The formal way to define many mathematical objects is careful not to assert the uniqueness of the object as part of the definition. For example, formally, we might define what it means for a number to have(adsbygoogle = window.adsbygoogle || []).push({}); "an"additive inverse and then we prove additive inverses are unique as a theorem.

Another approach to uniqueness is define a type of thing (e.g. "a" cyclic group of order 3) and then define what it means to have an "isomorphism" between such things and then define "the" thing" (e.g. "the" cyclic group of order 3) as an equivalence class with respect to this isomorphism.

What approach is used for familiar mathematical objects such as "the" Integers for "the" set of Real Numbers?

Most texts I have seen don't bother to apply such an approach to defining important mathematical objects like "the" Real Numbers. They are content to define "a" set that has the properties of "the" Real Numbers and introduce a symbol (e.g. ##\mathbb{R}## ) for that set. By using that symbol throughout the book, they refer to a unique thing.

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# Uniqueness of familiar sets

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