User blog comment:Ytosk/Trying to define TBMS/@comment-35470197-20191016101622/@comment-35470197-20191017005605

> When i refer to BMS, i mean the version that works without an infinite loop, which i think is BM2.3

Well, BM2.3 is not known to be terminating. It might have an infinite loop, although nobody has found it.

> Yes, i am currently assuming the termination of TBMS, though i probably should prove that at some point.

The point of my issue is that your definition seems to rely on a stronger assumption such as the linearity of the growth rates of functions asssociated to TBMS. Since BMS is based on the square-variant of Hardy hierarchy, it is quite sensitive even when I restrict the statement to BM2.3.

> I have no idea why i didn't write it in terms of formulae, and i'm not sure i even thought of it before reading this.

Actually we tend to express our ideas in natural languages. But when we deal with mathematical stuffs like transfinite objects, it is much better and more precise to talk in terms of mathematical formulae, which are free from language barrier.

> Could you specify which relation are you refering to?

I meant the binary relation x < y on TBMSs x and y. In order to define <, you need to specify the domain, i.e. what does a valid expression in TBMS first. Moreover, when you indentify specific expressions, then you need to define the equivalence relation which gives the identification. For example, if an expression in TBMS is just a formal string, (1,1,0) and (1,1) are different, and I and are different.