User blog comment:MilkyWay90/Help with ordinals/@comment-35470197-20180709220316/@comment-30754445-20180718021905

"sup" is shorthand for "the smallest number/ordinal which is still bigger than anything on this list"

So:

sup  {1, 1.9, 1.99, 1.999, ...} = 2

And

sup {1, 2, 3, 4, 5, ...} = ω