Q:

Language structure of $\mathbb{R}$ and $L_{\mathbb{R}}$

Ok, so, I'm reading up on some first order logic and am now studying languages and structures. If we define the language $L:= \{0,1,+,\cdot\}$, with $0,1$ the constants and $+, \cdot$ the $2$- place function symbols, then we should have that $\mathbb{R}$ is an $L$-structure. I'm having  some problems with that.
Obviously, from $L$, we can achieve the natural numbers by setting $1+1 = 2$, $2+1=3$ and so on. The integers can now be achieved by working with variables (which is permitted in any language): $x + 1 = 0$, $x + 2 = 0$ and so on. In a similar way, one can obtain the rationals, again by using variables: $2 \cdot x = 1$, $\ldots$ you get the idea. I can even see how you get some irrational numbers (like $x \cdot x = 2$ gives you $\sqrt{2}$), but I fail to see how one can obtain numbers like $\pi$ or $e$. So that's one thing I do not understand. 
Another problem to me is: Given that $L$-structure, one can consider the language $L_{\mathbb{R}}$, i.e. the language of the structure $\mathbb{R}$: $L_{\mathbb{R}}$ is $L$ together with, for each element $r$ of $\mathbb{R}$, an extra constant (also denoted by $r$). Here we assume that constants$(L) \cap \mathbb{R} = \emptyset$. So that there are no repetitions. So far so good. However, someone said to me that $L_{\mathbb{R}} = \{0,1,+,\cdot, \pi, e, \sqrt{2}, \ldots\}$. I understand why $\pi, e$ are in there, but not $\sqrt{2}$, as $\sqrt{2}$ can be achieved by $x \cdot x = 2$.
I'm probably mixing some stuff up, but can someone help me shine some light on this? 
Thanks in advance!

A:

A structure is not a syntactic object: we don't expect to be able to express every element of a structure for a language $L$ as a term in $L$ or even define every element by a formula of $L$.
To say that $\mathbb{R}$ is a $L$-structure for the language $L$ defined by the signature $(0, 1, +, \cdot)$ means that we can interpret $0$ and $1$ as elements of $\mathbb{R}$ and $+$ and $\cdot$ as binary operations on $\mathbb{R}$, i.e., functions from $\mathbb{R}\times\mathbb{R}$ to $\mathbb{R}$. If we do this in the obvious way: identifying $0, 1, +$ and $\cdot$ with the zero element, the unit element, addition and multiplication respectively, the resulting structure has many interesting properties. One of these interesting properties is that $\mathbb{R}$ contains many elements that cannot be defined in $L$: this doesn't stop $\mathbb{R}$ being an $L$-structure.
There is no constant for $\sqrt{2}$ in the signature of $L$, however we can extend that signature to an uncountably infinite signature which contains constants for every real number, not just $0$ and $1$. Your $L_{\mathbb{R}}$ is the language defined by this signature. Some of these constants like $\sqrt{2}$ are definable in $L$ and some like $e$ are not.

