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Section 22.5 Constructing the real numbers
Objectives
Explain why explicit models are needed to realize the real-number axioms.
Construct
\(\bb{R}\) using equivalence classes of Cauchy sequences.
Compare alternative constructions such as Dedekind cuts and nested intervals.
Axiomatic construction is nice, but it would be even nicer if we could exhibit an object that exhibits all those axioms, what would be called a model of the real numbers.
Could construct it as the equivalence classes of Cauchy sequences, where two sequences are equivalent if and only if their difference tends to zero. This generalizes to the completion of any metric space. Operations are defined termwise. This seems to be the easiest, and infinite decimals work well here.
Could construct it using Dedekind cuts, which are partitions
\((A,B)\) of
\(\bb Q\) such that neither
\(A\) nor
\(B\) is empty,
\(a<b\) for all
\(a\in A\) and
\(b\in B\text{,}\) and
\(A\) has no greatest element. Operations are defined elementwise, though multiplication is difficult.
Could construct it using nested intervals.
