Dedekind Cut - Construction of The Real Numbers

Construction of The Real Numbers

See also: Construction of the real numbers#Construction by Dedekind cuts

A typical Dedekind cut of the rational numbers is given by

This cut represents the irrational number √2 in Dedekind's construction. To establish this truly, one must show that this really is a cut and that it is the square root of two. However, neither claim is immediate. Showing that it is a cut requires showing that for any positive rational with, there is a rational with and The choice works. Then we have a cut and it has a square no larger than 2, but to show equality requires showing that if is any rational number less than 2, then there is positive in with .

Note that the equality b2 = 2 cannot hold since √2 is not rational.

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