Cardinal number
A number measuring the size of a set.
A cardinal number, or cardinal for short, is a kind of number that measures the cardinality of a set, i.e., how many elements there are in a set. Cardinality is defined in terms of bijective functions: two sets have the same cardinality if and only if there is a one-to-one correspondence between their elements. For finite sets, cardinal numbers correspond to natural numbers, but for infinite sets, cardinal numbers exhibit more complex behavior, as shown by Georg Cantor's work.
- field
- Mathematics
- known_for
- Measuring the size of sets, including infinite sets; introducing aleph numbers; Cantor's theorem on different sizes of infinity
Lore & Background
The cardinal number associated with a set is generally denoted by vertical bars (e.g., |A|), or by card(A) or #A. Cardinality is defined in terms of bijective functions: two sets have the same cardinality if and only if there is a bijection between them. For finite sets, cardinality can be found by counting elements; for example, the sets {1,2,3} and {4,5,6} both have cardinality 3.
Reader's Guide
The behavior of cardinalities of infinite sets is more complex than for finite sets. For instance, there exists a bijection between the set of natural numbers and the set of rational numbers, so they have the same cardinality even though the natural numbers are a proper subset of the rationals. This shows that a proper subset of an infinite set can have the same cardinality as the whole set, something impossible for finite sets. However, Cantor's theorem shows that two infinite sets can have different cardinalities; in particular, the cardinality of the real numbers is greater than that of the natural numbers. The cardinality of the natural numbers is denoted ℵ₀ (aleph-null), the smallest aleph number. The properties of other aleph numbers and infinite cardinal numbers depend on statements independent of Zermelo–Fraenkel set theory, such as the axiom of choice and the continuum hypothesis. For example, all infinite cardinal numbers are aleph numbers if and only if the axiom of choice is true. Cardinality is studied as part of set theory and is used in model theory, combinatorics, abstract algebra, and mathematical analysis. In category theory, cardinal numbers form a skeleton of the category of sets.
Did You Know?
- Two sets have the same cardinality if and only if there is a bijection between them.
- The cardinality of the natural numbers is denoted ℵ₀ (aleph-null).
- There exists a bijection between the natural numbers and the rational numbers, so they have the same cardinality.
- Cantor's diagonal argument shows that the power set operation always yields a strictly greater cardinality.
Frequently Asked Questions
What is a cardinal number in set theory?
A cardinal number is a mathematical object that quantifies how many elements a set contains. Two sets share the same cardinality exactly when their elements can be paired off one-to-one with no leftovers on either side.
What role do cardinal numbers play in the foundations of mathematics?
They serve as the fundamental measuring tool for comparing the sizes of sets, whether finite or infinite. For finite collections they reduce to the familiar natural numbers, but for infinite collections they reveal a rich hierarchy of distinct infinities.
How do cardinal numbers behave for infinite sets?
Georg Cantor demonstrated that not all infinite sets are the same size, introducing the aleph notation to label increasingly larger infinities. His diagonal argument, often called Cantor's theorem, proves that a set's power set always has a strictly larger cardinality than the set itself.
Why are cardinal numbers important to the study of set theory?
They provide the precise language for discussing the size of mathematical objects, which is essential for rigorous work in logic, analysis, and combinatorics. Without them, claims like 'there are more real numbers than integers' would lack formal grounding.
What distinguishes a cardinal number from an ordinal number?
A cardinal captures only the quantity of elements in a set, while an ordinal also encodes the specific ordering of those elements. Two sets can share the same cardinality yet carry different ordinal structures if their elements are arranged differently.
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