What does strength refer to in mathematics?

Suppose you have a theorem that says "If $X$, then $Y$." There are two ways to strengthen such a theorem:

1. Assume less. If you can reduce the number of hypotheses, but still prove the same conclusion, then you have proved a more "powerful" result (in the sense that it applies in more situations).
2. Prove more. If you can keep the same hypotheses, but add more information to the conclusion, then you have also produced a more "powerful" result.

Here is an easy example from Geometry.

Let $ABCD$ be a (non-square) rectangle. Then the internal angle bisectors of the vertices intersect at four points $WXYZ$, which are the vertices of a rectangle.

(You need the condition that $ABCD$ is not a square because if it is a square then all four angle bisectors coincide at a single point.)

Here are a few ways to strengthen the theorem:

1. The hypothesis "$ABCD$ is a (non-square) rectangle" can be relaxed to the more general "$ABCD$ is a (non-rhombic) parallelogram". The conclusion that $WXYZ$ is a rectangle still holds.
2. Alternatively, you can keep the original hypothesis that $ABCD$ is a (non-square) rectangle, and strengthen to the conclusion to say that $WXYZ$ is not just a rectangle, but a square.
3. Having done that, you can then strengthen the conclusion of the theorem even more, by noting that the diagonal of square $WXYZ$ is equal in length to the difference of the lengths of the sides of $ABCD$.
4. Once you know that, you can now strengthen the theorem even more by (finally) removing the hypothesis that $ABCD$ is non-square, and including the case in which the four angle bisectors coincide at a single point as forming a "degenerate" square with a diagonal of length zero.

Claims are said to be stronger or weaker, depending on the amount of information you can imply from the claim. For example, if $x$ is a positive solution of the equation $x^2 = 2$, then the claim $x > 0$ is weaker than $x > 1$ (the second is more precise).

Sometimes we try to make these implications strongest possible. For example, $1$ is the strongest integer lower bound on such an $x$.

In logic, saying that $\text{statement }x\text{ is stronger than statement }y$, is equivalent to saying that:

$$[\text{statement }x\text{ implies statement }y]\text{ but [statement }y\text{ does not imply statement }x]$$

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For example, suppose we conjecture the following statements on a set $S$:

• Conjecture $x$: all the elements in $S$ are divisible by $4$
• Conjecture $y$: all the elements in $S$ are divisible by $2$

It is obvious that conjecture $x$ is stronger than conjecture $y$.