Aleph-zero is a mathematical term that refers to the cardinality of the set of natural numbers. This number is also known as the countable infinity, as the set of all natural numbers can be listed and counted. There are several synonyms for aleph-zero, including the omega number, the first infinite cardinal, and the smallest transfinite cardinal. Other related concepts include the continuum hypothesis and the cardinality of larger sets of numbers, such as the real numbers. Understanding the concept of aleph-zero is important in various fields of mathematics, including set theory, topology, and analysis.