I am reading the book Analysis $I$ by Terence Tao, where the following is written: [...] $\mathbf{N}$ should consist of $0$ and everything which can be obtained from incrementing: $\mathbf{N}$ should consist of the objects $0,0++,(0++)++,...$ (with $++$ he denotes the increment/successor function). He then comes to
Axiom 2.1$0$ is a natural number.
Axiom 2.2 If $n$ is a natural number, then $n++$ is a natural number.
Later (on page 21) he writes: Assumption There exists a number system $\mathbf{N}$ whose elements we will call the natural numbers for which the Axioms $2.1-2.5$ (the Peano-Axioms) are true.
$(Q1)$ What does this mean? Does this mean that we assume that there is some system $\mathbf{N}$ that denotes natural numbers, where we don't know what exactly they are. In the context of set theory we thus have a set $\mathbf{N}$ whose objects are called natural numbers (whatever objects are; I have not seen any definition of this). Thus its objects could be anything in the beginning. Axiom $2.1$ now says that $0$ is a natural number. This should mean that $\mathbf{N}$ contains at least one object and one of those will be called $0$. We couldve however chosen any name or symbol here, as far as I know, such as for example $A$ or $a$ or even $5$. Now Axiom $2.2$ states that the set contains more than one element, namely infinitely many elements and the increment of the previous element will be called $n++$. Again the symbols here should be arbitrary and could be anything.
He now proceeds to define $1:=0++,2:=1++,3:=2++$ and so on. What this now does is it defines the underlying object of the symbol $1$ to be the same as the underlying object of $0++$ and so on. Thus the object that $1$ stands for is now a natural number. So what we have done is basically given the objects "simpler" (in terms of notation) names.
I could've also called the zero object $A$, $B:=0++$, $C:=B++$,... giving the objects different names. (Here I mean that this should also be done in a "good" way, such that essentially $A$ represents $0$, $B$ is $1$ and so on such that $J$ is $9$ and then $BA$ is $10$ and so on.
$(Q2)$ If this is the case, what is the difference between the natural numbers presented "commonly" by the symbols $0,1,2,3,...$ and the ones I defined by $A,B,C,...?$ Essentially they should be the same set, however, no one would say that the set $\{0,1,2,3,...,10,11,12,...\}$ and $\{A,B,C,...,BA,BB,BC,...\}$ are equal would they? Maybe I am wrong here, but are they actually the same set, because I defined them to mean the same underlying object, such that the symbols don't make a difference? Basically what confuses me is if the symbols matter, meaning that they make a difference, even when defined to mean the same object.
I think the above set should, in this context of defining them what they mean be literally equal. However, if I consider the set $\{0,I,II,III,IV,V,...,X,XI,...\}$ without defining anything, then $\{0,1,2,3,4,...\}$ should only be bijective to this set and not literally equal, right? This would mean that the difference is the context, in which I have defined the letters to be the objects that the symbols $1,2,3,...$ represent. In the book Tao says, that actually $\{0,I,II,III,IV,...,\}$ would be different from $\{0,1,2,3,...\}$, if one "wanted to be annoying" (guess I am kinda pedantic), despite them both referring to the natural numbers. How can this be? Do symbols make a difference, despite meaning the same underlying object?
Where I am coming from: I am basically confused about the concept of sets and "objects" and tried to give some "extreme" example in order to make sure I understand it correctly. As far as I understand, a set can contain objects, that I can't write down, which is why symbols are used. Different symbols can mean the same thing, however, which is why the context matters, as presented in the examples I gave. What caused this was my question whether $e^{2 \pi i} \in \{a+ib \ | a,b \in \mathbf{R}\}$ despite it not being a "fitting" symbol. I hope this helps making my question more clear.
