  The ordinal numbers have the following structure: every ordinal number has an immediate successor known as a successor ordinal; and for any infinitely ascending sequence of ordinal numbers, there is a limit ordinal which is greater than all the members of the sequence and which is not the immediate successor of any member of the sequence. Thus the following are successor ordinals: 5, 178, ω+12, (ω×5)+56, ω2+8; and the following are limit ordinals: ω, ω×2, ω2, (ω2+ω), etc. Given a limit ordinal η, a sequence S of objects is an η-long sequence if there is an object Sδ for every ordinal δ &lt; η. We will denote the class of ordinals as  On.   Any sequence S of objects is an  On-long  sequence if there is an object Sδ for every ordinal δ.