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.

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