Real Numbers and the Dedekind Cut
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The sets of rational and real numbers are ordered fields.
Equations are omitted for technical reasons - download the original pdf
However, the above axioms do not guarantee the existence of real numbers, which fill in the gaps between the rational numbers. In order to ensure that every real number exists a further axiom must be added. This is the axiom of completeness, due to Dedekind.
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Contents of
Real Numbers and the Dedekind Cut
1 Numbers as solutions to equations
2 The irrationality of root 2
3 The set of all rational numbers is dense
4 The Dedekind cut
5 The set of real numbers as an ordered field
6 The ordered field axioms
7 The sets of rational and real numbers are ordered fields.
8 Axiom of completeness
9 Bounded sets of numbers
10 Maximum and minimum
11 Bounded set
12 Least upper bound
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