Uniqueness of familiar sets

[Stephen Tashi]

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 "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.

 

[Hornbein]

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 "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.

So you are asking, is the set of integers unique? Hmmm. Usually the set of integers is constructed. A set is the set of integers if it contains all of those elements and no others. So yes, it is unique.

The set of real numbers can also be constructed as the set of limit points of converging series of rationals. So it would be similar and also unique.

Another way of looking at it, in each case we have a rule that tells us whether something is or is not a member of that set. So the set is unique. Defining by properties would be trickier.

 

[Stephen Tashi]

So you are asking, is the set of integers unique? Hmmm. Usually the set of integers is constructed.

In typical treatments "a" set is constructed and then it is called "the" Integers. So we must ask whether the technique used in the construction of a the set specifies a unique set. If we specify a set by saying that its elements must have certain properties, this doesn't rule out that there might be other different sets whose elements have the same properties.

 

[nomadreid]

Taking a model-theoretic approach: a definition is a sentence of the theory, and all objects (if there exist any) in the universe of your model that satisfy that sentence can be designated by the name you wish to give to this definition, but in most theories the objects which are the sets of real numbers will , by satisfying the same sentences, be equivalent to one another. So, you can choose to call them different (since they are not identical) or the same (since they are equivalent); that is a matter of taste. You can of course make a new model consisting of equivalence classes based on this equivalence; this is often more convenient, in which case the set of real numbers would be unique. Also, usually what is defined in analysis is not just the set of real numbers, but a structure (i.e., a set with an order on that set), as in the usual definition of an ordered field, etc., and a lot of effort is given in some of those texts to showing that all the definitions (in the theory) of a structure consisting of a set of real numbers along with appropriate orders define structures in the standard model that are isomorphic to one another, so they will all define objects that are equivalent to one another. Given all this, the choice of article ("the" or "a") is a matter of taste. (It is easier in those languages such as Russian or Chinese that don't have articles :-) . )

 

[pbuk]

If we specify a set by saying that its elements must have certain properties, this doesn't rule out that there might be other different sets whose elements have the same properties.

Indeed, however it is proven that any two sets that have the properties of the integers are isomorphic. So it doesn't matter whether you use, for example, Peano axioms to construct a set or Zermelo-Fraenkel, you can apply the label "The" integers to the resulting set.

 

[Fredrik]

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

For integers, it's fairly easy to single out a specific set and call it the set of natural numbers, because the ZFC axioms were chosen to (among other things) ensure that this is possible.

For real numbers, I think the alternative is far more elegant: Prove that all Dedekind complete ordered fields are isomorphic, and then say that any of them can be called "the" set of real numbers. (Of course, you will still need to do an explicit construction once, to prove that there exists a Dedekind complete ordered field).

By using that symbol throughout the book, they refer to a unique thing.

There are books that talk about $\mathbb R$ for hundreds of pages, and then define $\mathbb C$ as $\mathbb R^2$ with addition and multiplication defined by specific formulas. They will often claim that complex numbers of the form (x,0) are real numbers. But $\{(x,0)|\in\mathbb R^2\}$ isn't the same ordered field as the original $\mathbb R$. It's just isomorphic to it.