Real Numbers and the Dedekind Cut
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The set of real numbers as an ordered field
Equations are omitted for technical reasons - download the original pdf
The set of real numbers with the operations of addition [Equation] and multiplication [Equation] constitute an ordered field. That is, they satisfy the field axioms, and the axioms for an ordered number field. The field axioms. Let a, b, c, ... be elements of a set F. For F to be a field it must satisfy the following axioms. A1 If [Equation]; A2 [Equation]; A3 [Equation]; A4 There exists an element [Equation] such that for every [Equation], [Equation]; A5 For every [Equation], there exists an [Equation] such that [Equation]. This is denoted [Equation]; A6 If [Equation]; A7 [Equation]; A8 [Equation]; A9 There exists an element [Equation] such that for every [Equation], [Equation]; A10 For every [Equation], there is a [Equation] such that [Equation]. This is denoted [Equation]; A11 [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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