Oändligt minus 1?

Continuing to perform transfinite induction beyond Sω produces more ordinal numbers α, each represented as the largest surreal number having birthday α. (This is essentially a definition of the ordinal numbers resulting from transfinite induction.) The first such ordinal is ω+1 = { ω | }. There is another positive infinite number in generation ω+1:
ω−1 = { 1, 2, 3, 4, ... | ω }.
It is important to observe that the surreal number ω−1 is not an ordinal; the ordinal ω is not the successor of any ordinal. This is a surreal number with birthday ω+1, which is labeled ω−1 on the basis that it coincides with the sum of ω = { 1, 2, 3, 4, ... | } and −1 = { | 0 }.