Supplementary Notes on Dedekind Cuts
نویسنده
چکیده
Motivation: Our textbook discusses and even proves many properties of R, the field of real numbers; but it doesn’t define it. I felt that it would be rather awkward to discuss real numbers without knowing what they were and I decided to write some notes on the construction of R. The approach I am following is called ‘Dedekind cut’, discovered by a German mathematician, Richard Dedekind (1831-1916), and published in 1872. Idea: The idea behind Dedekind cut is roughly the following. Each rational number r can be naturally identified with {x ∈ Q|x < r}. However, if one looks at all similar subsets of Q, one will soon find ‘things’ other than rational numbers. For example, {x ∈ Q|x < 2 or x < 0}
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