User blog:P進大好きbot/Historical Background of the Ill-definedness of UNOCF

Since I am repeating the explanation about issues on UNOCF so many times, I summarise the historical background of UNOCF. For more a brief explanation, see this blog post.

= Original Definition =

UNOCF is originally introduced by Username5243 here. Although UNOCF is intended to form an OCF, the definition has many issues, which make UNOCF ill-defined.

Uncountable Ordinal
UNOCF uses an unspecified function \(f\) in the definition of \(\psi(\alpha)\) for an ordinal \(\alpha\) of cofinality \(\Omega\). The function \(f\), which depends on \(\alpha\) because it is required to satisfy \(f(\Omega) = \alpha\), is not unique at all, and the result of \(\psi(\alpha)\) heavily depends on the choice of \(f\). Even if we are allowed to apply any choice of \(f\), the resulting \(\psi\) can behave awfully different from usual OCFs because \(f\) is not required to be strictly increasing or continuous.

Stage Cardinal
UNOCF uses the stage cardinal \(T\), which is ill-defined by the reason explained here. It is intended to diagonalise the sequence "\((I_{\alpha},M_{\alpha},K_{\alpha},\ldots)\)" of large cardinal axioms, but there is no known reasonable sequence of large cardinal axioms whose first three entries are \(I_{\alpha}\), \(M_{\alpha}\), and \(K_{\alpha}\). Namely, what the creator did is just showing the first three entries and behaving as if they were portion of some existing sequence.

Guessed Expansions
Since UNOCF is not fully defined, many googologists tried to guess how it is supposed to work. However, there is also a problem. I recall that they often state that the benefit of UNOCF is that it is easier to understand than usual OCFs. Actually, few googologists understand usual OCFs based on higher inaccessibility, because those definitions are difficult. Therefore the guessed expansions are mainly given by googologists who do not understand the definitions of usual OCFs. As a result, expressions of UNOCF are guessed to expand in an awfully different way from usual OCFs. Although it is irrelevant to the original definition of UNOCF itself, the guessed unreasonable expansions have been regarded as "official" expansions. The situation is quite mysterious, but it is evidently due to the lack of the formality in the original definition of UNOCF.

For example, what is the worst "official" expansion is \(\psi(\varepsilon_0+1) = \psi(\varepsilon_0) + \psi(\varepsilon_0) + \psi(\varepsilon_0) + \cdots\). It requires UNOCF to be completely different from usual OCFs, although googologists who do not understand usual OCFs often state "It coincides with Buchholz's OCF in this realm". Needless to say, Buchholz's OCF does not satisfy such a weird property. Then why do they believe that UNOCF in this realm coincide with Buchholz's OCF? It is simply because they do not know usual OCFs such as Buchholz's OCF. Actually, they sometimes state that \(\psi(\varepsilon_0+1) = \psi(\varepsilon_0) + \psi(\varepsilon_0) + \psi(\varepsilon_0) + \cdots\) holds for Buchholz's OCF "because" it is the same as UNOCF. Right, this is a typical circular logic.

"Why does UNOCF expand in this weird way?"

"Simply because it is the same as usual OCFs such as Buchholz's OCF. Why couldn't you understand such a trivial thing?"

"You mean, Buchholz's OCF expands in that way, right?"

"Yes. It expands in this way, because it is the same as UNOCF! I WIN!!!!"

= Alternative Definition =

Since there are many issues in the original definition, several googologists are trying to formalise it.

As an OCF
There are no justification of UNOCF as an actual OCF, because weird expansions such as \(\psi(\varepsilon_0+1) = \psi(\varepsilon_0) + \psi(\varepsilon_0) + \psi(\varepsilon_0) + \cdots\) are unfortunately regarded as "official" expansions of UNOCF by googologists who believe that UNOCF can work. Why do they insist such expansions, which prevent a reasonable formalisation? Simply because such expansions are believed to be easier to understand than those of usual OCFs. The problem is that googologists who believe that UNOCF were the greatest OCF do not know usual OCFs well, and hence do not know a way to define an actual OCF.

As a Notation
There are many trials to formalise UNOCF as a notation, which can be defined in arithmetic. However, in order to defined an expansion of an expressions of the form \(\psi(\psi_{\Omega_{a+1}}(\Omega_{b+1}))\), we usually need to define a binary relation \(a < b\). Therefore when we deal with an OCF-like recursive notation above \(\psi_0(\Omega_{\omega})\) with respect to Buchholz's OCF, we need to equip it with an explicit algorithm to determine whether \(a < b\) holds or not.

There is a deeply related notion called an ordinal notation. Since the definition of an ordinal notation is widely misunderstood in this community, please see this blog post if you do not know the precise definition. In order to create an ordinal notation, we usually need an actual OCF or much more complicated mathematical theories. I recall that they are trying to formalise UNOCF into an arithmetic notation because it is really difficult to formalise it into an actual OCF. Therefore the usual strategy to create an ordinal notation is useless for this purpose.

The only well-defined stuff which formalises UNOCF is NIECF, which is created by Nayuta Ito here up to what is denoted by \(\psi(I)\) in UNOCF. How about other attempts? They are ill-defined by many reasons. For example, many of them lack the declaration of the set of expressions, include functions without declaration of domains, use overloaded equality, rely on unwritten rule sets based on pattern matching, and so on. At least, I have never seen an example other than NIECF such that an algorithm for \(<\) is completely defined. It is just kept WIP, or omitted as if it is trivial. I have seen circular logic on \(<\) so many times.

"Your rule sets of expansions includes \(<\). What is the definition of \(<\)?"

"Obviously, it is the comparison of ordinals. Why couldn't you understand such a trivial thing?"

"But you are formalising UNOCF into a notation, in which no ordinals are allowed to involved."

"Right. It precisely means the comparison of the ordinals corresponding to expressions. Why couldn't you understand such a trivial thing?"

"What does the ordinal corresponding to an expression in your notation mean?"

"It is the strength. More precisely, an expression \(s\) corresponds to an ordinal \(\alpha\) if the FGH at \(s\) is approximated by the FGH at \(\alpha\). Why couldn't you understand such a trivial thing?"

"But in order to define the FGH, you need to define the full rule set of expansions. Referring to the FGH in the definition of expansions is circular logic."

"To be more precisely, we do not need the FGH. I just used the FGH in order to make you understand well. For example, define a map \(o\) which assigns an expressing \(s\) to an ordinal \(o(s)\) in the following recursive way: If \(s = 0\), then \(o(s) = 0\). If \(s\) is the successor of \(t\), then \(o(s) = o(t) + 1\). When \(s\) is a limit, then \(o(s)\) is the limit of \(o(s[n])\) on \(n \in \mathbb{N}\). Quite obvious. Why couldn't you understand such a trivial thing?"

"But the definition of \(o(s)\) includes \(s[n]\), which should be defined by the rule set for expansions. Refering to \(o\) in the definition of expansions is a circular logic."

"Hey, stop it. Why should I help a beginner to understand such a trivial thing? Since UNOCF works, expressions in UNOCF corresponds to ordinals! I WIN!!!!"