Here is my attempt to explain in plain(er) English. I'll leave out some of the definitional fluff at the beginning and cut straight to the axioms themselves
- We assume that there is at least one natural number: 1 (some people prefer to start with 0, I'll just use the same convention as the one in the previous quotation). The set of natural numbers is not empty.
- For each number that is a member of our set (i.e. for each natural number) there exists a successor to that number. In other words, every natural number has a "next" number. The successor of 1 is 2, the successor of 2 is 3 and so forth. We'll label the successor of an unknown natural number x as x'.
- There is no natural number whose successor is 1. In other words, x'≠ 1, no matter what x is. In still other words, 1 is the first natural number. (It also works if you let 0 be this "first" natural number).
- Every natural number has one successor or no successor, i.e. the successor of a natural number is unique (if it has one). We write this symbolically as: x'=y' means that x=y.
- Let's make a special set of natural number that obeys the following rules. A: 1 is a member of this set. B: if a certain natural number is a member of the set, then its successor must also be a member of the set. THEN all the natural numbers have to be members of that set. This is the principle of induction.
In the meantime, I hope that helps you to understand Peano's axioms a bit better.
Oh, and by the way:
The coolest thing about the natural numbers is this: any set that obeys those five axioms (considered as rules) is, for all extent and purposes, essentially identical to the set of natural numbers.
This is deep. Think about it.