A number is an idea, nothing more than the result of axiomatic systems such as ZFC and Peano's Axioms

Philosophy students like to say things like "Would two mean anything if there weren't two objects for two to refer to." This is foolishness... Numbers need have no basis in the physical realm, they are ideas.

To state it simply, math is truth, truth that transcends what is observable. Math when done in an axiomatic sense has no "belief system." It is true.

A set theoretic derivation of the numbers...

let 0=null

Now, define the successor of a number as the successor ordinal of that number... that is, let x+1=p(x)=x U {x}

So, we have...


Now that we have the natural numbers, we have define that rationals...

for example, we can define 1/2 as the set of all ordered pairs {(1,2),(2,4),(3,6),(4,8),...}

Then, a given real number can be defined as the set of all Cauchy sequences convergent to that number.

Further, imaginary numbers can be defined as ordered pairs... for x=a+ib, let x=(a,b) where a and b are sets of Cauchy sequences convergent to a and b.<./p>

Further Notes

One good treatment of this construction is Enderton's Elements of Set Theory, published in 1977.

Instead of constructing the reals using Cauchy Sequences, it is also possible to use Dedekind Cuts, but I have always found that derivation to be more painful.

In 1908, Zermelo proposed to define the integers as:
von Neumann proposed the definition given above which has become standard because of the property that for all y<x,yεx. This leads to some nice things when doing arithmatic.