Final answer:
The Principle of Finite Descent is a mathematical method used especially in number theory to show that no solution to a problem exists by proving that an infinite descent in natural numbers is impossible.
Step-by-step explanation:
The Principle of Finite Descent is a method used in mathematics that is particularly useful in number theory. This principle states that if there is a non-empty set of natural numbers that satisfies certain conditions, then there must exist a smallest natural number within that set. The method involves assuming that there is a solution, and then showing that there must be a smaller solution, which leads to a contradiction because this descent cannot go on indefinitely in the set of natural numbers. By demonstrating that an infinite descent is not possible, one proves that no such solution exists.
An example of the application of the Principle of Finite Descent is in the proof of the impossibility of integer solutions for certain Diophantine equations, such as Fermat's Last Theorem for certain cases.