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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 Ref
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.