TITLE: What does strength refer to in mathematics?
QUESTION [63 upvotes]: My professors are always saying, "This theorem is strong" or "There is a way to make a much stronger version of this result" or things like that. In my mind, a strong theorem is able to tell you a lot of important information about something, but this does not seem to be what they mean. What is strength? Is it a formal idea?

REPLY [2 votes]: Terry Tao (Ask yourself dumb questions – and answer them!):

For instance, given a standard lemma in a subject, you can ask what happens if you delete a hypothesis, or attempt to strengthen the conclusion

To strengthen a conclusion is to say more.
We could have some lemma (or theorem) that says $p \to q$. To attempt to strengthen the conclusion is to see if we can say more than just $q$ so we would try to see if we could say $p \to q_1$ where $q_1$ is some proposition s.t. $q_1 \to q$.
For instance $x=1$ is stronger than $x=0$ or $x=1$. The former implies the latter.
So if we have some assumption that implies the conclusion '$x=0$ or $x=1$', oh let's say, '$x^2 = x$', we would try to see if we could try to strengthen the conclusion to '$x=1$'. We cannot because it is possible that '$x \ne 1$' while '$x^2 = x$' (namely when '$x=0$').
Let's try using a different assumption:
It is true that '$x+1=2$' implies '$x=0$ or $x=1$'. Here, we can strengthen the conclusion to $x=1$.