8.1 Definition and properties of cardinal numbers
3 min read•august 7, 2024
Cardinal numbers measure set size, from finite to infinite. They're crucial for comparing sets and understanding different types of infinity. Equipotent sets have the same size, even if their elements differ.
Transfinite cardinals extend this idea to infinite sets. Aleph numbers represent these infinite sizes, with aleph-null being the smallest. The continuum hypothesis deals with the gap between and infinities.
Cardinality and Cardinal Numbers
Defining Cardinality and Cardinal Numbers
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- represents the size or of a set
- Cardinality measures the number of elements in a set
- Two sets are equipotent (or have the same cardinality) if there exists a bijection between them
- Bijection is a between the elements of two sets
- Equipotent sets have the same cardinal number (e.g., the set of even numbers and the set of odd numbers are equipotent)
Transfinite Cardinals and Aleph Numbers
- Transfinite cardinals extend the concept of cardinality to infinite sets
- Aleph numbers represent the cardinality of infinite sets
- (aleph-null) represents the cardinality of countably infinite sets (e.g., )
- represents the smallest uncountable cardinal, which is the cardinality of
- Continuum hypothesis states that there is no set with a cardinality between and
Countability
Countable Sets
- A set is countable if it is either finite or has the same cardinality as the set of natural numbers
- Countably infinite sets can be put into a one-to-one correspondence with the natural numbers
- Examples of countable sets include the set of integers, rational numbers, and algebraic numbers
- Countable sets have a cardinality of
- The union of countably many countable sets is still countable
Uncountable Sets
- A set is uncountable if it has a cardinality greater than
- The set of real numbers is uncountable, as proven by Cantor's diagonal argument
- The power set of a countably infinite set is uncountable
- The power set of a set A, denoted as , is the set of all subsets of A
- The cardinality of the power set of a set A is always greater than the cardinality of A itself (e.g., )
Cantor's Theorem
Proving Cantor's Theorem
- states that the power set of any set A has a greater cardinality than A itself
- The theorem is proved by assuming the existence of a bijection between A and its power set, then deriving a contradiction
- The proof constructs a of A that is not in the assumed bijection, contradicting the assumption
- Cantor's theorem implies that there is no "largest" cardinal number, as the power set of any set will always have a greater cardinality
Implications of Cantor's Theorem
- Cantor's theorem establishes a hierarchy of infinite sets with increasing cardinalities
- The theorem proves that there are infinitely many different sizes of infinity
- Cantor's theorem has significant implications in set theory, mathematics, and logic
- The theorem is a fundamental result in the study of transfinite numbers and the foundations of mathematics
Key Terms to Review (20)
|a|: |a| represents the cardinality of the set 'a', which is a measure of the size of the set, indicating the number of elements it contains. This concept allows for comparisons between different sets and plays a crucial role in understanding both finite and infinite sets. The cardinality helps to classify sets as either countably infinite or uncountably infinite, connecting it to important mathematical ideas regarding set sizes and operations involving those sizes.
Addition of cardinal numbers: The addition of cardinal numbers refers to the process of combining two or more sets to determine the total number of distinct elements in the resulting set. This operation helps us understand the size of sets in terms of their cardinality, which is crucial when comparing different sets or establishing relationships between them. Cardinality focuses on how many elements are in a set, and addition serves as a foundational concept in set theory that showcases how cardinal numbers interact with one another.
Cantor's Theorem: Cantor's Theorem states that for any set, the power set of that set (the set of all its subsets) has a strictly greater cardinality than the set itself. This theorem highlights a fundamental aspect of the nature of infinity and implies that not all infinities are equal, leading to insights about the structure of different sizes of infinity.
Cardinal number: A cardinal number is a number that indicates quantity, representing the size of a set. They are used to compare the sizes of different sets, such as finite sets, infinite sets, and can even illustrate the concept of different 'sizes' of infinity. Understanding cardinal numbers is essential for grasping deeper concepts in set theory, such as infinite sets and their properties.
Cardinality: Cardinality refers to the measure of the 'number of elements' in a set, providing a way to compare the sizes of different sets. This concept allows us to classify sets as finite, countably infinite, or uncountably infinite, which is essential for understanding the structure of mathematical systems and their properties.
Cardinality of a Power Set: The cardinality of a power set refers to the number of elements in the power set of a given set. For any set with cardinality 'n', the cardinality of its power set is always $$2^n$$. This concept connects closely with how we understand sets, especially in terms of their structure and the relationships between different sets, like universal sets and their subsets.
