>>13928610Because there really are no constructions that actually work. Let's keep in mind the construction of the rationals. First you start with naturals as your first principle. Then you add a - sign in front of each number and include 0. Then you get the integers. Then from the integers you construct the rational numbers. Numbers of the form p/q where p,q are integers and q is non-zero. Then you say two rationals p/q = r/s if and only if p s - q r =0. It may be the case that he calculation of equivalence of rationals is complicated, but should always be doable in a finite number of steps. So with enough computing power you could potentially determine equivalence for some randomly chosen rationals.
Dedekind cuts are a cute attempt. But its a dream. Outside of the typical pet examples, you can't define the cuts. Can you define pi with a cut? The answer is no since you can't construct the cut for pi without having constructed an algorithm to calculate pi hence without knowing what pi. Can you calculate any numbers with it? No. The common theme here is that you have to have "constructed" a real number in order to construct a real number with this method. It's entirely fallacious circular reasoning. Not at all sound. At best, what you can do with this is assert that if you already constructed the real numbers, then you have a little relation that can be used to identify the real numbers with the power set of the rationals and maybe talk about cardinality in relation to the currently used set theory (to be discarded the moment the next paradox pops up). However, that's all you can really use it for. It's not actually a construction of the real numbers. Unfortunately the is what Rudin uses in his book.
