TITLE: Is it an abuse of language to say "*the* integers," "*the* rational numbers," or "*the* real numbers," etc.?
QUESTION [58 upvotes]: I'm finding that the more math I learn, the more concepts I thought were well-defined seem to be intuitive and naive. Here I'm asking about whether it's an abuse of language to refer to "the integers," "the rational numbers," or "the real numbers".
This is what I mean:
Suppose, for the sake of starting somewhere, that we have a set which we call "the integers" and denote by $\mathbb Z$.

I can define "the rational numbers," which I denote by $\mathbb Q$, as the set of equivalence classes of ordered pairs of integers, where $(a,b)$ is equivalent to $(c,d)$ if $ad = bc$. But now when I say "the integers," do I mean the original set $\mathbb Z$, or the image of this set in $\mathbb Q$, i.e., the collection of equivalence classes of ordered pairs of the form $(a,1)$ where $a\in \mathbb Z$?
I can define "the real numbers," which I denote by $\mathbb R$, as the set of equivalence classes of Cauchy sequences of rational numbers, where $(a_n)$ is equivalent to $(b_n)$ if $a_n - b_n \to 0$.
But now when I say "the rational numbers," do I mean the original set $\mathbb Q$, or the image of this set in $\mathbb R$, i.e., the collection of equivalence classes of sequences of the form $(q,q,q,\dotsc)$ where $q\in \mathbb Q$?
I can define "the complex numbers," which I denote by $\mathbb C$, as the set of equivalence classes of the polynomial ring $\mathbb R[X]$, where $f(x)$ is equivalent to $g(x)$ if $(x^2 + 1) \mid (f(x) - g(x))$.
But now when I say "the real numbers," do I mean the original set $\mathbb R$, or the image of this set in $\mathbb C$, i.e., the collection of equivalence classes of constant polynomials in $\mathbb R[X]$?

And then of course I can take sets further up on the list and try to ask where they fit in lower down, e.g., I can ask whether or not by $\mathbb Z$, I mean a specific subset of $\mathbb R$, or a specific subset of $\mathbb C$, etc.
Going further, I can also ask, given an arbitrary integral domain $D$, for example, and given its field of fractions $F$, whether or not when I say "$D$" I am referring to the original integral domain or its image in $F$, etc.
What is going on? Which set I should think of when I hear someone say "the integers" or "the rational numbers"?
Is this a matter of learning to no longer think of these sets or structures as unique, but rather unique up to isomorphism? Or unique up to unique isomorphism? So when someone says "the rational numbers," should I be thinking of any field isomorphic to the rational numbers? Feel free to modify tags or provide comments.

REPLY [2 votes]: Short answer:
There are many ways to define a thing. As long as we have a consistent way to declare the definitions as equivalent, we can choose our preferred definition and use that as the "operating definition". As long as we have machinery for declaring equivalency of definitions, we just say, if only as a matter of practicality, that the things being defined are indeed the same thing. 
This doesn't mean that in some deep existential/ontological/philosophical sense they are actually the same thing. It just means they can be identified with each other in a logically consistent and helpful way. 
Long answer:
As was noted in the original post, a real number can be thought of as an equivalence class of Cauchy sequences of rationals (once we accept the existence of the rationals). Call these equivalence classes $\mathbb R_{Q.seq}.$ But this is only one way to define the real numbers.
We may posit the existence of a continuum and define each real number as a specific point on that continuum. Call these $\mathbb R_\frak c.$ Or we might say that a real number is identical to an equivalence class of sequences natural numbers upon which the concatenation of all but the first with placement of the decimal according the the first (as long as we accept the existence of the naturals, a decimal point, and our ability to concatenate and place such a decimal, etc.). Call these $\mathbb R_{N.cat}.$
Do we have $\mathbb R_{Q.seq}=\mathbb R_{\frak c}=\mathbb R_{N.cat}?$
Consider $x,y,z$ in $\mathbb R_{Q.seq}$, $\mathbb R_{\frak c}$, $\mathbb R_{N.cat}$ respectively. $x$ is an equivalence class of sequences of rationals. $y$ is a point on a continuum. $z$ is an equivalence class of sequences of natural numbers. (Note that although we might like to think of $z\in\mathbb R_{N.cat}$ as being a collection of sequences of rationals as well, those sequences are not necessarily Cauchy.)
How can we even check if $x=y=z$ when, structurally, they are three distinct types of objects? Certainly they are all pair-wise non-identical (again since they are from three distinct classes of objects). 
However, we are able to define one-to-one and onto mappings $f:\mathbb R_{\frak c}\rightarrow\mathbb R_{Q.seq}$ and $g:\mathbb R_{N.cat}\rightarrow\mathbb R_{Q.seq}$ so that when $f(y)=x$ and $g(z)=x$, we say that $x\overset{f}{=}y$ and $x\overset{g}{=}z$ hence $y\overset{g^{-1}\circ f}{=}z.$ Of course, we have to have some "belief" the existence of such mappings!
In practice, we drop the mappings indicated in the equivalences and just say $x=y=z=r$ where $r$ is our chosen object for defining a specific "real number" (potentially from none of the given definitions of $\mathbb R$).
A simpler example: 
To distill the argument into much simpler terms, consider: $\frac{1+2}{3}=1$ and $10-9=1.$ I'll call attention to two ideas:
Quantitative equivalence/identity.
$$\frac{1+2}{3}=10-9$$
Specific 'structural' identity.
$$\text{Equation}\left[\frac{1+2}{3}\right]\neq\text{Equation}\left[10-9\right]$$
Where the latter is calling attention to the fact that both sides are distinct collections of mathematical operations. (Note: I'm just making up the terminology.)
We have two classes: 
$[1]_3=$ a specific equivalence class of arithmetic operations on $3$ integers. Specifically, $(n,m,k,o_1,o_2)\in[1]_3$ where $n,m$ are finite sequences of $1$'s, $k$ is a natural number, $o_1$ is removal or concatenation, and $o_2$ is dividing the result into $k$ equivalent finite sequences of ones.
 $[1]_2=$ an equivalence class of operations on $2$ integers defined similarly with elements $(n,m,o_1)$ where $n,m$ are finite sequences of ones and $o_1 $ is either concatenation or removal.
We say $[1]_3=[1]_2$ because we have a mapping which allows us to calculate/evaluate them and get the number $1$ as the result. Of course, I'm likely brushing aside many foundational and technical issues.