 Cantor used lowercase Greek letters to represent infinite ordinal numbers, with \(\omega\) representing the order type of the standard ordering of the natural numbers. Addition of ordinal numbers corresponds to the order type that results from taking an ordering of the first type, followed by an ordering of the second type. Thus, \(\omega+\omega\) represents the order type of the integers with the non-negative numbers first and the negative numbers afterwards, while \(\omega+1\) represents the order type of the natural numbers with just one element put at the end. Note that \(1+\omega\) is the order type of a single element, followed by a copy of the natural numbers, which is in fact the same as the order type of the natural numbers! Thus, \(1+\omega=\omega\), which is not equal to \(\omega+1\). So ordinal addition is not commutative.