User blog:P進大好きbot/Proposal to Choose an Official Standard OCF in This Wiki

Although I am afraid that few are interested in this topic now, there have been many arguments on an official standard OCF in this community. Right, there is no common agreed-upon choice of a standard OCF which is "the most standard", while many googologists use "\(\psi\)" without specifying what \(\psi\) they are referring to. Then we have many problems as I recall in the first section.

Issues
First, average googologists comfound distinct OCFs, because many of us do not know more than three well-defined OCFs. For example, when we see an occurrence of \(\psi\) in an analysis, then we need to guess what \(\psi\) is used. If the behaviour of the \(\psi\) is quite similar to that of a specific \(\psi\) which we know, then it might be the one. However, it might not be the one.

Is it problematic? Isn't it sufficient for us to ask the definition of the \(\psi\) to the author of the analysis? It might have been correct, if it were the day when many experts were active in this community. They would appropriately answer definitions or references of the \(\psi\) which they used. Unfortunately, it is not the case. Many analysts in the current googology even do not know definitions or references of the \(\psi\) which they are using. How many time did I experienced to receive wrong informations such as "It is competely the same as Buchholz's OCF in this realm" and "It is Rathjen's standard OCF based on the least weakly Mahlo cardinal"? When I replied "but it does not satisfy such a weird property", then they just said "Then it is not completely the same as Buchholz's one" or "Then it is not Rathjen's one". Where is the definition? In order to be responsible for our analysis, I recommend us to use what we actually know. For this purpose, it is good to specify what \(\psi\) we are using and what hypotheses we are assuming without reasonable evidences.

The existence of fake OCFs, which are ill-defined and non-sense, also makes the situation worse. They are not strong, or even not weak. Simply they do not make sense. Estimations like "My OCF is significantly stronger than Rathjen's \(\psi\)" for your OCF which you have never written down my well-defined OCF simply because it is tiresome and not interesting for you is worse than doing nothing, because beginners might believe such a non-sense statement through the process of looking around what they need to study. Then they might feel your OCF so attractive, and study it through finitely many written examples. After then, when they need to apply it to a specific notation, they finally notice that they have never read the original definition, which is needed in order to analyse it. They ask "where can I read the precise definition?", but will be disappointed to understand that what they learned is just non-sense. How long time do they spend? It might be more than years, because even "professional" googologists do not understand the issue. For more details on these issues, see this, in which I explained how dangerous analyses based on unspecified OCFs without associated ordinal notations and how non-sense analyses based on ill-defined tools such as UNOCF, Pi-notation, BEAF, and Catching function are.

Also, it is important to choose an OCF instead of computable notations such as BM2.3 or SAN. Although many analysts misunderstood the relation between the computaility and the well-foundedness, a computable notation is not necessarily well-founded. Statements like "BM2.3 is terminating because it expands in a natural way" or "SAN is terminating because there has never been a known infinite loop" are wrong. The naturality of an expansion or the non-existence of a known infinite loop does not ensure the termination. On the other hand, an ordinal notation associated to an OCF is always well-founded, and a system of fundamental sequences on an ordinal notation is always terminating. That is why it is good to use an OCF in analyses.

Proposal to Choose One Standard OCF
Isn't it good to officially decide to choose one standard OCF as "the most standard" OCF? If all analysts in the community using OCFs get accustomed to it, then we do not have to suffer from that issue. Here, the notion of a "standard OCF" is quite ambiguous, because several googologists state "my OCF is standard!" or something like that. In order to clarify the requirement, I would like to consider a well-defined OCF satisfying the following conditions: Of course, when we decide to choose one standard OCF, then we need to check whether the definitions and the algorithms actually work as intended. If you do not know the relation between OCFs and ordinal notations, see this.
 * 1) The defintions of the data below are published in an external source. For example, it should not be a work which is defined only in a wiki article which can be freely updated.
 * 2) The OCF \(\psi\) itself. (It should be stable, and hence need an external source.)
 * 3) The relation \(=_{\textrm{NF}}\) characterising a normal form expression with respect to \(\psi\). (It should be stable, and hence need an external source.)
 * 4) Definition of the date below are published in some places which are freely accessible.
 * 5) A system \([ \ ]\) of fundamental sequences below the least ordinal which does not admit a normal form expression with respect to \(\psi\). (Since the difference of fundamental sequences rarely causes problems in actual analyses, there can be multiple choices of them.)
 * 6) An ordinal notation \(OT\) associated to \(\psi\) is published somewhere accessible. (Since the difference of specific algorithms to compute \(\in\) rarely causes problems in actual analyses, there can be multiple choice of them.)
 * 7) An algorithm to compute the system \(\textrm{Expand}\) of fundamental sequences on \(OT\) associated to \([ \ ]\). (Since the difference of specific algorithms to compute \([ \ ]\) rarely causes problems in actual analyses, there can be multiple choice of them.)
 * 8) Informally, \(\textrm{Expand}\) can be computed in terms of copying and nesting of expressions. For example, the system of fundamental sequences directly defined by using the length or the complexity of expressions, which always exists as we fix \(OT\) as I explained here, is not so useful in actual analyses.

