Hi, infinite-set reader. Are you ready to Set Theory?
This time we will discuss about immediate successor and immediate predecessor and its properties. Keep in mind that is . Happy reading.
If the immediate successor of two natural numbers is equal then the two natural numbers are the same.
Take such that . Will be proven that . Suppose . Because of then or . On the other hand, because of then or . According to assumptions then it is only occur and . According to the Theorem (x) in this link, we get and equivalent to . There was a contradiction then it should be .
In the previous discussion has introduced the notion of immediate successor. Using the same sense, can be recovered an element if it has been known to the immediate successor of . Furthermore is called the immediate predecessor of the immediate successor of . In other words, is called immediate predecessor of natural numbers if .