Question

Dedekind felt that arithmetic lacked a scientific foundation

Answer 1

I always felt as if the two approaches demonstarate a difference in philosophy as to how one might approach real analysis.

The cantor construction is more geometric, it works for any metric space, even being characterized by a univeraal property. The cantor construction solves the issue of having missing points in order to carry out geometric arguments, and is constructed essentially in a way that fills those holes and gives a "universal" geometric space over Q.

On the contrary, dedekind construction is far more analytical, it relies heavily on the ordered properties of the field and introduces the imporant analysis formalism of infimums and supremums that is crucial to give a well assgined numerical quality to many measurements like the integral and to allow onr to perform arguments like continous induction.
 
In my head, the cantor construction sees the reals as points on a line, whereas the dedekind construction sees them as numbers with value. And one of the beauties of real analysis is that both of those approaches go hand in hand and lead to the right things we want.

Answer 2

My analysis class began by rigorously defining decimal expansions...Operations were painful. But I still think it's an acceptable way to introduce reals to freshmen

Answer 3

Yeah Dedekind cuts are basically black magic. Terrence Tao admits as much in his Analysis course.

Answer 4

The issue I have with Cantor's construction (which really isn't a big issue) is that you first need to define the concept of convergence in the rationals that only relies on the rationals, a Q-convergence, if you will. If I remember correctly, this is the approach Tao takes in his introductory real analysis texts. To me, this is inelegant because it's immediately abandoned once you complete Q, in favor of the usual convergence in R. It's almost as though Cantor knows what he wants to have (for convergence), but since R hasn't been invented yet, he can't say "for all epsilon > 0" unless he only means rational epsilons.

Answer 5

Are you going to participate in the SOME2 perhaps?

Answer 6

Dedekind cuts good

Cauchy sequences boring

simple as