Dedekind Cuts Question (Conceptual)

[mindarson]

Hi,

I have just begun a self-study program in analysis and have a question about Dedekind cuts.

My question is this: My understanding is that cuts are constructions by which to define the reals in terms of the rationals. I.e. the reals are a set of cuts, each of which is a set of rationals. And a cut is the set of rationals to the left of (or less than?) a certain point.

So my real question is: If we must use cuts to construct the reals, how do we know WHERE to cut in order to construct the cut that corresponds to an irrational, real number like sqrt(2)? How is this reasoning not circular? If we DO NOT HAVE the number yet, then how can we even talk about objects TO THE LEFT of the number (or oriented in any way with respect to it)?

Thanks for taking the time to read and consider.

 

[conquest]

What you actually do is consider all limits in the rationals. For instance to construct the square root of 2 just take all cuts such that all rationals in it are smaller than 2 when squared. Here no irrational number is used in any way just rationals and being able to square.

There can be no knowing where to cut, because there is in this construction nowhere to cut. You construct the reals as a set of subsets if the rationals namely all the subsets where is a rational is included also all rationals to the left of it are included.