> There is no infinity at the end of a number line. There is a process that says how to extend that number line ever further.

Sure, but ordinal numbers exist and are useful. It's impossible to prove Goodstein's theorem without them.

https://en.wikipedia.org/wiki/Ordinal_number

https://en.wikipedia.org/wiki/Goodstein%27s_theorem

The statement and proof of the theorem are quite accessible and eye-opening. I think the number line with ordinals is way cooler than the one without them.