User blog:LittlePeng9/Fundamental sequences for large ordinals

This is just a short blog post in which I'll describe how to, consistently, construct fundamental sequences for ordinals far beyond \(\omega^\text{CK}_1\).

This method makes use of Hamkins' infinite time Turing machines. To make everything unambiguous, blog:LittlePeng9/ITTM galore here are all technicalities connected with machines themselves, and blog:LittlePeng9/Encoding ordinals using binary strings here is a way of representing ordinals as binary strings.

Up to \(\lambda\)
\(\lambda\) is my favorite greek letter and smallest ITTM ordinal at the same time, so we will start with it.Here is the definition:


 * Let \(\alpha\) be a limit ordinal such that \(\omega<\alpha\leq\lambda\). Then we define \(\alpha[n]\) to be the largest ordinal smaller than \(\alpha\) writable by some \(n\)-state ITTM.

Why does this definition work? We know that writable ordinals don't have gaps (if you haven't seen a proof, try it yourself! It's not too hard) and \(\lambda\) is, by definition, supremum of all writable ordinals, then all infinite ordinals below \(\alpha\) are writable. But, as \(\alpha\) is limit, and there is only finitely many \(n\)-state ITTMs, only finitely many of these ordinals are candidates for \(\alpha[n]\). But we nevertheless can write unbounded in \(\alpha\) number of ordinals, so we indeed get a fundamental sequence for \(\alpha\).

Up to \(\gamma\)?
One could think that analoguous definitions for ordinals below \(\gamma\) would be possible:


 * Let \(\alpha\) be a limit ordinal such that \(\omega<\alpha\leq\gamma\). Then we define \(\alpha[n]\) to be the largest ordinal smaller than \(\alpha\) clockable by some \(n\)-state ITTM.

The problem is: clockable ordinals have gaps. First gap starts at \(\omega^\text{CK}_1\) and then \(\omega^\text{CK}_1+\omega\) is clockable. So, even if we limit ourselves to clockable \(\alpha\), this won't work. For example, every sequence unbounded in \(\omega^\text{CK}_1+\omega\) would have elements of form \(\omega^\text{CK}_1+k\) for \(k\) finite. But no such ordinal is clockable. So this won't work.

Fortunatelly, \(\gamma=\lambda\), so we don't have to worry.

Up to \(\zeta\)

 * Let \(\alpha\) be a limit ordinal such that \(\omega<\alpha\leq\zeta\). Then we define \(\alpha[n]\) to be the largest ordinal smaller than \(\alpha\) eventually writable by some \(n\)-state ITTM.

Does this work? Yes it does! Eventually writable ordinals also don't have gaps (argument similar to one for writable ones, but a bit more tricky!), so reasoning just like in case of \(\alpha\leq\lambda\).

Up to \(\Sigma\)?

 * Let \(\alpha\) be a limit ordinal such that \(\omega<\alpha\leq\Sigma\). Then we define \(\alpha[n]\) to be the largest ordinal smaller than \(\alpha\) accidentally writable by some \(n\)-state ITTM.

Does this work? Yes it doesn't. Wait, what

Accidentally writable ordinals don't have gaps either (argument just like for eventually writable), true, but there is a major difficulty: for some \(n\), we get infinitely many ordinals accidentally writable by machines with \(n\) states. Simple example is a machine writing \(\omega+k\), erasing it, writing \(\omega+k+1\) and looping like that. One could replace "largest ordinal" by "upper bound of blah blah", but this won't solve above problem, as we'd have \(\omega2[n]=\omega2\), which is nonsense. However, I have an idea on how to fix this:

Coming spoon