The
integers are the ring of non-fractional numbers:
... -3, -2, -1, 0, 1, 2, 3, ...
These are the
natural numbers,
with the addition of their additive inverses. In other words, the
integers extend the natural numbers into an abelian group under
addition. We will start by stating axioms which describe the
integers, and then we'll construct a model in which we can show that
the axioms are true, which in turn provides a limited proof of the
consistency of the axioms.
Axioms for the Integers
(
A.1) The integers form an Abelian group under the operation of addition, written as '+'. Hence they obey the
Abelian group axioms:
(
G.1)
[Closure under addition]
(G.2) [Associativity]
(G.3) [Commutativity]
(G.4) There is an identity element, called
"0", which has the
property:
(G.5) [Existence of inverses]
For convenience, we will define subtraction as addition of the additive inverse:
We now discuss the axioms which distinguish the integers from other Abelian groups:
(
A.2)
is
totally ordered by the less-than relation, '
<'.
The relation has the properties:
(R.1) [total order] Any pair of elements of may be compared
(R.2) [Transitivity]
(R.3) [Anti-symmetry]
For convenience, we will use "a>b" to mean "b<a".
(A.3) "<" is compatible with addition:
Axiom (A.3), along with the total ordering, is needed to distinguish the integers from a finite group. A set of the form satisfies the group axioms and can be ordered, but the order won't be compatible with addition.
(A.4) There is a distinguished element "1" in , and 0 < 1.
(A.5) [Induction axiom] If a subset is nonempty, and if it is closed under the operation of adding or subtracting 1,
then .
Axiom (A.5) says you can get from 0 to any element in by adding or subtracting 1
a finite number of times. Without this axiom, a set containing
multiple copies of the integers laid end to end would satisfy the
axioms. This axiom is also needed to assure that every subset of
the integers which is bounded below is well ordered.
We have, as yet, said nothing about multiplication. That can be
defined
as repeated addition, however, and its behavior derived from axioms
(A.1) through (A.5); no additional axioms are needed for it. We
will do that later on this page, after we present a model for the
axioms given here.
A Model of the Integers
How do we know the integers exist -- how can we be sure
any set will satisfy the axioms? We can construct a model for them, and then prove the axioms within the model.
Start with the model of the
natural numbers. Then construct two "flag objects", using pieces of that model:
Our model of the natural numbers, as laid out on the
natural numbers page, consisted of the sets
We can see by inspection that
F_{1} and
F_{2} are not equal to any
of those. We can, therefore, use the flags to ''tag'' two copies of the
natural numbers. We now define the objects used in the model of the
integers, shown here with primes attached, to distinguish them from
the similar objects used in modeling the natural numbers:
We also define the successor function,
succ', with a prime attached
to distinguish it from the (unprimed)
succ function which operates on the
natural numbers:
In the remainder of this page, we will use
to refer to this set:
We will refer to members of this set as "
nonnegative integers".
We can now define the
negative integers as:
We also define the predecessor function on the negative integers,
pred', with a prime attached to distinguish it from the (unprimed)
pred function on the natural numbers:
The entire set of integers, positive, negative, and zero, will be referred to as
.
We'll need negation later so we define it now:
and we define > and <, with respect to zero
only, as:
For
negative integers -- elements of
-- we
define the
succ' function implicitly, by the relation:
along with the special case
and finally we define
pred' for values in
:
For
, we can now define the
n^{th} successor to a number
inductively, as
With these definitions in hand, we can define a general "≤" as
As with the
natural numbers we again can define addition and multiplication
in the obvious way.
This completes the basic model: We have comparisons, addition, and
multiplication. The natural numbers,
, are contained within our new
model, and comparison in the integers is clearly an extension of comparison in
the natural numbers. It's also not hard to show that addition and
multiplication are also extensions of the same operations on the natural
numbers.
We could now go on to
prove the axioms which define the integers as
theorems within our model, including particularly the fact that each integer
has a unique additive inverse, thus showing that
really is a model of the
integers.
Page not complete yet. We
plan to add the proof that the model satisfies the axioms, the
definition of multiplication in terms of the axioms alone, and a
handful of theorems to show that the axioms really do lead to what we
think of as the integers, but as of 11/2007 these have not yet been
written up.
This article was originally posted on Anarchopedia, in (even) less complete form, in 2004.
Page
created on 11/18/2007