Given a definition of the natural numbers $N$ define multiplication

Question

This link has a good answer to how to define the set of natural numbers with only knowledge of set theory. how to express the set of natural numbers in ZFC However, he depends on the operator * for multiplication. I was wondering if it was possible to define multiplication from within set theory without the need of that nor the need to define addition first. My idea was to try and do a Cartesian product, but it ends up being non-commutative. Could anyone tell me if there's a better way according to these criteria?

Answer 1

In fact, in ZFC you can define the cartesian product of two sets (using e.g. Kuratowsky's definition of pair), that is $$A\times B:=\left\{x\in P\left(P\left(\bigcup\{A,B\}\right)\right)\ \Big|\ \exists a\exists b\colon a\in A\land b\in B\land x=\left\{\{a\},\{a,b\}\right\}\right\} $$ then for $n, m\in\mathbb N$ define $n\cdot m$ as the unique element of $\mathbb N$ that can be bijected with $n\times m$. As you note from even writingdown the definition, a lot of other axioms are involved (powerset, union, pairing, comprehension). Also, have fun showing existence and uniqueness and the arithmetic properties (not to mention defining the concepts of function and bijection first). Honestly, recursion is the method of choice for definitions for the set $\mathbb N$ or the class of ordinals. Ordinals are so rich and powerful even in the context of weirdest set theories (e.g. they are always well-founded even if a theory allows non-well-founded sets), it would be a shame to define an operation on them with heacvy machinery instead of recursion.