Full Idea
Compact consequence undergenerates - there are intuitively valid consequences which it marks as invalid, such as the ω-rule, that if A holds of the natural numbers, then 'for every n, A(n)', but the proof of that would be infinite, for each number.
Gist of Idea
Compactness blocks the proof of 'for every n, A(n)' (as the proof would be infinite)
Source
Stephen Read (Thinking About Logic [1995], Ch.2)
Book Reference
Read,Stephen: 'Thinking About Logic' [OUP 1995], p.44