Could someone explain this statement about logic? I'm literally new to logic, yet, I'm interested in Godel's Incompleteness Theorem. Reading a "pdf proof" in the internet, I encounter the following sentence:
"We assume a theory of arithmetic, say
$N
= (
ℕ
,
+
,
×
)$ to be consistent"
I tried to research this and if I understood correctly, a consistent theory is one that doesn't lead to contradictions...right?
However, I couldn't get to find the meaning of "$N
= (
ℕ
,
+
,
×
)$" and I was wondering if someone could explain it assuming that I know nothing more than high school math.

I would truly appreciate any help/thoughts!
 A: A theory $T$ is typically a set of statements that is logically closed, meaning that all statements that are logical consequences of $T$ are already included in $T$. If the contradiction is not in $T$, then $T$ is said to be consistent.
Now, for it to be a theory of arithmetic, we want the statements in the set to use a language that includes arithmetical symbols like $0$, $+$, and $\times$.  
So, I suppose you could refer to a theory using such symbols as $T=(0,+,\times)$. This would not tell us exactly what statements are in $T$ though, merely what symbols it is using.
The author denoting a theory as $N = (\mathbb{N}, + , \times)$ is really strange: $\mathbb{N}$ is most likely not a symbol that the author wants to use as part of the language to express statements about arithmetic, but is instead the intended domain. In fact, $\mathbb{N}$ is the domain of what is typically called the 'standard interpretation' or 'standard structure' of arithmetic, and this interpretation maps the symbol $+$ to the arithmetical operation of addition, $\times$ to multiplication, and the symbol $0$ to the number $0$.
In fact, I wonder if that is what the author tried to denote by $N$: not a theory, but the standard interpretation ...
