"The lack of aftereffect" is supposed to capture the idea that given any initial segment of a sequence, one cannot predict the probability of the next event any better than predicting it based on the limiting relative frequency of events in the infinite sequence.
If μ is sufficiently long then there is an initial segment σ of μ such that C < |σ| − k.
By the very nature of the notion of lawless sequence, the choice of the number $n$ for which $A(\alpha,n)$ holds has to be made after only a finite initial segment of $\alpha$ is known.
Thus ON would be similar (order-isomorphic) to a proper initial segment of itself, the one determined by Ω, contradicting a well-known theorem about well-ordered sets.
We thus find that the order type of the restriction of the natural order on the ordinals to the ordinals less than Ω is T2, whence we find that T2 (as the order type of a proper initial segment of the ordinals) is strictly less than Ω (which is the order type of all the ordinals).
