infinite descent

Infinite descent is a method of mathematical proof. It is used to show that a statement is true for all positive integers. The method works by assuming that the statement is false for some positive integer and then showing that this assumption leads to a contradiction. This contradiction arises because the assumption implies the existence of an infinite sequence of decreasing positive integers, which is impossible.

The principle of mathematical induction is closely related to infinite descent. Both are used to prove statements about integers. However, mathematical induction typically proceeds by showing that if a statement is true for an integer n, it is also true for n + 1. Infinite descent, on the other hand, shows that if a statement is false for some integer, it must also be false for a smaller integer, leading to an endless chain of smaller integers.

While infinitesimals were once used in calculus, they are not considered real numbers. However, mathematicians have developed ways to construct objects that behave like infinitesimals, using logic and nonstandard models of arithmetic.1