Real Numbers and the Dedekind Cut
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The ordered field axioms
Equations are omitted for technical reasons - download the original pdf
If in addition to the above A1 – A11, F satisfies the following, then F is an ordered field. O1 For every [Equation] one, and only one, of the following holds [Equation] O2 If [Equation] and [Equation], then [Equation] O3 If [Equation] then [Equation] O4 If [Equation] and [Equation], then [Equation]
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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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