"Note that two sets with different intensions might nevertheless have the same extension. Suppose O is “the set of all people over 25 years old” and R is “the set of all people who wear wedding rings.” If our Ω is the Davies family, then O and R have the same extension, namely, Mom and Dad. They have different intensions, though: conceptually speaking, they’re describing different things. One could imagine a world in which older people don’t all wear wedding rings, or one in which some younger people do. Within the domain of discourse of the Davies family, however, the extensions happen to coincide. We say two sets are equal if they have the same extension. So it is totally legitimate to write O = R since, by the definition of set equality, they are in fact equal. This is no weirder than saying “the number of years the Civil War lasted = Brett Favre’s jersey number when he played for the Packers.” The things on the left and right side of that equals sign refer conceptually to two very different things, but that doesn’t stop them from both having the value 4, and thus being equal."