This is a list of common misconceptions related to Rathjen's \(\psi\) based on the least weakly Mahlo cardinal. Since old members irresponsibly spread wrong informations on \(\psi\) without knowing the precise definition, there are so many common misconceptions, which I repeated to correct so many times. One starting point is to check the precise definition before stating wrong informations. Seriously, please do not believe that you can understand an OCF without reading the definition. See User blog:p進大好きbot/List of common mistakes in googology#I fully understand it although I do not know the precise definition! for details about the cheating.


I first wrote this list in an old version of the article on OCFs, and moved it to Rathjen's psi#Common misconceptions. Since now it is not welcomed to drastically update the article not for its main purpose (due to an admin's proposal, although the new rule has not been officially validated yet), I update the list in my blog post instead.


Contents

Is \(\chi_{\alpha}(\beta)\) defined for any ordinals \(\alpha\) and \(\beta\)?

No. Read the definition.


Is \(\chi\) the same as the binary \(I\)?

No. Although the definition of the binary \(I\) depends on the author, \(\chi\) does not coincide with the binary \(I\) with respect to any reasonable formulation. See also #Is \(\chi_{\alpha}(\beta)\) defined for any ordinals \(\alpha\) and \(\beta\)?.


Does \(\chi_{\alpha+1}(0)\) coincides with the limit \(\chi_{\alpha}(\chi_{\alpha}(\cdots(\chi_{\alpha}(0))\cdots))\) of finite iteration of \(\chi_{\alpha}\)?

No. Read the definition of the regularity.


Is \(\chi_{\alpha+1}\) the enumeration of fixed points of \(\chi_{\alpha}\)?

No. See #Is every value of \(\chi_{\alpha+1}\) a fixed point of \(\chi_{\alpha}\)?.


Is every value of \(\chi_{\alpha+1}\) a fixed point of \(\chi_{\alpha}\)?

No. Read the definition of the closure.


Does the map \(\alpha \mapsto \chi_{\alpha}(0)\) have a fixed point?

No. The non-existence immediately follows from the definition.


Is \(\chi_{\omega}(0)\) the limit of \(\chi_n(0)\) with \(n < \omega\)?

No. Read the definition of the regularity.


Is the limit \(\chi_{\chi_{\cdot_{\cdot_{\cdot_{\chi_0(0)}\cdot}\cdot}\cdot}(0)}(0)\) of finite iterations of the map \(\alpha \mapsto \chi_{\alpha}(0)\) a fixed point of the map?

No. See #Does the map \(\alpha \mapsto \chi_{\alpha}(0)\) have a fixed point?.


Is the least weakly hyper inaccessible cardinal a fixed point of the map \(\alpha \mapsto \chi_{\alpha}(0)\)?

No. See #Does the map \(\alpha \mapsto \chi_{\alpha}(0)\) have a fixed point?.


Is \(\chi_M(0)\) ill-defined?

No. Read the definition.


Is \(\chi_M(0)\) bigger than \(M\)?

No. Read the definition.


Does \(\chi_M(0)\) coincide with the limit \(\chi_{\chi_{\cdot_{\cdot_{\cdot_{\chi_0(0)}\cdot}\cdot}\cdot}(0)}(0)\) of finite iterations of the map \(\alpha \mapsto \chi_{\alpha}(0)\)?

No. Read the definition of the regularity.


Is \(\chi_{\chi_M(0)}(0)\) ill-defined?

No. Read the definition.


Does \(\chi_{\chi_M(0)}(0)\) coincide with \(\chi_M(0)\)?

No. See #Does the map \(\alpha \mapsto \chi_{\alpha}(0)\) have a fixed point?.


Can ordinals between the least weakly hyper inaccessible cardinal and \(M\) be ignored in an analysis?

No. Since ordinals between them drastically contribute to the strength of the whole system of Rathjen's \(\psi\), it is quite unreasonble to ignore them. Altough many googologists who do not know the definition frquently ignore ordinals like \(\chi_{\chi_M(0)}(0)\), \(\chi_M(1)\), \(\chi_{M+1}(0)\), \(\chi_{\varepsilon_{M+1}}(0)\), but it is perhaps simply because they do not know the definition of Rathjen's \(\psi\).

