From Dedekind to Zermelo versus Peano to Gödel
The author compares Dedekind's and Peano's approach to the characterization of the natural numbers. He points out the problems emerging from the set theoretical frame of Dedekind's theory and states also that the mathematical arguments don't depend on that problematic set theoretical frame. After sketching Peano's axiomatic approach, the author refers to the decisive difference: Peano's axioms can be restricted to the first-order logic but Dedekind's theory of natural numbers can't. In the following, the author discusses the relation between the first-order logic and Gödel's incompleteness theorem, non-standard models of the (first-order) Peano arithmetic, Dedekind's refusal of the possibility to characterize his approach in first-order logic, and some works of Zermelo in which he followed the ``tradition of Dedekind''.
non-standard models of the (first-order) Peano arithmetic
scientific article; zbMATH DE number 6835167
scientific article