Single Idea 10213

[catalogued under 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers]

Full Idea

Real numbers are either Cauchy sequences of rational numbers (interpreted as pairs of integers), or else real numbers can be thought of as Dedekind cuts, certain sets of rational numbers. So π is a Dedekind cut, or an equivalence class of sequences.

Gist of Idea

Real numbers are thought of as either Cauchy sequences or Dedekind cuts

Source

Stewart Shapiro (Philosophy of Mathematics [1997], 2.5)

Book Reference

Shapiro,Stewart: 'Philosophy of Mathematics:structure and ontology' [OUP 1997], p.53


A Reaction

This question is parallel to the question of whether natural numbers are Zermelo sets or Von Neumann sets. The famous problem is that there seems no way of deciding. Hence, for Shapiro, we are looking at models, not actual objects.