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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:

  1. https://math.stackexchange.com/questions/3995486/whats-the-difference-between-a-singleton-set-and-its-member
  2. https://softwareengineering.stackexchange.com/questions/421361/represent-individual-object-as-a-class
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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.

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  • 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." – Lee Daniel Crocker Jan 22 at 20:14

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