If you find analyses ignoring all of those ordinals, it is good to suspect that the authors of the analyses do not know the definition of Rathjen's \(\psi\) although they might be trying to behave as if they were experts who knew highly difficult functions such as Rathjen's \(\psi\). It is just a cheating for them to use a notion whose definition they do not know as if they knew it very well when they state an analysis or a conjecture. See also #User blog:P進大好きbot/List of common mistakes in googology#I fully understand it although I do not know the precise definition! and #User blog:P進大好きbot/List of common mistakes in googology#I conjecture this!.


Is it sufficient to only consider a correspondence between expressions of cardinals in an analysis?

No. Since the strength of an OCF heavily depends on how to collapse cardinals, it is really meaningless to talk about an analysis based only on a correspondence between "famous" cardinals. Indeed, we can express a new cardinal by using a large countable ordinal, and hence forcussing on famous cardinals does not ensure the strength at all.

This mistake is based on the lack of the knowledge of the non-injectivity of an OCF and the predicate for normal forms, and is frequently made by UNOCF users, who strongly believe that googology is a game to list famous cardinals without setting a specific fundamental sequences. Even if a notation can "express" a cardinal larger than \(M\), it does not mean the whole system is stronger than Rathjen's \(\psi\). If you find such analyses, it is good to suspect that the authors of analyses are just trying to behave as if they were experts. See also #Does the simple extension using the enumeration of Mahlo cardinals work?.


Does the countable limit coincide with \(\psi_{\Omega}(\Omega_{M+1})\)?

No. The latter value is ill-defined. Read the definition.


Does the countable limit coincide with \(\psi_{\Omega}(\Gamma_{M+1})\)?

No. The latter value is ill-defined. Read the definition.


Does \(\psi_{\Omega}(M)\) coincide with \(\psi_{\Omega}(\chi_{\Gamma_{M+1}}(0))\)?

No. The latter value is ill-defined. Read the definition.


Does the countable limit coincide with \(\textrm{PTO}(\textrm{KPM})\)?

No. See also a related issue based on the confusion with Rathjen's another OCF.


Does Rathjen's simplification work in completely the same way as the original one does?

No. See the difference.


Does Rathjen's function symbol \(\psi\) in "Proof-theory of KPM" work in completely the same way as the original one does?

No. First, an OCF is different from an ordinal notation. Secondly, even if we "ignore" the difference of an OCF and an ordinal notation, they do not work in completely the same way as the original one does.


Does the OCF in "2000 steps" work in completely the same way as the original one does?

No. There are many known misconceptions in "2000 steps", and the OCF is irrelevant to Rathjen's \(\psi\).


Does \(\psi_I(0)\) coincide with the least omega fixed point?

No. It is much greater than the least omega fixed point \(\Phi_1(0)\).


Does \(\psi_{\Omega}(\psi_I(0))\) coincide with the countable limit of Extended Buchholz's function?

No. It is much greater than the countable limit of Extended Buchholz's function.


Is my simplification cooler than the original one?

Usually no. Almost all OCFs created by googoogists as "simplifications of Rathjen's \(\psi\)" (except for Denis' great simplification) do not work by one or more of the following reasons:

