From our list of example objects above, we would intuitively consider
the number \(2\) to not be a set;
the real number line to be a set as it is a collection of points, each representing a different real number;
a monkey to not be a set; and
a basket of tennis balls to be a set as it is a collection of tennis balls (though the basket itself is not part of this set, just the container for the objects making up the set).
However, the answers above may depend on your point of view. For example, a monkey could be considered a collection of cells. Even the number \(2\) is sometimes defined to be a set! (See Example 11.4.2.)
Remark9.1.3.
Formally, we leave object and set as primitive terms in the axiomatic system of set theory. The reason for leaving these terms undefined is because any attempt to define them would lead us down a never-ending rabbit-hole of definitions: what is an “entity”? what is a “collection”?
We will not discuss any axiomatic basis for set theory, but instead rely on naive set theory.
naive set theory
whatever axioms for set theory the experts decide upon, we are safe (usually, see Warning 9.7.7) to assume that all the mathematical objects that we would like to be sets, will be
We need one more primitive term to make set theory workable.
membership
a property of sets relative to other objects: given object \(x\) and set \(S\text{,}\) exactly one of the statements “\(x\) is a member of \(S\)” and “\(x\) is not a member of \(S\)” is true
