Pfhorrest wrote:Looking back on the earlier parts of this discussion about "year omega" raises a math question that I guess might belong here:
Can you not do subtraction on transfinite ordinals? "Omega minus one" seems like something that couldn't exist, because omega is by definition the first number after all the finite ordinals, so one less than that would be "the last finite ordinal", and there is no such thing.
ω - 1 does not exist, 1 + ω = ω, and ω + 1 > ω. It's not completely intuitive. You can describe a partial function for subtraction, but it will never work if the implied result would be less than a limit ordinal. In other words "there exist ordinals α,β such that α≥β and that, for no ordinal γ, we have γ+β=α." Note that this implicitly allows 0 to be an ordinal; it's still true. However, "[for] every ordinals α,β such that α≥β, there exists a unique ordinal γ such that β+γ=α). That's why ordinals have unique representation in Cantor Normal Form.
In the example where the only ordinals in question are in [0, 2ω), we could say that ω + 10 happened 5 years before ω + 15, but not how long after year 20 it happened, which is the idea. That's why Tom was talking about disconnected timelines and stuff.