i don't have much background in set theory and mathematical logic besides isomorphisms thus i can't quite understand(justify) the way of "identifying" integers with naturals in Tao's analysis.
That's how i interpret what i have read so far about integers: he constructs integers from naturals(integers are elements of the set $N×N$ so they are not the same objects as naturals he defined earlier, he uses a notation $(a,b):=a-b$ for them). He defines equality $"="$ relation between integers(based on equality between naturals), and operations of additions and multiplication for integers(again in terms of natural numbers) After that I have a problem with understanding his next paragraph:
The integers $n−0$ behave in the same way as the natural numbers n; indeed one can check that $(n−0) + (m−0) = (n + m)−0$ and $(n−0) × (m−0) = nm−0$.
I know it should mean something like this$:$ if we map $(n,0)$ with $n$, then we have a function $f:A⊆N×N↦N$, where $A$ consists of integers of the form $(n,0)$ that has properties that if $a+b=c$ ($"+"$ and $"="$ are those defined for integers), then $f(a)+f(b)=f(c)$($"+"$ and $"="$ are those defined for naturals) and if $a×b=c$ then $f(a)×f(b)=f(c)$(same thing with $"×"$ and $"="$)
Furthermore, $(n−0)$ is equal to $(m−0)$ if and only if $n = m$.
I guess that means that $f$ is injection. Though it's a surjection too.
(The mathematical term for this is that there is an isomorphism between the natural numbers $n$ and those integers of the form $n−0$). Thus we may "identify" the natural numbers with integers by setting $n ≡ n−0$;
From here it begins: What exactly does $"≡"$ sign mean? Does it stand for my function $f$? Or is it some new relation for integers like our already defined relation $"="$ but he just doesn't want to overload the sign $"="$ or something?
this does not affect our definitions of addition or multiplication or equality since they are consistent with each other. Thus for instance the natural number $3$ is now considered to be the same as the integer $3−0: 3 = 3−0$.
Hey now i wonder what $"="$ sign means, because we have $"="$ for naturals, $"="$ for integers but we don't have $"="$ for integer-naturals(did he define it implicitly?)(is it the same sign as $"≡"$?)
In particular $0$ is equal to $0−0$ and 1 is equal to $1−0$. Of course, if we set $n$ equal to $n−0$, then it will also be equal to any other integer which is equal to $n−0$, for instance $3$ is equal not only to $3−0$, but also to $4−1$, $5−2$, etc.
We can now define incrementation on the integers by defining $x++ := x + 1$ for any integer $x$; this is of course consistent with our definition of the increment operation for natural numbers.
How does this operation work? It looks like it has only one argument from $N×N$ but then computing an output it uses $1$ from $N$ so operation $"+"$ has one argument from $N×N$ and the other from N and how it supposed to react to this?!
So basically all my questions are about what we can do with isomorpisms and why we can do it.