Return to Revisions

There are many uses of infinite sets and their properties. Let me just give you a specific one from computer science. One important task in computer science is proving or verifying that programs do what they are supposed to do. When such programs involve loops and recursive calls (self-reference), we need methods for showing that the loops and recursive calls terminate, i.e., that the program won't run forever. The usual induction principle for natural numbers suffices for showing that a single loop terminates, but we need double induction for double loops, triple induction for triple loops, etc. The whole business can get very complicated when the program is more than just a simple combination of loops. Set theory helps sort it all out with the principle of transfinite induction and the calculus of (infinite) ordinal numbers. Transfinite induction covers all possible ways in which one could show that a program terminates, while the ordinal numbers are used to express how complex the proof of termination is (the bigger the number, the more complicated it is to see that the program will actually terminate).