It's two different processes. Normal counting is saying, "take a thing and add one" (a successor ordinal). It's how we get from 42 to 43. If a number can be rea

It's two different processes. Normal counting is saying, "take a thing and add one" (a successor ordinal). It's how we get from 42 to 43. If a number can be reached from zero using just this kind of counting, we call it finite. The new type of counting also allows you to take a different step and say, "take a collection of things and add a new thing at the end" (a limit ordinal). It's how you get from the set of finite numbers to ω. There's nothing that's "one before" ω, but all the finite numbers are before it. Repeat these steps as much as you want (making sure none of your sets are circularly defined), and you're counting with ordinals.