> I don’t really know how to work with the definition of ordinal arithmetic in terms of successors and limits.<p>What a shame... I think the dude is overcomplicating stuff just by not understanding this.<p>As an analogy, the naturals are the closure of 0 being a natural number, and to each natural number, having a function that gives you another number not in the set, as the "next" natural number (called the successor of that original number).<p>The class of ordinals are just the closure of having 0 be an ordinal, and to each *set* of ordinals being able to get the "next" ordinal, (i.e. the smallest ordinal that comes (not necessarily immediately) after all elements in the set). Then omega is the "next" ordinal wrt the natural numbers (aka finite ordinals), and that's what's meant by a limit ordinal, defined by exclusion, as not the "next" ordinal of any set of ordinals that has a maximum element (the natural numbers have no maximum element, but the natural numbers and omega together have omega as the maximum element); the "next" ordinal of a set that has a maximum element is equally the "next" ordinal of the set containing just that maximum element, and we can do a +1, just as with natural numbers, hence we call it successor ordinal.