Understanding Dedekind Cuts and the Construction of Real Numbers in Analysis

[mmmboh]

Hi, I am independently learning analysis, I found videos online and Rudin's textbook, though I am unclear on one thing. I thought the point of dedekind cuts was to construct the reals without explicitly talking about the irrationals, which is why to get at the square root of 2 you let the cut be all rationals with x²<2, and x<0...But then how do you get numbers like pi, and e, or 2^1/2-1? If for example pi you said the cut where all rationals are less than pi, you would be explicitly talking about an irrational, which I thought goes against the purpose?
Would you instead take the power series representations?

 

[Hurkyl]

There are lots of ways you could go about defining an explicit cut for pi or for e, but it's not something you would typically bother doing. Generally speaking, constructing the Dedekind reals are not something you do for the purpose of calculation, or even for the purpose of proving theorems -- it's something you do to gain confidence that we haven't made any silly mistakes in setting up the foundations: so you can prove that the assumed consistency of set theory implies the consistency of real analysis.

Once you have the framework in place, you generally won't ever think about a Dedekind cut again -- the closest you'll get is invoking the Least Upper Bound property.