Having done a lot of hard work putting the natural numbers on a sound footing the next step, to build , is rather more straightforward. The most direct method is perhaps to define as a disjoint union of two copies of , one copy being and the other copy being . A slightly more sophisticated method is to use equivalence relations.
Let be the following relation defined on : we define to mean .
in an equivalence relation on .
We write for the equivalence class of . The set is the set of equivalence classes.
The function defined by is a one-to-one function mapping into .
Let with . Then hence hence hence .
We identify each with its image under the map . In particular, is the element , and is .
We define addition, multiplication and order relations on by
The operations and relation on are well-defined, i.e., the definitions above do not depend on the particular choice of representatives .
The embedding is a homomorphism respecting and :
for all .
Having constructed the set of integers and its main operations, we need to prove that it satisfies the key axioms we expect. We list here the key axioms and in a few cases sketch the proof that the set satisfies the axioms.
Firstly, , with the addition and multiplication operations forms a nice algebraic structure. Division is not available but almost everything else you'd hope works doe so. In particular the integers forms a commutative ring with .
The set , with the addition and multiplication operations, is a commutative ring with . I.e.,
The ring of integers has a further nice property concerning multiplication.
The ring of integers is an integral domain. That is, it has no zero divisors, i.e, satisfies the additional axiom
So far we haven't yet mentioned the order structure on the integers. This is easily remedied, and the axioms look very similar to those for Archimedean ordered fields.
As for , etc., we can work with or with the relation defined to mean . We shall work with here.
defined by is a linear order on , i.e., satisfies
We also need to add axioms stating the relationship between the order and the arithmetic operations.
The integers with forms an ordered ring, i.e., also satisfies the following axioms:
In addition, this order is discrete, that is, there are no elements between and :
We have been presenting all the properties as they most naturally arise. However, our axiomatisation is not optimal in the sense that some axioms could be omitted since they follow from others. A case in point is where the new axioms for the order automatically implies the no-zero-divisor property.
Show that any ordered ring is an integral domain.
The se of integers forms an Archimedean structure in the same sense as in Archimedean ordered fields. The direct translation of this axiom from the earlier web page is perfectly correct, though looks somewhat silly when one recalls the correct picture. It says that no single integer is greater than all natural numbers. Phrased in a slightly more meaningful form for the integers, this becomes
No integer is greater than all natural numbers. In other words, the set of nonnegative integers is exactly .
By the least number principle in , this can be rephrased in an equivalent but rather more powerful form.
Let hold for some integer and suppose that there is such that . Then there is a least in such that .
Use the least number principle on the property of natural numbers where says that holds.
For an example where the conditions of the last result are not met and where the least number principle fails, consider the property that says that .
We now have all the information we need about the integers. The next task is to combine these to make an Archimedean field, the field of rationals, containing the integers.
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