Full Idea
The result of joining Hume's Principle to second-order logic is a consistent system which is a foundation for arithmetic, in the sense that all the fundamental laws of arithmetic are derivable within it as theorems. This seems a vindication of logicism.
Clarification
Hume's Principle involves one-to-one correlation
Gist of Idea
Neo-logicism founds arithmetic on Hume's Principle along with second-order logic
Source
B Hale / C Wright (Logicism in the 21st Century [2007], 1)
Book Reference
'Oxf Handbk of Philosophy of Maths and Logic', ed/tr. Shapiro,Stewart [OUP 2007], p.169
A Reaction
The controversial part seems to be second-order logic, which Quine (for example) vigorously challenged. The contention against most attempts to improve Frege's logicism is that they thereby cease to be properly logical.