User blog comment:MilkyWay90/Help with understanding Veblen array notation/@comment-30754445-20180811202716/@comment-30754445-20180820005609

Not sure what's your problem here is. Limit ordinals, as their name implies, are the limit of some infinite sequence of smaller ordinals. This is exactly how the concept of "fundamental sequences" work. So the statement "If you tended for x<c to go higher and higher and higher, it'd eventually reach c, and then it'd be equal to c" is not true. You can count 1,2,3,4,... until you're blue in the face and even until the universe ends, but you'll never reach the supremum of all these numbers which is ω.

Here are a couple of explicit examples of rule 3:

φ(1,2,ω) is the smallest ordinal bigger than φ(1,2,n) for all finite n (which is equivalent to saying n<ω). So it's the limit of φ(1,2,1), φ(1,2,2), φ(1,2,3), φ(1,2,4), ...

φ(1,2,ε₀)  is the smallest ordinal bigger than φ(1,2,α) for all α<ε₀. So it's the limit of φ(1,2,ω), φ(1,2,ωω), φ(1,2,ωω ω )...

Does this clarify the issue?