# What is the difference between an object and its singleton set?

I have read in books on Mathematical Logic that we have things called "Sets" and Set Theory that correspond to classes of objects in Ontology. for example { Barack Obama, Donald Trump } is a set (or class) of Presidents.

My question is about Singletons in particular. Singletons are sets with only 1 member, for example { Barack Obama } where Barack Obama is the only member.

Does this mean we can reguard the set { Barack Obama } as THE Barack Obama member? Does it mean that the properties of the set { Barack Obama } and member Barack Obama are the same?

This is important because in computer science we can model Classes (~sets) but not the individual members, however, if the above holds and the singleton set is "the same" as its individual member then we can model the individual i.e. Barack Obama as a class.

Cross-references:

• Classes in CS is more like type not set in philosophy. In CS we can also flexibly model an individual as a prototype object without any usual class template in some languages... Commented May 6, 2021 at 19:01
• "Does this mean we can reguard the set { Barack Obama } as THE Barack Obama member?" A question is: what does "regard" mean in this sentence? Math is full of examples where we "regard" one thing as another. For instance, we can "regard" a degree-zero real polynomial as a real number. We can "regard" a vector as a linear form, and we can "regard" a matrix as a linear map. So, yes, we can "regard" a singleton set as its element.
– Stef
Commented May 16, 2022 at 9:20

A set is a mathematical object; a singleton set is a set with only one element.

The set N of natural numbers has infinite many elements.

The singleton set { N } has only one element.

In general, the properties of an object and those of the set with that object as single elements are not the same.

As per example above, the singleton { N } has only one elements while its (only) element have infinitely many element.

But we may consider the singleton { emptyset }; again, it has one element, while its element has no elements.

• This is important to computer science: if you have a function that takes a number as an argument, and you give it a set of numbers that happens to have only one element, it will complain. "I asked for a number, and you gave me set." Commented Jan 22, 2021 at 20:14
• Taking the example in this answer. Also the operations we apply to sets are usually different to the operations we apply to the elements of a set (unless the elements are themselves sets). Consider the set A = {2} where 2 is the natural number, 2. (If we take usually definition of operations) A ∩ A has meaning, whereas 2 ∩ 2 has no meaning (2 is a natural number) . So one aspect that a singleton set is not the same as its members is the operations on them are not the same. Commented May 16, 2022 at 9:38