The reason why I think that it is good to fix one standard OCF is because choosing two or more standard OCFs can cause confusion. For example, something like "In this analysis, we use Buchholz's OCF up to its limit, and use Rathjen's OCF based on the least weakly Mahlo cardinal after then." is troublesome because expressions such as \(\psi(\Omega)\), which is a common abbreviation for \(\psi_0\(\psi_1(0))\) when we deal with Buchholz's OCF and also is a common abbreviation for \(\psi_{\Omega}(\chi_0(0))\) when we deal with Rathjen's OCF, can possess two distinct meanings in a single analysis. It might not be so problematic for average analysts, but beginners will get confused.

Of course, even if we officially decide to choose one standard OCF, we can use other OCFs, because there should not be any restrictions on our sincere activities in googology. This is just a recommendation to study and use the chosen standard OCF. Moreover, even if we use the chosen standard OCF in analyses, it is better to clarify that we are using the chosen standard OCF in order to help beginners to study the results.

Candidates
Many googologists love UNOCF, but UNOCF is known to be ill-defined. Therefore it and can never be a candidate. Using UNOCF in analysis is something like stating "My number is the greatest, because it is the ultimate infinity such that no greater object can exist". For more detals on the historical background, see this. Many googologists also love 2000 steps analysis, but the definitions of the OCFs appearring in the table are not available in English. According to the descriptions in author's textbook in Russian, the bahaviour of the OCFs seem to differ from other well-known OCFs, and hence it is not so easy to guess the actual definitions from the table. Moreover, the table itself includes errors, and hence can never be an alternative of definitions. Bashicu invented two OCFs, but they seem to have never been defined. At least, Bashicu estimated (0,0,0,0)(1,1,1,1) in BM2.3 using his newer OCF, then it would be the strongest OCF in the world if it were actually well-defined. Even if it is intended to be stronger than any other OCFs, the lack of the precise definition is crucial. Hyp cos has defined many OCFs, but ordinal notations associated to them have never been defined according to him. Deedlit also introduced several OCFs, but ordinal notations associated to them seem to have never beed defined, either. Moreover, at least one of them has turned out to be intensionally completely the same as Rathjen's OCF except for the position of an index, although Deedlit somewhy did not refer to the original source and many googologists believed that it was his own work. In order to keep the community sound, we should respect the original creators. Thus none of them can be candidates in the context in this blog post even if they have external sources.

How about Madore's OCF, Buchholz's OCF, Rathjen's OCFs, Stegert's OCFs, and Arai's OCF? Among many great candidates, I propose extended Buchholz's OCF. It is sufficiently strong, as almost all computable notations which are known to be well-founded is dominated by the limit of extended Buchholz's OCF. Another merit is that its definition is quite sophisticated and is pretty simple, while stronger OCFs such as Jager's and Rathjen's are seriously complicated. On the other hand, since it does not employ Veblen function as a base function, it is good if we have an official analysis on extended Buchholz's OCF versus Veblen function without using non-normal form expressions such as \(\zeta_0\) and \(\Omega^2\). I explain the importance of this restriction in the next section.

Proposal to Use Normal Form Expressions in Analyses
In order to analyse a computable notation by an OCF, we need to consider "the constructibility", which is formalised by the maps dnoted by \(B\) or \(C\), in order to compute the actual value of \(\psi\). Otherwise, we will be suffered from wrong equalities such as \(\psi_0(\varepsilon_0+1) = \psi_0(\varepsilon_0) \times \omega\) and \(\psi_0(\psi_1(\psi_2(\psi_3(0)))) = \psi_0(\psi_3(0))\) for Buchholz's OCF. Since we do not have to care about such traps along as we only use normal expressions by precisely checking the constructibility, it is reasonable to only use normal form expressions, isn't it?

There might be some troubles. For example, forbidding to use \(1\) or \(\Omega\), which are not portions of normal form expressions, for Buchholz's OCF causes the problems that the result of the analysis will be stressfully redundant. In order to make analyses comfortable, I recommend to use "semi-normal expression" in the following way: For example, if you are working on Buchholz's OCF, then you can abbreviate \(\psi_0(0)\) to \(1\), and resolve the abbreviation by replacing all the occurence of \(1\) outside the index of \(\psi\) by \(\psi_0(0)\). For more detail see the introductions of the notions of semi-standard expressions for various OCFs here.
 * 1) Explicitly define abbreviations of normal form expressions. (There should be an algorithm to resolve the abbreviations.)
 * 2) Define the notion of a semi-normal expression as an expression such that the expression given by resolving all the abbreviations is of normal form.

I have heard an analyst to say "I do not know how to replace the expressions in my analysis by the corresponding normal form expressions". Then how could he justify the accuracy of the analysis? We need to check the constructibility in order to compute values of \(\psi\) instead of "pattern matching" of expressions.