Full Idea
The identification of geometric points with real numbers was among the first and most dramatic examples of the power of set theoretic foundations.
Gist of Idea
Identifying geometric points with real numbers revealed the power of set theory
Source
Penelope Maddy (Naturalism in Mathematics [1997], I.2)
Book Reference
Maddy,Penelope: 'Naturalism in Mathematics' [OUP 2000], p.26
A Reaction
Hence the clear definition of the reals by Dedekind and Cantor was the real trigger for launching set theory.