Nevertheless there is this useful notion of addition for ordinal numbers, as well as multiplication, exponentiation and so on. And so the 'omega + 1' notation i

Nevertheless there is this useful notion of addition for ordinal numbers, as well as multiplication, exponentiation and so on. And so the 'omega + 1' notation is common. It is an ambiguous notation though -- you need to know whether it's ordinal arithmetic, or cardinal arithmetic, being used. Usually it's clear from context; and of course the two coincide on the plain old natural numbers. http://en.wikipedia.org/wiki/Ordinal_arithmetic http://en.wikipedia.org/wiki/Ordinal_arithmetic for anyone interested in investigating further. One interesting fact is that while ordinal arithmetic can satisfy a lot of the properties you'd expect, it's not commutative in general: omega+1 > 1+omega = omega. As for whether or not ordinals have the right to be considered 'magnitudes' in a sense, or whether only cardinals deserve that name. I think this stuff already strays so far from the realms of normal human intuition about magnitude that this would be more just an argument over definitions than anything else.