I am studying the logical fundaments of mathematics, but very often I have trouble to understand Peano's and ZFC/ZFC axioms.
In Tao's book Analysis I, I found very helpful when he points out what would go wrong if we did not have an axioms, e.g. if we ignore Peano's axiom that states that the successor of two numbers is equal implies that the numbers themselves are equal, we could define 4=0, which would obviously break math.
However some axioms are so intuitive, that I fail to imagine how we could mess up maths if we did not have it. Especially Peano's scheme for mathematic induction and the choice axiom (finite and infinite) for set theory.
Also how do we know where to draw the line on what needs to be "axiomized" and what is common sense?
I understand Axioms are not provable, but could someone recommend me a literature that approaches both, Peano's axioms and ZF/ZFC with counter examples to justify its necessity?
Take for instance Peano Axioms. Before Peano, there were many elementary facts about numbers. For instance, multiplication is commutative. Another example is that every natural number is either odd or even. These facts could be considered as common sense. Yet, these facts might have been given proofs. For instance, multiplication is commutative might be proven by saying that the area of a rectangle is the invariant upon flipping the length and breath. That every natural number is either odd or even could be proven by saying that if you keep subtracting by $2$, you will either get $0$ or $1$.
The role of Peano's Axiomatisation could then be regarded as a way to choose a possibly minimal collection of assumptions from which all known elementary facts about numbers could be deduced from.
You asked about the necessity of Peano's Axioms. Let me interpret Peano Arithmetic as being necessary to mean that we need the entire axiom system of Peano Arithmetic to derive all known elementary facts about numbers. Under this interpretation, Peano Arithmetic is not necessary. Indeed, Mathematical Induction in Peano Arithmetic is axiomatised as an axiom schema, meaning that there is one induction axiom for each elementary formula. As not every elementary formula has been used to do induction in a number theory proof, you can drop at least one axiom from the induction schema from Peano Arithmetic, yet not lose any presently known theorem.
Using ZF, you can prove that the set representing the natural numbers satisfy Peano Arithmetic.