Part Four - Euclidean Set Theory

Summary

There is perhaps something mysterious in the fact that we seem to know instinctively what the natural numbers actually are. For as children (or adults) we are provided with just a comparatively small number of descriptions as to what “zero”, “one”, “two”, “three”, etc., mean (“three oranges”, “one banana”, etc.); yet we can grasp the entire concept despite this inadequacy. In some Platonic sense, the natural numbers have an absolute conceptual existence independent of ourselves.

Roger Penrose, Shadows of the Mind

What is essential is to regard the natural numbers as mental constructions, generated in determinate manner by repeated application of the successor operation to zero. Considered as an infinite structure, the totality N of natural numbers is uniquely determined: there cannot be non-isomorphic structures each with an equally good claim to represent N.

Michael Dummett, Elements of Intuitionism