Hyperfinites

Hyperfinites are a conceptual mathematical object that represents a quantity "infinitely larger" than any finite number, yet "infinitely smaller" than the concept of infinity itself. They do not exist within the standard framework of real numbers or the extended real number line. The idea of hyperfinites arises in non-standard analysis, a branch of mathematics that rigorously deals with infinitesimals and infinite quantities. In non-standard analysis, there are numbers called hyperreal numbers, which include both standard real numbers and quantities that are infinitely small or infinitely large. Hyperfinites are a particular class of these infinitely large hyperreal numbers. They are not the same as the transfinite numbers used in set theory, such as aleph-null (ℵ₀) or aleph-one (ℵ₁), which represent sizes of infinite sets. Instead, hyperfinites represent a different kind of infinity, one that can be manipulated in some ways similar to finite numbers. For instance, a hyperfinite number plus one is still a hyperfinite number, just as a finite number plus one is a larger finite number. Their existence is established through model theory, specifically using the transfer principle, which allows properties of the standard real numbers to be extended to the hyperreal numbers. The concept of hyperfinites allows for a rigorous development of calculus and other areas of mathematics using intuitive infinitesimals and infinitely large quantities.