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?
Given a definition of the natural numbers $N$ define multiplication
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elementary-set-theory
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0I think I would just use the normal approach in Peano Arithmetic, using recursive definition in terms of addition; where addition is similarly defined recursively in terms of the successor function; and where the successor function (*e.g.* in the case of the von Neumann ordinals) is just the function $s(x) = x \cup \{ x \}$. – 2012-09-13
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0$m \cdot n = k$ iff $k \in \omega \wedge ( \exists f ) ( f \text{ is a bijection from } k \text{ onto } m \times n )$. Of course, you now have to prove that $m \cdot n$ always exists. (And I would hardly call this better than the usual recursive definition.) – 2012-09-13