# Is it an abuse of language to say “*the* integers,” “*the* rational numbers,” or “*the* real numbers,” etc.?

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 $\mathbb Z$.

1. 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 $a\in \mathbb Z$?

2. 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 $q\in \mathbb Q$?

3. 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.

How you probably should think about this, most of the time — and certainly the approach that seems to be most productive for most ordinary mathematics — is that the integers, real numbers and even complex numbers exist in and of themselves in some “Platonic” sense, independently of what we think of them.

The set-theoretic constructions you find in textbooks construct models of the Platonically existing numbers within a pure set theory. Knowing how to do this — and in particular knowing that it can be done — is important and brings with it many technical advantages, but you should not let it trick you into thinking that these set-theoretic models are “what numbers really are”. Doing so would make a mockery of the centuries and millennia where mathematicians reasoned about numbers without set theory even having been invented. It would be absurd that Dedekind, Cantor, Zermelo, and other mathematicians working in the 19th and 20th centuries were for the first time discovering what Euclid, Euler, Gauss and so forth had really been talking about.

Thus for ordinary mathematics the most useful approach is to continue thinking that all of the real numbers (and the complex ones too, if you can bring yourself to it) just exist, that you can put them all into a set, and that the rationals and integers are particular subsets of them.

Of course this “naive” view is not sufficient in order to work in axiomatic set theory, logic, or similar foundational areas of mathematics. There you’re interested in how well the set-theoretic formalism can capture our naive ideas about integers and real numbers, and perhaps even in to which extent the usual Platonic belief in the existence of numbers can be upheld when one really thinks about it.

It it good to be able to ask such philosophical questions, and to know enough to begin answering them.

All I’m saying here is that it is not productive to let such philosophical uncertainty paralyze you when doing mathematics outside the foundational domain. One of the benefits about knowing about foundational theories is that it allows you to relax and know that whichever philosophical objections one might have to such-and-such argument in everyday mathematics can be handled another day, because we have a good model of the fundamental assumptions that everyday mathematics depends on.