Originally Posted by

**DrRocket**
You can in fact push the Peano axioms all the way to construct the natural numbers, integers, rational, reals and complex numbers with all of the usual algebraic operations.

Very well, let’s do it!

Define an equivalence relation ~ on

by

. Then ~ is an equivalence relation, and the integers are defined to be the set of all equivalence classes under ~:

, where I have written

to denote the equivalence class containing

. (Intuitively,

represents the “difference” between the natural numbers

*m* and

*n*.)

Now define addition

, multiplication

and order

on

as follows:

,

,

. Yes, the two binary operations and the order relation are all well defined, as you can check for yourself.

You can also check for yourself that

, with zero element

and unity

, satisfies all the axioms listed by

**Faldo_Elrith** in this post:

http://www.thescienceforum.com/viewtopic.php?t=8466. We have therefore constructed a well-ordered ring from the natural numbers.

Moreover, the mapping

,

is bijective and has the property that

,

,

,

. Hence the structure

is ismorphic to a substructure of

. Therefore we shall re-define our natural numbers as a subset of our integers, treating the integer

as the natural number

*n*. We will also rewrite

,

and

more familiarly as

,

and

(or even drop the multiplication symbol altogether). 8)