Construction of The Real Numbers
See also: Construction of the real numbers#Construction by Dedekind cutsA 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.
Read more about this topic: Dedekind Cut
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