I am working through Landau’s “Foundations of Analysis” and have been stopped in my tracks as early as theorem 4 (attached below, for reference). I spent a few days attempting to formulate my own proof- to no avail. I then spent a couple more days attempting to understand his proof- and only successfully comprehend part A).
Nonetheless, after consulting some other resources and realizing most use the recursion theorem to define addition of naturals, I became curious as to why Landau does not follow suit. Recently, I found in the preface some pages criticizing the invalidity of Peano’s addition proof (also attached).
Is anyone able to explain why Peano’s proof is invalid and, in turn, how Landau’s proof including the “construction” of addition works? In particular, parts B I) and B II) are murky (“the number… is as required..”).
Thank you!
