What is $Th(\mathbb{N})$? How to correctly reason about it? So I am curious how to correctly reason about $Th(\mathbb{N})$.
Is it a set of constants 0,1 and relations on them?
E.g can we say that $(1+1+1+1) * (1+1+1)$ is in $Th(\mathbb{N})$ because we take a constant $(1+1+1+1) \in \mathbb{N}$ and constant $(1+1+1) \in  \mathbb{N}$ and apply relation $*$ on them that produces new number $n$ such that $n \in \mathbb{N}$?
Am I reasoning in a right way?
Can we say that relations $ +, -, *, /, \sqrt{} $ etc are all relations in  $Th(\mathbb{N})$?
 A: $Th(\mathbb{N})$ is the set of sentences in the first-order language of arithmetic (usually understood to be $\{+, \times, 0, 1\}$) which are true of the natural numbers.
So, for example:


*

*"$\forall x(x=0)$" is not in $Th(\mathbb{N})$, because it is not true that every natural number is zero.

*"$1+1=2$" is not in $Th(\mathbb{N})$, because there is no primitive symbol for "2".

*"$1+1+1$" is not in $Th(\mathbb{N})$, because it is a term, not a sentence - what would it mean for "$1+1+1$" to be false?

*"$\forall X\subseteq\mathbb{N}(0\in X\implies 0\not\in X)$" is not in $Th(\mathbb{N})$, since in first-order logic we can't quantify over sets.

*But "$\forall x\forall y(x+y=y+x)$ is in $Th(\mathbb{N})$. It is a first-order sentence, using only (nonlogical) symbols from among $\{0, 1, +, \times\}$, which is true of $\mathbb{N}$.

EDIT: in first order logic, in addition to the nonlogical symbols provided by the specific context we're working in (in this case, $+, \times, 0, 1$), we always have: parentheses $(, )$, Boolean connectives $\wedge, \vee, \neg, \implies$, quantifiers $\forall,\exists$, variables $x_0, x_1, . . .$, and equality $=$.
A: We have to start with a language $\mathcal L$, e.g the first-order language for arithmetic (or elemntary number theory) :

Constant symbols: $0$
One-place function symbols: $S$ (for successor)
Two-place function symbols: $+$ (for addition) and $\times$ (for multiplication).

Then we have to consider a structure $\mathcal N =(\mathbb N, 0, S, +, \times)$ for that language.
Finally, we define the theory of $\mathcal N$, written $\mathsf {Th} \mathcal N$, as the set of all sentences true in $\mathcal N$.
Examples : $0=0 \in \mathsf {Th} \mathcal N$; $\exists n (S(n) = 0) \notin \mathsf {Th} \mathcal N$ (recall that $\mathsf {Th} \mathcal N$ is a set of sentences).
The first sentence is clearly true, while the second one is false ($0$ is not a successor).
But we have to pay attention to the language : we can take into account the expessive capability of it.
We cannot say, e.g. : $\exists n (n= \sqrt 4)$ but we can say : $\exists n (n \times n = S(S(S(S(0)))))$.
We cannot say : $0 < 2$ but we have to say : $\exists n (0+n = S(S(0)))$.
In order to use the "usual" expressions, we have to "enlarge" the original language adding suitable definition for the new terms and relations.
