Lesson 1

Positive integers and the natural order on them possess many remarkable properties, and we have used them repeatedly. For example, the method of mathematical induction works for them. And if some positive integer parameter decreases with each iteration of a loop (this parameter is then called a semi-invariant), then we can immediately conclude that the number of iterations of this loop will be finite. In this section, we will learn for which other ordered sets such properties remain valid and what other useful properties exist.