**Peano Postulates** 29.1:139 (refers to Dodge, section 29.1, page 139)

[1] 1 **N**

[2] Each x **N** has a unique sucessor x'**N**

[3] If x, y **N** and x' = y' then x = y

[4] 1 is not the successor of any element of **N**

[5] The only natural numbers are those given by [1] and [2]

[5] is due to S. C. Kleene and is equivalent
to Dodge's [6].

[6] If S **N** such that 1 S, and
k S (k+1) S,
then S = **N** N8:110

[7] Definition of Addition: Let a + 1 = a' and a + b' = (a + b)'

[8] Definition of Multiplication: Let a1 = a and ab' = ab + a
29.3:140

[9] CLOSURE + a, b **N**,
a + b is defined 29.4:140

[10] CLOSURE a, b **N**,
ab is defined 29.5:140

[11] INDEPENDENCE The Peano postulate set is independent
29.6:140

[12] TRICHOTOMY For given a, b **N**,
exactly one of the following three statements is true:

a + x = b
a = b
a = b + y
for some x **N** or
some y **N**
N7:110

The following properties of Natural Numbers can also be proved:

[13] COMMUTATIVE + If a, b **N**,
then a + b = b + a

[14] COMMUTATIVE If a, b **N**,
then ab = ba

[15] ASSOCIATIVE + If a, b, c **N**,
then (a + b) + c = a + (b + c)

[16] ASSOCIATIVE If a, b, c **N**,
then (ab)c = a(bc)

[17] DISTRIBUTIVE if a, b, c **N**, then
a(b + c) = ab + ac

[18] IDENTITY
1 **N** such that n **N**,
n1 = 1n = n

Extension to integers (**Z**):

[19] Definition: Let **Z'** denote the set of all ordered pairs (a,b) of natural
numbers a and b 32.1:154

[20] Definition: We define two elements (a,b) and (c,d) of **Z'** to be
__equivalent__, and we write (a,b) (c,d),

iff a + d = b + c 32.2:154

Dodge proves the preceding definition satisfies reflexive, symmetric,
and transitive properties and is therefore an equivalence relation. 32.3:154

[21] Definition: The __sum__ of members (a,b) and (c,d) of **Z'** is
defined to be the ordered pair (a + c, b + d) 32.4:155

[22] Definition: The __product__ of members (a,b) and (c,d) of **Z'**
is defined to be the ordered pair (ac + bd, ad + bc) 32.5:155

[23] Definition: Let the set of all equivalence classes of **Z'** be called integers
and be denoted by **Z** 32.10:156

Dodge proves the following theorems:

[24] CLOSURE, COMMUTATIVE, ASSOCIATIVE **+**

**Z** is closed under addition, which is commutative and associative.
33.1:158

[25] CLOSURE, COMMUTATIVE, ASSOCIATIVE

**Z** is closed under multiplication, which is commutative and associative.
33.2:158

[26] IDENTITY **+** The equivalence class of (m,m) is the additive
identity and is unique.

Denote this equivalence class by "0".
33.3-5:159

[27] INVERSE **+** Each z **Z** has a unique additive inverse,
-z, such that z + (-z) = 0 33.12-6:160

Extension to Rational Numbers (**Q**)

[28] Definition: Denote by **Q'** the set of all ordered pairs [a,b] of
integers a and b with b 0 38.1:179

[29] Definition: We define two elements [a,b] and [c,d] of **Q'** to be
equivalent, and write:

[a,b] [c,d] iff ad = bc 38.2:179

Dodge shows this is an equivalence relation. 38.3:179

[30] Definition: The __sum__ of members [a,b] and [c,d] of **Q'** is
defined to be the ordered pair [ad + bc, bd]. 38.4:180

[31] Definition: The __product__ of the members [a,b] and [c,d] of **Q'**
is defined to be the ordered pair [ac,bd] 38.5:180

[32] Definition: Let the set of all equivalence classes of **Q'** be called
rational numbers and be denoted by **Q** 38.9:180

Dodge shows that **Q** satisfies all properties of a field:

[33] CLOSURE **+**

[34] CLOSURE

[35] COMMUTATIVE **+**

[36] COMMUTATIVE

[37] ASSOCIATIVE **+**

[38] ASSOCIATIVE

[39] IDENTITY **+**

[40] IDENTITY

[41] INVERSES **+**

[42] INVERSES

[43] DISTRIBUTIVE PROPERTY

[44] Definition: For given x, y **Q**, if there is a unique d
**Q**, such that x = y + d, then we define d = x - y to be the __difference__
of x and y. The operation is called __subtraction__. 40.2:184

Dodge shows this is equivalent to the definition:

[45] x - y = x + (-y)

[46] Definition: For given x, y **Q**, if there is a unique q
**Q**, such that x =yq, then we define q = x y to be the
__quotient__ of x and y. We also write q = x/y. The operation is called __division__.

Dodge shows this is equivalent to the definition:

[47] If y 0, then x y = xy*, where y* is
the multiplicative inverse of y.

Extension to real numbers ()

**Q** can be extended to the field of real numbers, , in many ways.
The exact method, whether by Cauchy sequences, Dedekind Cuts,
or other method, is not of importance here. The important thing
to note is that satisfies all field properties [33] - [43]
as did **Q**. The definitions of subtraction [45] and division
[47] are also relevant.