# Are there any objects which aren’t sets?

What is an example of a mathematical object which isn’t a set?

The only object which is composed of zero objects is the empty set, which is a set by the ZFC axioms. Therefore all such objects are sets.

Objects composed of many objects are obviously sets.

What about objects composed of exactly one object? Are there any which aren’t sets?

Textbooks in set theory will happily tell you how to use sets to represent numbers, often using the Von Neumann scheme in which the set $\{\{\},\{\{\}\}\}$ represents the number two. They will often, for convenience, even use the symbol $2$ to stand for that set, with the understanding that every formula in the book’s formalism is about sets, so taking this context into account there’s no risk of the symbol $2$ to be misunderstood as the actual number two.
In particular, second-order arithmetic works for formalizing large parts of mathematics — and while second-order arithmetic does have sets, the integers are explicitly not sets there, and $\{\{\},\{\{\}\}\}$ doesn’t even exist in this theory.