D. Pierce, Induction and Recursion, The De Morgan Journal, 2 no. 1 (2012), 99-125.
From the Introduction:
In mathematics we use repeated activity in several ways:
- to define sets;
- to prove that all elements of those sets have certain properties;
- to define functions on those sets.
These three techniques are often confused, but they should not be. Clarity here can prevent mathematical mistakes; it can also highlight important concepts and results such as Fermat’s (Little) Theorem, freeness in a category, and Goedel’s Incompleteness Theorem.
The main purpose of the present article is to show this.
In the `Preface for the Teacher’ of his Foundations of Analysis of 1929, Landau discusses to the confusion just mentioned, but without full attention to the logic of the situation. The present article may be considered as a sketch of how Landau’s book might be updated.