What exactly makes the ordinals an indefinitely extensible concept?

Question

I understand the principles of generation that cantor used to create the ordinals but I cannot see what exactly is the property that makes the ordinals an indefinitely open plurality and not the natural numbers for example. We say that the cardinality of the set of natural numbers is aleph0 but this is not a natural number, if it was a natural number then to my understanding we would run into the same paradox that burali forti discovered about the ordinals themselves. Why don't we say that the natural numbers form a proper class and they form an indefinitely extensible concept but we do that for ordinals. What is the difference?

Answer 1

The intuitive generating principles are different.

For natural numbers, the generating principle is: Given any natural number $n,$ there's a least natural number greater than $n.$ (You need to seed the process by starting with $0,$ or $1$ if you prefer.)

For ordinal numbers, the generating principle is: For any set $X$ of ordinal numbers, there's a least ordinal number that doesn't belong to $X.$ (And no seeding of the process is even needed here, since you can start things going by taking $X=\emptyset.)$

In both cases, the intention is that you include only those things that are generated by the specified process.

This is all just the desired intuition though. It's not meant to be a rigorous way of defining these things. (Nor can these concepts be fully expressed in first-order logic.)