How do we formally "identify" objects using isomorphisms?

Question

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.

Answer 1

Lots of questions here. I will suggest an answer to the last one

So basically all my questions are about what we can do with isomorphisms and why we can do it.

in hopes that it helps with the rest.

What Tao is doing is to show you how to construct formally what you intuitively understand as the integers $\{ \ldots, -3, -2, -1, 0, 1, 2, \ldots \}$ assuming that you know everything necessary about the natural numbers $\{0, 1, 2, \ldots \}$,

Since he wants to write with formal set theory vocabulary, unravelling the notation can be challenging.

The idea he wants to capture is that a "missing" negative number like $-5$ can be defined by the expression "$2-7$" even though that expression has no meaning in the natural numbers. But you must be careful, because "$2-7$" and "$96-99$" and "$0-5$" all capture the essence of the missing $-5$. So he tells you just when two of those expressions (each constructed from a pair of natural numbers) should count as the same integer, using the ordinary arithmetic properties of the natural numbers. The $\equiv$ sign between two such expressions says the represent that "same integer". In formal terms, $\equiv$ is an equivalence relation and Tao defines the integer represented by any of those pairs as the equivalence class - the set of all the pairs. It's how you might define the rational numbers once you know the integers as pairs of integers, where $(1,2)$, $(2,4)$ and $(75, 150)$ all represent the same rational, one half. (Tau may well do this next.)

Having done this and checked all the arithmetic facts about these new things called "integers" using only the properties of the natural numbers. he wants to back away from the formal structure. To that end he shows you that there's a faithful copy of natural numbers inside the "integers" he's constructed. That's the essence of the function $f$.

Once that's done you can forget that $5$ is (formally) the set of all the pairs that are equivalent ($\equiv$) to $(5,0)$ and that $-5$ is (formally) the set of all the pairs that are equivalent ($\equiv$) to $(0,-5)$. Then can go about business as usual with $\{ \ldots, -3, -2, -1, 0, 1, 2, \ldots \}$.

Edit:

In a comment you ask if this is "overloading" the natural numbers. That's a formal term from computer science, describing a situation where (say) the meaning of an operator symbol like "$+$" depends on the context. That operator is overloaded here, since it's used both for adding natural numbers and for adding the integers defined as sets of pairs of natural numbers. I'm not sure whether you'd say the natural numbers are themselves overloaded. It's more the opposite: two different representations of the same thing. The $5$ and the $0$ in the integer represented by $(5,0)$ are natural numbers. When you identify that pair with $5$ the symbol "5" stands for both $5$ and the equivalence class of $(5,0)$. Since the embedding $f$ is injective you won't get into trouble reusing the name. You've actually done this kind of thing before. When you work with polynomials you unthinkingly use the natural embedding of the integers into the ring of polynomials. "5" can mean the constant polynomial $5$ or the coefficient in the polynomial $5x$.

Answer 2

Your basic understanding is correct. In your paragraph under the first colored box we need $f$ to be a bijection. He claims that in the second colored box with the if and only if. The sign $\equiv$ is read "equivalent to". Once you have shown that $f$ is an isomorphism we can consider the two things related by $f$ as the same for whatever purpose we have in mind. Constructing the integers as ordered pairs of naturals is formally nice, but representing the integers as equivalence classes of naturals is very clumsy. We would like to get back to the notation we are used to with the positive integers and zero using the same symbols as naturals and the negative integers using the naturals with a minus sign prefixed. The equivalence sign shows which integer as an ordered pair corresponds to which natural. In the part you quote he never points out that the integers as ordered pairs are really equivalence classes of ordered pairs, though it is implied when he talks about $3-0$ being equal to $4-1, 5-2, $etc. I am sure that point is made in the article. I am not crazy about writing $3=3-0$ as he does because (as you say) this is asserting equality between two different sorts of objects. What he has done is give the traditional names to the integers as single numbers which may be preceded by a minus sign instead of the equivalence classes of ordered pairs. He asserts that the formal incrementation operator works on the integers just as you would expect. The basic point is that once you have an isomorphism you can think of it as having two different descriptions of the same object. You have the formally defined integers as equivalence classes and you have the informally defined integers as the naturals plus the negatives. He is showing that the informally defined ones with the rules we are used to work the same as the formally defined ones, then will say that it is much easier to write the informally defined ones so we will use that notation in the future.