Are algebraic axioms actually just definitions?

Question

Euclidian postulates are statements about points, lines, angles, etc. which exist separate from their definitions, created solely based on observation of our universe. Consequently, we have been able to model universes with hyperbolic space, for example, where euclid's postulates don't hold, for the same definitions of points, lines, angles.

However, when I look at the ZFC axioms, or the Peano axioms, they seem to me just definitions of sets, succesors, and equality. Do we have complete definitions of these terms separate from the axioms? If so, can we model algebraic theories where these axioms do not hold, for the same definitions?

Answer 1

A definition would be the same as an axiom, apart from the fact that you don't need definitions, you can always rewrite a sentence with defined terms using their definientia (that which was used to define the term).

Answer 2

A proper definition should satisfy 2 criteria (Suppes, 'Introduction to Logic', chapter 8):

1.

criterion of eliminability: if a new symbol $D$ is introduced (the definiendum) by: $D \iff S$ then $S$ ($S$ is called the definiens) should contain only primitives and $D$ should be derivable from axioms and previous theorems of the theory.

Thus one can not define: $x\leq x \iff (x=x) \lor (x<x)$ in the theory of arithmetic where the primitives are $=, >$ since one would not be able to eliminate: $\leq$ for 2 variables.

2. a definition can not be used as an axiom to prove things that can not be proved by axioms, theorems and primitives of the theory.

this is the criterion of non-creativity.

Thus, definitions are not axioms and usually in mathematics they require a previous theorem.