User blog comment:Ecl1psed276/Star Notation Revamped! - Introduction and Analysis/@comment-35470197-20190424003652/@comment-35470197-20190424024948

> Why do you not doubt the termination of Dropping array notation and Taranovsky's C?

I do not believe that DAN terminates.

On the other hand, it is more reasonable to believe that TON terminates, because it forms an ordinal notation system under an extension of ZFC according to Taranovsky's statement. Mathematicians widely believe the \(\Sigma_1\)-consistency of reasonable set theories. Under the assumption, Taranovsky's statement implies that the pointwise (non-total) termination of TON can be verifiable under ZFC.

It is not the same case when you consider DAN, BMS, and your notations. They do not form ordinal notations by definition, I do not have reasons why they terminate.

I know that many googologists believe their termination, but when I heard their statements, I noticed that they were working on wrong "understanding" of OCFs and ordinal notations. For example, BMS is often said to be an ordinal notation. Moreover, hyp cos, who is one of the greatest googologists using OCFs, actually misunderstood how to define an ordinal notation system associated to an OCF. Since statements on terminations under wrong "understanding" is not relaible, I do not believe their termination. UNOCF, which has been used for "analysis", has never been formalised above sufficiently large ordinals.

> Why do you not doubt the termination of those notations, but request evidence or a proof of termination for mine?

I do not intend to say that I believe that they have infinite loops. I just mean that I have no idea to verify whether they terminate or not. Also, I was interested in why you can believe their termination.

At least, I personally hope that they can be verified to terminate under ZFC set theory, because then I can compare them to my stuffs and safely say that one of my stuff actually goes beyond them.

> Even if a notation is ill-defined or has an infinite loop, we can still analyze it as if it doesn't.

But it is non-sense, because analysis based on a contradiction (i.e. termination of a system with infinite loop) can generate any desired results according to an elementary fact of proof theory. You can state that BEAF goes beyond any computable functions as long as you have "desired expansions" but no formalisation. Of course, such results are useful to find infinite loops or bugs, as long as you explicitly declare what assumptions you are assuming on your analysis.

If you do not declare assumptions, then beginners will assume that you are talking about facts. I remember that you said that FOOT is the fastest function. even though there is no justification of the well-definedness. (Actually, it is ill-defined.) Isn't it so much responsible? If you assume something which none has verified, it is good to declare it.