Cardinal Number
Overview
Cardinal number (기수; 基數) is a term referring to two different mathematical concepts depending on context. First, it is a number representing the number of elements in a set, that is, the size of a set, and is a core concept in set theory that deals not only with finite sets but also with the sizes of infinite sets. Second, in positional numeral systems, it means the base (radix) that determines the weight of each place, and defines the system itself for representing numbers, such as 10 in decimal or 2 in binary. In everyday language, it is also used as a homonym referring to a horse-racing jockey (騎手) or a go/janggi player (棋手).
Main Content
Cardinal Numbers in Set Theory
The cardinality of a set is introduced by defining two sets to have the same size when a one-to-one correspondence exists between them. The cardinality of a finite set is expressed by a natural number, but infinite sets produce results different from intuition. The set of natural numbers, the set of integers, and the set of rational numbers all have the same cardinality (aleph-null, ℵ₀), which is called countably infinite. On the other hand, the set of real numbers is proved by Cantor's diagonal argument to be impossible to put into one-to-one correspondence with the natural numbers, so it has a larger cardinality, and its size is called the cardinality of the continuum 𝔠.
According to Cantor's theorem, for any set, its power set always has a larger cardinality. Therefore, cardinal numbers have no upper bound, and there exists a hierarchy of infinitely increasing cardinal numbers. Comparisons of size and operations (addition, multiplication, exponentiation) on cardinal numbers are elaborated together with the axiom of choice, and the continuum hypothesis (whether another cardinal number exists between ℵ₀ and 𝔠) was proved by Cohen in 1963 to be independent of the Zermelo–Fraenkel axiom system. Since then, combinatorics of infinite cardinals, large cardinal theory, and forcing have become major research areas of modern set theory.
Base (Radix) in Numeral Systems
In positional numeral systems, the base b refers to a system that represents numbers using b digits from 0 to b-1. For example, decimal uses base 10, binary uses base 2, octal uses 8, and hexadecimal uses 16. When an arbitrary number is represented in base b, each place value is a power of b, and this property allows arithmetic operations and base conversions to be carried out systematically.
In computer science, binary is the basic representation, and hexadecimal, which groups 4 bits for human readability, is widely used alongside it. Color codes, memory addresses, character encoding (UTF-8), and cryptographic key representation are representative examples. In addition, base 60 is used for time and angle units, base 12 for traditional counting, and base 64 (Base64) for text encoding of binary data. Variants such as non-integer bases (e.g., golden ratio base) and negative bases have also been studied.
Other Homonyms
A horse-racing jockey (騎手) is an athlete who rides a horse in a race, and weight and riding skill greatly affect performance. A gi-sa (棋手) refers to a professional player of board games such as go, janggi, and chess, and in Korean it is often written as '기사' to avoid confusion.
Recent Trends
As of 2024-2025, the concept of cardinal number/base is attracting attention along two axes. First, in software and artificial intelligence education, understanding binary and base conversion and bit/byte units is emphasized as essential literacy, and positional numeral systems are treated as a core concept in the informatics subject of the 2022 revised curriculum. Qubits in quantum computing utilize superpositions of 0 and 1, so they are explained as an extension of the base-2 system, and base-related thinking in quantum algorithms frequently appears in educational materials.
Second, as infinite cardinals and the continuum hypothesis are formalized in systems such as Lean/mathlib, a formal mathematical proof tool, the reproducibility and verifiability of set theory research have increased. In addition, analyses are increasing that when the tokenizer of a large language model numerically encodes vocabulary, base conversion and integer representation affect performance.
Related Topics
- [[기수법]]
- [[이진법]]
- [[십진법]]
- [[집합론]]
- [[칸토어]]
- [[연속체 가설]]
- [[알레프 수]]
- [[경마]]