambda$ smaller sets
  • Regular cardinals are those whose cofinality is equal to themselves, while singular cardinals have a smaller cofinality

Cardinal Arithmetic in Set Theory

• Cardinal arithmetic is an essential part of set theory, as it allows for the comparison and manipulation of the sizes of sets
• The axiom of choice is often used in proofs involving cardinal arithmetic, as it ensures that every set can be well-ordered
• The continuum hypothesis and its generalization are important topics in cardinal arithmetic, although they are independent of ZFC
• König's theorem states that if $\kappa$ is an infinite cardinal and $\lambda$ is a cardinal with $\lambda < cf(\kappa)$, then $\lambda^{<\kappa} < \kappa$
  • Here, $\lambda^{<\kappa}$ represents the cardinality of the set of all functions from a set of cardinality $<\kappa$ to a set of cardinality $\lambda$
• The singular cardinal hypothesis (SCH) is a generalization of König's theorem, stating that for any singular cardinal $\kappa$, $\kappa^{cf(\kappa)} = \kappa^+$
• Large cardinal axioms, such as the existence of inaccessible, measurable, or supercompact cardinals, have significant implications for cardinal arithmetic and the structure of the universe of sets

Applications and Examples

• Cardinal numbers are used in various branches of mathematics, including topology, algebra, and analysis
• In topology, the cardinality of a topological space can be used to classify spaces and study their properties
  • For example, separable spaces are those with a countable dense subset (cardinality $\leq \aleph_0$)
• In algebra, cardinal numbers are used to study the sizes of algebraic structures, such as groups, rings, and fields
  • The cardinality of a group can determine its properties and behavior
• In analysis, cardinal numbers are used to study the sizes of function spaces and the properties of real and complex numbers
  • The cardinality of the continuum ($2^{\aleph_0}$) is a fundamental concept in real analysis
• In computer science, cardinal numbers are used to analyze the complexity of algorithms and data structures
  • The cardinality of input sets can affect the running time and space requirements of algorithms

Common Pitfalls and Misconceptions

• It is important to distinguish between cardinal numbers and ordinal numbers, which represent the order type of well-ordered sets
• The arithmetic of infinite cardinal numbers does not always behave like the arithmetic of finite numbers
  • For example, $\aleph_0 + 1 = \aleph_0$ and $\aleph_0 \cdot 2 = \aleph_0$
• The continuum hypothesis is a statement about the possible cardinalities between $\aleph_0$ and $2^{\aleph_0}$, not a proven fact
• Not all infinite sets have the same cardinality, as demonstrated by Cantor's theorem and the existence of uncountable sets (real numbers)
• The axiom of choice is required for certain proofs in cardinal arithmetic, but it is independent of the other axioms of ZFC
  • Some results, such as the well-ordering theorem, depend on the axiom of choice
• Cardinal exponentiation is not always commutative, especially for infinite cardinals
  • For example, $2^{\aleph_0} > \aleph_0$ but $\aleph_0^2 = \aleph_0$