kardinaaliluku

Kardinaaliluku, known in English as a cardinal number, is a fundamental concept in set theory and mathematics. It is used to quantify the size or cardinality of a set. Essentially, a cardinal number answers the question "how many?" for a given collection of objects. For finite sets, the cardinal number is simply the number of elements in the set. For example, the set {apple, banana, cherry} has a cardinal number of 3.

The concept of cardinal numbers is extended to infinite sets as well. Georg Cantor, a pioneer in set theory, demonstrated that there are different sizes of infinity. The smallest infinity is the cardinality of the set of natural numbers, denoted by aleph-null (ℵ₀). This is the cardinality of countably infinite sets. Sets that have the same cardinality as the natural numbers are called countably infinite.

Cantor also proved that the set of real numbers has a larger cardinality than the set of natural numbers. This cardinality is denoted by the continuum (c) or sometimes as aleph-one (ℵ₁), assuming the continuum hypothesis. Sets with a cardinality larger than countably infinite are called uncountably infinite. The study of cardinal numbers allows mathematicians to compare the sizes of even the largest infinite sets. Cardinal arithmetic deals with addition, multiplication, and exponentiation of cardinal numbers.