User blog:LittlePeng9/Levels of ITTMs

I got so much into these ITTM levels and \(\tau\) ordinal I defined that I decided to create whole separate blog post for them, partially because blog post linked above doesn't really fit that purpose.

Definitions
I assume familiarity with infinite time Turing machines and all related topics, like accidentally writable ordinals.

Let's call an ordinal \(\alpha\) a level if there exists an ITTM M such that \(\alpha\) is an upper bound of all ordinals accidentally writable by M.

Let's call an ordinal \(\alpha\) achievable if there exists an ITTM M such that \(\alpha\) is accidentally writable by M but no larger ordinal is.

Define \(\tau\) to be the smallest ordinal which isn't a level of any ITTM.

Simple lemmata
Coming noon