Ordinals and fusibles - why write $2\omega^2$? I was reading through this pdf on fusibles, and on page 16, ordinals are introduced.
I understand the concept, but my issue is with the notation. Rather than $2\omega^2$, should the author not have written $\omega^22$? From what I've understood by reading the Wikipedia page, $2\omega^2$ is the same as $\omega^2$, as there is an isometry between their elements. On the other hand, $\omega^22$ is bigger.
 A: To address the title of your question: it really just depends on how you choose to define ordinal multiplication.
In the majority of the literature (certainly that which I've seen), ordinal multiplication is defined as a lexicographic ordering on the Cartesian product of two ordinals where the least significant position is compared first. 
Under this definition, it makes sense to denote the ordinal $\sup\{\omega + n \mid n \in \omega\}$ as $\omega\cdot2$ since we would order the set $\omega \times 2$ as: 
$$(0, 0), (1, 0), (2, 0), (3, 0),\ \dots,  (0, 1), (1, 1), (2, 1), (3, 1)\ \dots$$
Writing $2\cdot\omega$ would really imply the ordering of $\omega \times 2$ to be:
$$(0, 0), (1, 0), (0, 1), (1, 1), (0, 2), (1, 2), (0, 3), (1, 3),\ \dots$$
which just has order-type $\omega$.
MathWorld's article on ordinal multiplication notes a couple of sources that use a lexicographic ordering where the $most$ significant position is compared first, i.e.:
$$(0, 0), (0, 1), (0, 2), (0, 3),\ \dots,  (1, 0), (1, 1), (1, 2), (1, 3),\ \dots$$
So using this definition of lexicographic order, the ordinal $\sup\{\omega + n \mid n \in \omega\}$ would sensibly be denoted $2\cdot\omega$. Perhaps  this is what the author of the presentation had in mind. 