Comparison of cardinalities: Comparison of cardinalities refers to the process of determining the size of two sets and understanding how they relate to each other in terms of their elements. It involves identifying whether two sets have the same number of elements, or if one set is larger or smaller than the other. This concept is crucial in understanding the properties of cardinal numbers, as it helps categorize infinite and finite sets based on their sizes.
Countable: A set is called countable if its elements can be put into a one-to-one correspondence with the natural numbers, meaning that there exists a way to list all the elements of the set without missing any. Countable sets can be either finite or infinite, and understanding this concept is essential for discussing the properties of cardinal numbers and the implications of various axioms in set theory, especially in terms of determining the size of different types of infinite sets.
Finite cardinal: A finite cardinal is a number that represents the size of a finite set, indicating how many elements are in that set. Finite cardinals are essential in understanding the concept of counting within set theory, as they help to establish a foundation for comparing sizes of sets and performing arithmetic operations on them.
Georg Cantor: Georg Cantor was a German mathematician known for founding set theory and introducing concepts such as different sizes of infinity and cardinality. His work laid the groundwork for much of modern mathematics, influencing theories about infinite sets, real numbers, and their properties.
Infinite cardinal: An infinite cardinal is a type of cardinal number that represents the size of an infinite set. Unlike finite cardinals, which count a specific number of elements, infinite cardinals measure the size of sets that do not have a last element. They play a crucial role in understanding different sizes of infinity and the properties that arise when performing arithmetic operations on these quantities.
Multiplication of cardinal numbers: Multiplication of cardinal numbers refers to the operation that combines two cardinal numbers to produce a new cardinal number, reflecting the total number of elements in the Cartesian product of two sets. This operation allows for the quantification of sizes when dealing with infinite sets, highlighting properties such as commutativity and associativity, similar to traditional multiplication.
One-to-one correspondence: One-to-one correspondence is a relationship between two sets where each element of the first set is paired with exactly one unique element of the second set, and vice versa. This concept is crucial for understanding functions, specifically injective and bijective functions, as it establishes that no two elements in one set can map to the same element in another. This relationship also underpins the comparison of cardinal numbers, allowing us to determine if two sets have the same size.
Richard Dedekind: Richard Dedekind was a German mathematician known for his contributions to abstract algebra, number theory, and the foundations of set theory. His most significant work involves the concept of infinite sets and Dedekind-infinite sets, which fundamentally shaped how mathematicians understand different sizes of infinity and the definition of cardinal numbers.
Subset: A subset is a set where every element of that set is also contained within another set. Understanding subsets is crucial because they form the basis for defining relationships between sets, including set membership, unions, intersections, and various operations performed on sets.
Superset: A superset is a set that contains all elements of another set, meaning if set A is a superset of set B, then every element of B is also an element of A. Understanding superset relationships helps clarify concepts like membership and subset relations, as well as the interactions between sets through operations like union and intersection. Recognizing supersets also plays a crucial role in visualizing relationships using Venn diagrams and grasping the nature of cardinal numbers when comparing sizes of sets.
The set of natural numbers: The set of natural numbers is a collection of positive integers starting from 1 and going upwards indefinitely, represented mathematically as {1, 2, 3, ...}. This set forms the foundation for counting and ordering in mathematics and is crucial when discussing cardinality, which measures the size of sets.
The set of real numbers: The set of real numbers is a collection that includes all the rational and irrational numbers, which can be represented on the number line. This set encompasses integers, fractions, and non-repeating decimals, providing a comprehensive framework for measuring continuous quantities and addressing many mathematical problems.
Uncountable: In set theory, a set is described as uncountable if it has more elements than the set of natural numbers, meaning it cannot be put into a one-to-one correspondence with the natural numbers. This concept highlights the existence of different sizes of infinity, showcasing that some infinite sets are larger than others, particularly in discussions involving cardinal numbers and foundational aspects of mathematics.
ℵ₀ (aleph-null): ℵ₀ (aleph-null) is the cardinality of the set of natural numbers, representing the smallest infinite cardinal number. It is a fundamental concept in set theory that establishes a distinction between finite and infinite sets. Understanding ℵ₀ helps in comparing the sizes of different infinite sets and is crucial for grasping the concept of different levels of infinity.