I have to present about the Peano axioms and the ZFC for my introductory seminar. It's one of the first topics presented, so I can't refer to more advance topics like cardinality, the only things introduced are the ZFC-axioms, the peano axioms and how functions/tupels are defined (and russles antiome, but i don't see it being useful here ;) ).
I would like to give an (impractical) practical example of how Von Neumann-Ordinals and functions work by defining an addition function (a set of ((x,y),z) representing x+y=z or +(x,y)=z).
This is my first idea in predicate logic:$$\exists A:((\emptyset,\emptyset),\emptyset)\in A \wedge (\forall ((x,y),z) \in A:(S(x),y),S(z)) \in A \wedge (x,S(y)),S(z)) \in A)$$
My problem with this definition is that i don't know how i can demand that it's only containing the things i want it to. The Peano axioms eliminate this by having $$S(m)=S(n) \to m=n$$ (i think), but i don't see how i can provide a similar construct here. Ideally i would like to have a set-comprehension, but i am not able to find/create one that does not use the not formally introduced concepts like cardinality.