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**Constructing the "real number system" by Dedekind cuts?**

Hi all,

I've just been working through baby Rudin, and I've almost finished chapter 1. I am currently working through the appendix of chapter 1, which constructs the real number system from the rational number system by Dedekind cuts.

Here is my problem: the elements of the set of "real numbers, R" defined look nothing like a real number!

The elements of R are defined as cuts which are subsets of Q. My question is "How can a set of rational numbers be a real number?"

For instance, I know that 5 is a real number, so where is the "5" in that set R? The R defined in there has other sets as elements - not numbers.