Számosság

Számosság is a Hungarian term often translated as "cardinality" in English mathematics. It refers to the number of elements in a set. For finite sets, the számosság is simply the count of its members. For example, the set {apple, banana, cherry} has a számosság of 3.

The concept of számosság becomes more nuanced when dealing with infinite sets. Georg Cantor introduced the idea that some infinite sets can have a "larger" számosság than others. He demonstrated that the set of natural numbers (1, 2, 3, ...) has the same számosság as the set of even numbers (2, 4, 6, ...), despite the latter appearing to be only half the size. This is because a one-to-one correspondence can be established between the elements of both sets. This is denoted by the Hebrew letter aleph, with the számosság of the natural numbers being denoted as $\aleph_0$ (aleph-null).

Cantor also proved that the set of real numbers has a strictly larger számosság than the set of natural numbers. This larger infinity is often denoted by c (for continuum) or $\aleph_1$ under the assumption of the continuum hypothesis. The study of different sizes of infinity is a fundamental part of set theory. The comparison of cardinalities is done through the concept of bijections. If a bijection exists between two sets, they have the same számosság. If a set has a subset with a larger számosság than itself, then the set must be infinite.