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Published online by Cambridge University Press: 07 January 2010
Ordered fields
A system which satisfies the axioms of arithmetic is called a field. A system which satisfies both the axioms of arithmetic and the axioms of order is called an ordered field. The real number system ℝ is an ordered field which satisfies a further axiom called the continuum axiom. This is discussed in the next chapter. For the moment we only wish to comment on the fact that, while there are many distinct ordered fields, the system ℝ is unique (See §9.21.)
In this chapter our aim is to introduce the ordered field ℚ of all rational numbers (or fractions) and to discuss the reasons why this ordered field is not adequate for the purposes for which we require the real number system ℝ. As a preliminary to this objective, it is necessary to begin by providing a precise, formal definition of the system ℕ of natural numbers (or whole numbers) and to investigate the properties of this system. The main tool in this investigation is the exceedingly important principle of induction.
The sets ℕ and ℚ (and also the set ℤ of integers) will be defined as subsets of ℝ. Note, however, that, since no use at all will be made of the continuum axiom in this chapter, it follows that any ordered field contains subsets with the same structure as ℕ, ℚ and ℤ. In particular, any ordered field contains an ordered subfield with the same structure as ℚ. In some sense therefore, ℚ is the ‘simplest’ possible ordered field.
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