I am trying to get my head around the two different definitions of multiplication of ordinals.
The first definition is with transfinite induction, which I found pretty easy to understand, as well as all the properties that derive from it.
The other definition is with antilexicographic order, it states that $a \cdot b=ord(A\times B,<_{alex})$. More precisely, we take the cartesian product $A \times B$ and it says that: $$(x_1,y_1)<(x_2,y_2) \iff (y_1<_By_2) \lor (y_1=y_2 \land x_1<_Ax_2)$$ Does that mean $(\omega, 2) <(2,3)$ as $2<3$? If so, is there any way to make this clearer than just using the definition? cause so far the $2$ different definitions for multiplication seem very distant to me.... I'm sure I have misunderstood something about the connection of the multiplication and the antilexicographic order, but there were no further examples on my notes on this topic and every book I searched just mentioned the properties and moved on.
As an example consider $\omega \times 2$, and lay it out as a sequence:
$$(0,0) < (1,0) < (2,0) < \ldots < (0,1) < (1,1) < (2,1) <\ldots$$
What we have done essentially is take $2$ copies of $\omega$ and lay them one after the other (in order type $2$).
Similarly: to visualise $\alpha \times \beta$: Lay-out the ordinals of $\beta$ in their order type, and replace every ordinal in $\beta$ by a copy of $\alpha$ (that copy in itself is ordered just as $\alpha$, and members from different copies are compared based on the ordinal they're "replacing" as it were).
So we see that $2 \times \omega = \omega$ (lay out the natural numbers and replacing every one by two discrete copies, makes for the same order type), while $\omega \times 2$ is just the limit of $\omega+n$ as $n$ grows.