Full Idea
Gödel proved the completeness of standard formalizations of first-order logic, including Frege's original one. However, an implication of his famous theorem on the incompleteness of arithmetic is that second-order logic is incomplete.
Gist of Idea
Gödel proved that first-order logic is complete, and second-order logic incomplete
Source
report of Kurt Gödel (works [1930]) by Michael Dummett - The Philosophy of Mathematics 3.1
Book Reference
'Philosophy 2: further through the subject', ed/tr. Grayling,A.C. [OUP 1998], p.136
A Reaction
This must mean that it is impossible to characterise arithmetic fully in terms of first-order logic. In which case we can only characterize the features of abstract reality in general if we employ an incomplete system. We're doomed.