Is my understanding of Peano Axioms as mentioned below correct? I’ll also be grateful if questions below are answered definitively

Question

I have concluded the reading of second chapter of Prof. Tao’s Analysis books in which he covers natural numbers and defines addition and multiplication operation on them,

He states the following axioms,

1. 0 is a natural number.
2. If n is a natural number, then n++ is also a natural number.
3. 0 is not the successor of any natural number; i.e., we have n++ ̸= 0 for every natural number n.
4. Different natural numbers must have different successors; i.e., if n, m are natural numbers and n ̸= m, then n++ ̸= m++. Equivalently, if n++ = m++, then we must have n = m.
5. (Principle of mathematical induction). Let P(n) be any property pertaining to a natural number n. Suppose that P(0) is true, and suppose that whenever P(n) is true, P(n++) is also true. Then P(n) is true for every natural number n.

There is also an assumption here makes:
Assumption (Informal) - There exists a number system N, whose elements we will call natural numbers, for which Axioms 1-5 are true.

Here is what I inferred from these,

1. 0 (Zero) is just a symbol here with no inherent meaning, other than the fact that its a natural number as defined by Axiom 1. Its behaviour is gradually defined when the addition, multiplication operations are defined.(Example, Prof. Tao later when mentioning addition defines 0+m=m).
2. ‘++’ or ‘increment’ operator also has no inherent meaning. Its behaviour is defined by Axioms 2-5.

Questions;

1. Is my inference correct?

2. If I have would have called the ‘Increment’ operator the ‘decrement’ operator instead with the same aforementioned axioms, would they result in the same conclusion i.e. the same number system?

3. Does the assumption imply that there could be a possibility that the natural no. system following the axioms won’t exist? Or could there be other number systems following the same set of axioms?

4. Also as this is my first exposure to something abstract, my reading so far has led me to conclude that math. is to be treated like a language. We assign numbers say 0, 0++, (0++)++ etc. to physical entities say distance. They aren’t part of the nature. I may be wrong or sound stupid but am I right?

5. Are there other such abstract things in math.? Is everything abstract in math.?

Answer 1

Speaking a bit imprecisely here, but here goes…

Yes, the zero and successor function have no inherent meaning, except what is defined in the axioms.

1. Yes
2. Yes, the name of a function doesn't matter.
3. No, once you have the induction axiom, there is only one system that satisfies all the axioms: the natural numbers.
4. Yes, in formal logic you write everything in a formal language. Very much like programming, but much more precise :)
5. Everything in math that can be formalized with formal logic is just as abstract as this.