 Multiplication of ordinal numbers corresponds to replacing each element of the second ordering by an entire ordering of the first type. \(\omega\cdot\omega\) represents the order type of an \(\omega\)-sequence of \(\omega\)-sequences, which we can have by taking the positive rational numbers and sorting them first by denominator, and then by numerator: 1, 2, 3, … 1/2, 3/2, 5/2, …, 1/3, 2/3, 4/3, 5/3, …, 1/4, 3/4, 5/4, …. Note that \(\omega\cdot 2\) is 2 copies of \(\omega\), while \(2\cdot\omega\) is \(\omega\) copies of 2. \(\omega\cdot 2\) is thus \(\omega+\omega\), while \(2\cdot\omega\) is \(2+2+2+\dots\), which is \(\omega\). (This is the difference between the orderings 1, 2, 3, …, −1, −2, −3, … and 1, −1, 2, −2, 3, −3, ….) So again, ordinal multiplication is not commutative.