Peano's Axioms.

Created 980625. Last change 980728.

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Peano's Axioms.

The first part of these pages about numbers will use the Peano's axiom system as a foundation. The arithmetic created
using this is called Peano Arithmetic, PA. Further down will you find a look at the numbers defined using the
The Zermelo-Fraenkel's system, ZF.

When Peano created his axioms he wanted to catch the spirit of the natural numbers is a small set of rules. He created a
starting point by axiom number 1, a way to get more naturals by axiom 2, a way to ensure that 0 really is the starting point
by axiom 5, and so on. A informal way to write these axioms could be :

1 : 0 is a natural number.

2: If a is a natural number then so is a+1.

3: If you can prove something about a and that implies that you can prove it for a+1, and if you can
prove the very same thing for 0 , then will this hold for all natural numbers.

4: If a+1=b+1 then a=b.

5: You can not add 1 to a natural number to get 0.

In the following pages will we look at the usage of this system.

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