2 Fixed typo "countable" for "computable"

This isn't an answer, but it's too long for a comment.

I don't think the countable computable ordinals are well enough defined for the function $f(n)$ to work. Suppose you give me a system mapping {$0,1$}$^*$ into the computable ordinals. I'll give you a system, which is your system together with a new symbol, $2$, which stands for the smallest ordinal you can't define in your system. I can then map the numbers {$0,1,2$}$^*$ back into {$0,1$}$^*$; my system reaches a computable ordinal that's not defined in your system, and it even defines it with length 2.

So the computable ordinals are a concept that makes sense, but it is impossible to have a single encoding that gives you all of them. Thus, I don't see how your function $f$, which is defined using the phrase

Next, let $\alpha(n)$ be the largest computable ordinal that can defined (in the sense above) by a Turing machine with at most $n$ states.

works. You should be able to get a Turing machine with an oracle that corresponds to any countable computable ordinal, but that's where it stops.

1

This isn't an answer, but it's too long for a comment.

I don't think the countable ordinals are well enough defined for the function $f(n)$ to work. Suppose you give me a system mapping {$0,1$}$^*$ into the computable ordinals. I'll give you a system, which is your system together with a new symbol, $2$, which stands for the smallest ordinal you can't define in your system. I can then map the numbers {$0,1,2$}$^*$ back into {$0,1$}$^*$; my system reaches a computable ordinal that's not defined in your system, and it even defines it with length 2.

So the computable ordinals are a concept that makes sense, but it is impossible to have a single encoding that gives you all of them. Thus, I don't see how your function $f$, which is defined using the phrase

Next, let $\alpha(n)$ be the largest computable ordinal that can defined (in the sense above) by a Turing machine with at most $n$ states.

works. You should be able to get a Turing machine with an oracle that corresponds to any countable ordinal, but that's where it stops.