User blog comment:Edwin Shade/A Small Question/@comment-30754445-20171029013412

x=x+1 has no fixed point. The successor of any ordinal is always bigger than the ordinal itself.

And on an unrelated note: is still countable. "countable" and "recursive" are two different terms.

An ordinal x is recursive if you can (in principle) write a computer program that can do arithmetics on any ordinal smaller than x.

An ordinal x is countable if there's a 1-to-1 correspondence between the set of ordinals less than x and the natural numbers. Or putting it differently: An ordinal x is countable if you can "tag" every ordinal smaller than x with a unique natural number.

Now, it is easy to see that any recursive ordinal must be countable. Why? Because in order to write a computer program that can manipulate "any ordinal smaller than x", we must have a way to represent these ordinals as numbers in the computer memory. So by definition, any such ordinal must be countable.

The reverse, however, is not true. The fact that we can "tag" every ordinal below x with a unique number, does not gurantee that we could do actual computations with these numbers. It can be proven, for example, that no computer program would be capable of analyzing the set of ordinals below . Indeed, the existence of such a program would lead to a logical paradox.

Yet the ordinal  is still countable. For an example of how to "tag" all the ordinals below with numbers, see "Kleene's O" notation. Note that since is non-recursive, this tagging system is noncomputable: No computer program could ever tell you which Kleene's O index represents which ordinal.