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?

Answer

The number two is not a set.

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.

This does not, however, mean that the number two is its set-theoretical representation. It is convenient, technically useful, and interesting to be able to express reasoning about numbers in a formalism made for reasoning about sets, but one should not confuse the model for the things it models.

It is perfectly possible to reason about numbers without committing to the philosophical baggage of set theory. Within mathematical logic, it’s a sort of default assumption that Peano arithmetic rather than set theory is the standard vehicle for reasoning about numbers — but both weaker and stronger non-set theories for aritmetic than this are studied for various purposes.

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.

Attribution
Source : Link , Question Author : user325165 , Answer Author : hmakholm left over Monica

Leave a Comment