4Sets within sets

A general point I should also make about sets before we go further is that in general it does make sense to ask if a set is itself an element of a further set. While sets are fully defined by and constituted by their elements, this does not mean that “a set is nothing more than its elements”. From a logical perspective, the set itself counts as a further object. Thus, for example, there is a set X containing precisely the following elements:

• The number 7
• The number 8

And there is also a set Y containing precisely the following elements:

• The number 7
• The number 8
• The set X

Set Y is then a set containing three elements, and it is not the same set as set X (because set X contains only two elements).

Sets can contain anything, including other sets. This feels strange at first, but you’ll soon get used to it.