In a club of 40 members, 30 prefer a picnic and 25 prefer a party. Find the greatest number of members who prefer neither picnic nor party.

Total number of members, $N = 40$.
Number of members who prefer a picnic, $|P| = 30$.
Number of members who prefer a party, $|A| = 25$.

The number of members who prefer at least one activity is given by the union of the two sets: $|P \cup A|$.
The formula for the union is:

To minimize $|P \cup A|$, we must maximize $|P \cap A|$ (the number of members who prefer both).

The number of members who prefer both activities cannot exceed the size of the smaller group. In this case, the smaller group is those who prefer a party (25 members).
So, the maximum number of members who prefer both a picnic and a party is 25.

Substitute the maximum overlap into the union formula:

This is the total number of members minus the minimum number of members who prefer at least one activity.

The greatest number of members who prefer neither picnic nor party is $\boxed{\text{10}}$.