more from David Bostock

### Single Idea 18095

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

Full Idea

In addition to cuts, or converging series, Cantor suggests we can simply lay down a set of axioms for the real numbers, and this can be done without any explicit mention of the rational numbers [note: the axioms are those for a complete ordered field].

Gist of Idea

Instead of by cuts or series convergence, real numbers could be defined by axioms

Source

David Bostock (Philosophy of Mathematics [2009], 4.4)

Book Reference

Bostock,David: 'Philosophy of Mathematics: An Introduction' [Wiley-Blackwell 2009], p.100

A Reaction

It is interesting when axioms are best, and when not. Set theory depends entirely on axioms. Horsten and Halbach are now exploring treating truth as axiomatic. You don't give the 'nature' of the thing - just rules for its operation.