**2.3 All the reals from the positive reals**

This is more like the passage from the natural numbers to the positive
rationals.

There we were closing up under the operation of division , here we

are closing up under the operation of subtraction. There we used a familiar

simplification to make sure that only one quotient was used to represent each

positive rational , here we use a simplification which is no more complicated

but will not be familiar. Each real number can be expressed in the form

r - s where r and s are positive reals in just one way - if we require that

min(r, s) = 1.

So we define a real number as an ordered pair (r, s) where r and s are

positive real numbers and min(r, s) = 1.

The positive real r will correspond to the general real (r + 1, 1), the

negative real -r will be represented as (1, r + 1), (1, 1) represents 0 (which

now appears for the first time as a first-class citizen of the world of numbers).

We define simp(a, b) as (a+1-min(a, b), b+1-min(a, b)): notice that these

subtractions can be carried out entirely in the positive reals, and that for any

a and b, simp(a, b) will be the real number (according to our definition) which

we expect to represent a - b.

We define (r, s) + (t, u) as simp(r + t, s + u) and (r,
s)(t, u) as simp (rt +

su, ru+st). We define (r, s) ≤ (t, u) as r+u ≤ s+t. Everything here makes

perfect sense if one interprets (r, s) as r - s (that's the whole point). We

leave to the reader the project of showing that all the axioms of sections 3

and 4 hold for the resulting system of real numbers.

**3 We make everything go away?**

We have explained the entire system of real numbers in terms of the natural

numbers, the notion of ordered pair, and the notion of set.

In this section, we show how to explain the pairs and the natural numbers

as sets, and we give the basic axioms of set theory which are required to

make everything work. In modern mathematics, everything is reduced to the

theory of sets!

We give a first axiom: sets with the same elements are equal.

You probably recall the notation {a, b} for the set whose only elements

are a and b, and {a} for the set whose only element is a.

We define the ordered pair (a, b) as {{a} , {a, b} }. There is a gap here:

one needs to prove that if (a, b) = (c, d), then a = c and b = d. This is a

little involved, but can be proved just from the definitions of the sets and

the axiom given above.

To make sure that we can always build ordered pairs, we introduce another

axiom: for any objects a and b, there is a set {a, b} which has a and

b as its only elements. If a = b, we get the set {a} (so we do not need a

separate axiom for this case).

We introduce another axiom: there is a set Ø, with no elements (it is

straightforward to prove that there can be only one such set).

We introduce a further axiom. For any sets A and B, there is a set A∪B

such that for all x, x ∈ A ∪ B x ∈ A
x ∈ B. This is the union of A and

B.

We define 0 as Ø, 1 as 0∪{0} = {0}, 2 as 1∪{1} = {0,1}, 3 as 2∪{2} =

{0,1,2}, and so forth.

The "and so forth" should make you suspicious (it is a verbal ellipsis!)

To explain it we introduce a definition and an axiom. We say that a set

I is inductive if it contains {Ø} as an element and contains x ∪ {x} as an

element for each x ∈ I. Notice that Ø is our 1 and for each of our natural

numbers x we have x+1 = x ∪ {x}. The axiom asserts that there is a set N

which has as its elements exactly those sets n which
belong to all inductive

sets: of course we call the elements of N natural numbers.

Notice that 1 belongs to any inductive set, and so do 2,3, and so forth.

Of course we could easily have included 0.

Here is a gap: it can be proved that the axioms N1-5 are true if we take 1

to be the singleton of the empty set, define n+1 as n∪{n}, and take the set

of natural numbers to be defined as our N. (The power set and separation

axioms below are actually needed for this development: defining addition

and multiplication is a bit tricky).

To construct our number systems we need two more axioms, both of which

have to do with subsets. Recall that for sets A and B, we say that A is a

subset of B (A B) if and only if all elements of A also belong to B.

Power set axiom: for any set A, there is a set P(A) whose elements are

exactly the subsets of A.

Separation axiom: for any set A and property P (x) (this is a sentence

about an object x) there is a set {x ∈ A l P(x)} whose elements are exactly

the elements x of A such that P(x) is true.

Theorem ( cartesian products ): for any sets A and B, the set A × B of

all ordered pairs (a, b) with a ∈ A and b ∈ B exists (hint: it is a subset

of P (P(A ∪ B)): to see this you have to look at the internal details of the

definition of the ordered pair - for the last time!)

Now we can see how the whole construction is done. N is the system of

natural numbers. The system of positive rationals is a subset of N × N which

can be picked out using the separation axiom. The system
of positive real

numbers is a subset of (it is a set of sets of positive rationals) which

can be picked out using separation. The system R of all reals is a subset

of which can be picked out using separation. The operations on

the various systems of numbers can also be constructed using the basic set

theory axioms.

The set of axioms I have given is not quite the usual one. The axiom

providing A ∪ B is enough for our purposes, but the usual union axiom

asserts the existence of ∪A, the set of all elements of elements of A, for any

set A: A∪B = ∪{A,B} is provided by this axiom. There are a couple more

axioms which usually appear but which are not needed for this construction.

But the exact system of axioms I have given is adequate for the construction

of almost all of mathematics, including all of mathematics which has

any significant application in the sciences. Isn't that interesting?

A final observation: what about the "geometric intuition" I keep men-

tioning? Notice that we can "construct" the Euclidean plane as R ×R, the

set of ordered pairs of real numbers, and similarly for Euclidean three-space

- and just as easily for higher dimensional spaces... What happened?