Dtjytyk

I'm struggling with $overline{mathbb{Q}}$ and $mathbb{R}$.
We know that $overline{mathbb{Q}} = mathbb{R}$, but $overline{mathbb{Q}}$ is countable and $mathbb{R}$ is uncountable. How two equal sets could be equal if one is countable when the other one isn't ?

edit 1: $overline{mathbb{Q}}$ the closure of ${mathbb{Q}}$

edit 2: I am in the construction of $mathbb{R}$ (with Dedekind complete field)

$overline{mathbb{Q}}$ is not $mathbb{Q}$ itself, it is its closure. These are two completely different things. The closure of $mathbb{Q}$ is the set of all points in $mathbb{R}$ which are the limit of some sequence of elements from $mathbb{Q}$. And yes, it is equal to $mathbb{R}$.

Your edit remains unclear, and your comment does not particularly help. The problem is that the "closure" means different things in different settings.

The notation $overline{mathbb Q}$ can be used for either the topological closure of $mathbb Q$ inside some larger topological space such as $mathbb R$, or for the algebraic closure of $mathbb Q$.

If you use it to mean the topological closure inside $mathbb R$ then $overline{mathbb Q}$ is equal to $mathbb R$ (as the answer of @Mark says), and it is uncountable.

If you use it to mean the algebraic closure then this is really a topic in number theory or abstract algebra, where $overline{mathbb Q}$ is used (sometimes... usually...) to refer to all complex numbers $z in mathbb C$ which are roots of polynomial equations with coefficients in $mathbb Q$. In this case $overline{mathbb Q}$ is a countable subset of $mathbb C$. The proof of countability is that there are only countable many polynomials with coefficients in $mathbb Q$, and each has only finitely many solutions in $mathbb C$.

It's a regular thing, in fact, it's quite normal. And much like those two terms the term "closure" and the notation $overline A$ has plenty of meanings throughout mathematics.

1. In the topological space $Bbb R$, the closure of the set $Bbb Q$, which is often denoted by $overline{Bbb Q}$ is indeed $Bbb R$. This is set of all the limit points of $Bbb Q$ in $Bbb R$, and by the fact that $Bbb Q$ is dense in $Bbb R$ we get that $overline{Bbb Q}=Bbb R$.




2. In the topological space $Bbb Q$, the closure of the set $Bbb Q$ is just $Bbb Q$ itself. Because given any space $X$, it is closed in itself, so its closure with respect to itself is itself again.




3. In the algebraic context, the algebraic closure of $Bbb Q$, also sometimes denoted by $overline{Bbb Q}$, is the collection of all the complex numbers which are roots of polynomials with rational coefficients. This is not even a subset of $Bbb R$, since $i^2=-1$, and so $iinoverline{Bbb Q}$, but $i$ is not a real number.




4. In a different algebraic context, one might want to look at the real closure of $Bbb Q$, which is not usually denoted by $overline{Bbb Q}$, which is all the real numbers which are roots of polynomials with rational coefficients. This is of course a subset of $Bbb R$, and in fact we can say more from a model theoretic point of view: it has the same first-order theory as the real numbers.

So we get four different closures. But exactly the first one is uncountable, and indeed $Bbb R$, whereas the others are countable.

This is yet again a testament to the importance of context. When I tell you that we need to talk about Kevin, your first question would normally be "who's Kevin? I don't know you sir." or "Which Kevin?". Because maybe I want to talk about award winning actor Kevin Costner, or maybe I want to talk about award winning writer-director Kevin Smith, or maybe your best friend is Kevin and I want to talk about him. Context!