User blog comment:Edwin Shade/Understanding The Infinite/@comment-32213734-20171109082956/@comment-32213734-20171110065258

ОК, let's deal.

sup is supremum. Here is Wikipedia article: Infimum and supremum

Definition of supremum from there: "The supremum (abbreviated sup; plural suprema) of a subset S of a partially ordered set T is the least element in T that is greater than or equal to all elements of S, if such an element exists."

(That is,

if S has maximum then supremum of S is the maximum of S;

if S has not maximum then supremum of S is least element of T larger than all elements of S).

In case of Rules 2, 3 T is all ordinals; S is set of (α + δ) + 1 for all δ < β.

Let β is successor ordinal: β = γ + 1

sup{S} is least ordinal such as sup{S} ≥ λ for all λ ∈ S (by definition)

S has a maximum: max{S} = (α + γ) + 1 since γ is largest ordinal lesser than β

sup{S} = max{S} (any ordinal < max{S} cannot be supremum since max{S} ∈ S; and max{S} satisfies the condition: max{S} ≥ λ for all λ ∈ S)

sup{S} = (α + γ) + 1

α + β = (α + γ) + 1

α + (γ + 1) = (α + γ) + 1

We get Rule 2.

Similarly for Rules 5, 6 and 8, 9. I hope everything is right.

By the way, let α = 0, then from Rules 2, 3

β = sup(δ + 1), δ < β

(looks like a definition...)