User blog comment:Ytosk/Trying to define TBMS/@comment-35470197-20191016101622

In order to define a binary relation, you need to define its domain. Otherwise, it is still ill-defined.

> Finite columns compare in the same way as in BMS

There are many versions of BMS. If you are referring to the original BMS, i.e. BM1, then it is known to have an infinite loop. Therefore you need to specify the version.

> (n1,n2…nm,I…)

What is I? Your notation does not include I. Is is any natural number or the empty string? When you use a new symbol without any specific definition, you need to quantify it with any conditions.

> "..." either means completing an obvious sequence or that there can be anything that doesn't cause a syntax error

Does ... include only numbers? It can never be obvious, because you have not defined the domain of the binary relation. Moreover, such an ambiguous abbreviation is not allowed in a mathematical definition.

> it is greater than l,

What is the definition of the comparison to I? For example, when I is the empty string (namely, I = ), the comparison is ill-defined.

> For sequences without subscripts, we compare them by choosing a natural number n greater than 1, changing the sequences into matrices and solving them with n as input. The sequence with the bigger output represents a larger transfinite length.

Are you assuming not only the termination of your TBMS, but also the linearity of the growth rate? Otherwise, such a comparison does not give a total ordering. Please clarify what assumptions are you employing in the definition.

Moreover, using the resulting function in the definition of the comparison is a circular logic, because the function uses the full definition of the comparison as long as you set a suitable ordering of the simultaneous recursive definitions.

> formal enough to not be ill-defined

I believe that you have a formal definition inside your mind, but I think that you have not fully expressed what you think. I mean, it is a little informal. How about writing the rules in terms of actual algorithms or mathematical formulae?