An infinite natural sum

As far as algebraic properties are concerned, the usual addition on the class of ordinal numbers is not really well behaved; for example, it is not commutative, nor left cancellative etc. In a few cases, the natural Hessenberg sum is a better alternative, since it shares most of the usual properties of the addition on the naturals. A countably infinite iteration of the natural sum has been used in a recent paper by Väänänen and Wang, with applications to infinitary logics. We present a detailed study of this infinitary operation, showing that there are many similarities with the ordinary infinitary sum, providing an order theoretical characterization, and finding connections with certain kinds of infinite mixed sums.