What's the relationship between ZFC and Peano axioms? Are they overlapping, complementary or separate?
I'm particularly wondering whether natural number stuff like order can expressed in ZFC at all. Perhaps there's some way to associate "smaller" and "larger" numbers as cardinals or something.
Best Answer
The two system are not related in a direct way. Peano axioms come to axiomatize arithmetic, whereas Zermelo–Fraenkel (with or without choice) come to axiomatize the properties of the set theoretic universe.
Neither pretends to "express what is a natural number". For arithmetic, it is us who said that we have a concept of a natural number, and it seems to satisfy these properties. And the properties were later distilled into Peano axioms, second-order and then the first-order version.
Set theory can be used to encode the natural numbers, usually this is done via the von Neumann ordinals: the finite ordinals satisfy all the axioms of Peano. But since ZFC provides us with an infinite set, which might as well be the set of all finite ordinals (and if not, then it provides us with the means of proving that the collection of all finite ordinals is in fact a set), it lets us do more. For example, since there is a model of Peano axioms, the theory is proved to be consistent, something you cannot do from Peano itself.
Even more than just that. ZFC lets us interpret the second-order Peano axioms. And in that sense, it is much stronger in defining what is a natural number. Of course, encoding "objects" into "sets" is not by any means unique. There are many ways we can think about the natural numbers as being sets, some are more natural and others less, but they are all valid methods.
What ZFC does prove, however, is that any two models of second-order PA are isomorphic in a unique way. And in that sense, ZFC can actually say what is a natural number.
One last thing, perhaps, is the fact that replacing the Axiom of Infinity with its negation, one transforms ZFC into a theory which is equivalent to first-order Peano axioms in a very strong sense.