megszámlálhatóság

Megszámlálhatóság, a set theory concept, refers to the property of a set whose elements can be put into a one-to-one correspondence with the natural numbers. In simpler terms, a set is countable if its elements can be listed in an infinite sequence, even if the set itself is infinite. This means that we can assign a unique natural number (0, 1, 2, ...) to each element in the set, and every element will eventually be assigned a number.

The set of natural numbers itself is the archetypal example of a countably infinite set. Other examples include the set of integers (..., -2, -1, 0, 1, 2, ...) and the set of rational numbers (fractions). While it might seem counterintuitive, these infinite sets have the same "size" or cardinality as the natural numbers. This cardinality is denoted by the Hebrew letter aleph-null, $\aleph_0$.

A set that is not countable is called uncountable. The most famous example of an uncountable set is the set of real numbers. It is impossible to create a list that includes all real numbers, no matter how long the list is. This was famously proven by Georg Cantor using his diagonalization argument. The cardinality of the real numbers is denoted by $c$ or $\aleph_1$ (under the continuum hypothesis). Understanding the distinction between countable and uncountable sets is fundamental in various areas of mathematics, including set theory, topology, and computability theory.