  1. The OCF is just ill-defined.
    1. There is no explicit definition of the domain, which should be a set of ordinals when you are defining a function, and should be a definable class of ordinals when you are defining a definable function. (See User blog:p進大好きbot/List of Common Failures in Googology#Lack of the Clarification of the Domain for more details.)
    2. The definition refers to an ill-defined expression.
      1. You should not refer to \(\min X\) for \(X = \emptyset\). The failure frequently occurs when googologists try to extend Rathjen's \(\psi\) to allow \(\psi_{\pi}\) for a singular ordinal \(\pi\) or \(\chi_M(\alpha)\) for more complicated a large cardinal \(\alpha\) than \(M\).
      2. You should not refer to \(\kappa^-\) for a cardinal \(\kappa > M\). The failure frequently occurs when googologists try to extend Rathjen's \(\psi\) to allow the function \(x \mapsto \Omega_x\) without any restriction or \(\chi_M(\alpha)\) for a higher cardinal \(\alpha\) than \(M\).
      3. You should not refer to the well-definedness of values of \(\psi\) itself to define \(\psi\). The failure sometimes occurs when googologists believe meaning phrases like "I assume the well-definedness" and "I consider \(\psi_{\pi}(\alpha)\) only when it is well-defined" magically solve anything.
    3. There is no written definition.
      1. Googologists sometimes cheat by omitting the definition and stating how it is strong.
      2. "My OCF in my mind is simpler than the original one, because I can simultaneously define \(\chi\) and \(\psi\) without changing the value!"
      3. "My OCF works for experts, and hence I do not have to write down the definition for beginners who do not understand how it should work!"
    4. There are many other errors listed in User blog:p進大好きbot/How to Create a Recursive Notation#General Setting.
  2. The OCF is weaker than the original one.
    1. Simple dropping of \(\varphi\) does not preserve the strength, as the countable limit is perhaps weaker than \(\textrm{PTO}(\textrm{KPM})\) depending on your simplification. You might state that collapsing powers of \(M\) plays an alternative role of \(\varphi\), but even powers of \(M\) is not expressed if you drop \(\varphi\).
    2. Simple dropping of \(\{+, \varphi, \Phi\}\) is awful.
    3. A non-trivial extension might not work as intended. See #Is any OCF collapsing larger cardinals than \(M\) stronger than Rathjen's \(\psi\)? for a specific example.
  3. The OCF does not allow a simpler ordinal notation associated to it.
    1. Googologists frequently confound an OCF, which is uncomputable, and an ordinal notation, which is computable, and state something like "I created an ordinal notation simpler and stronger than the original one!" See User blog:p進大好きbot/Relation between an OCF and an Ordinal Notation.
    2. Even though what we need in computable googology is an ordinal notation, googologists sometimes state "My OCF is obviously computable, and hence works in computable googology" without understanding what the computability means. When googologists are pointed out the misunderstanding of the computability, they tend to get angry. (Why...?)
      1. Dropping several conditions such as the one given by the normal form predicate from the definition of \(C\) might not change the ordinal notation, but I have never seen a proof although several googologists state it as an obvious fact.
      2. Some googologist states as an obvious fact that \(B\) and \(C\) can be simultaneously defined in a simpler way without changing the behaviour of the whole system, but I have never seen a proof or even a specific definition. See also the branch 132.
    3. Even though the OCF possibly yields an ordinal notation, it is meaningless to omit the definition, because defining an ordinal notation is one of the most difficult parts in computable googology using an OCF. One biggest problem is that the ordinal notation might be much more complicated than Rathjen's ordinal notation.
      1. Dropping the condition \(\beta \in C_{\pi}(\beta)\) from the application of \(\psi\) in the definition of \(C\) makes the comparison algorithm much more complicated, although googologists who do not know the definition of Rathjen's ordinal notation tend to state that the condition is useless and freely drop.
      2. Dropping the condition \(\delta, \eta < M\) from the application of \(\Phi\) in the definition of \(C\) makes the comparison algorithm and the standardness algorithm more complicated, although googologists who do not know the definition of Rathjen's ordinal notation tend to try to drop the condition after replacing \(\Phi\) by the enumeration of uncountable cardinals. See also the branch 122.


Does my extension/simplification in my mind work?

Clarify the definition. The answer heavily depends on the precise definition. For example, stating "I can simultaneously define \(\chi\) and \(\psi\) without changing the values" is meaningless unless you specify the way. See User blog:p進大好きbot/List of common mistakes in googology#I have already done it! for details about the cheating.


Does the simple dropping of \(\varphi\) preserve the strength?

No. See the branch 21 in #Is my simplification cooler than the original one?.


Does the simple extension using the full enumeration of cardinals work?

No. See the branches 122 and 332 in #Is my simplification cooler than the original one?.


Does the simple extension using the enumeration of Mahlo cardinals work?

No. See the branch 121 in #Is my simplification cooler than the original one? and also #Is it sufficient to only consider a correspondence between expressions of cardinals in an analysis?.


Does the simple dropping of the condition \(\beta \in C_{\pi}(\beta)\) from the application of \(\psi\) in the definition of \(C\) make the notation simpler?

No. See the branch 331 in #Is my simplification cooler than the original one?.


Is any OCF collapsing larger cardinals than \(M\) stronger than Rathjen's \(\psi\)?

No. Let us observe hyp cos's OCF collapsing Mahlo cardinals introduced here.

We denote by \(\textrm{On}\) the class of ordinals, by \(\textrm{Reg} \subset \textrm{On}\) the subclass of uncountable regular cardinals, and by \(\textrm{Mah} \subset \textrm{Reg}\) the subclass of weakly Mahlo cardinals. We assume that \(\textrm{Mah}\) is a proper class, and denote by \(\alpha \mapsto M_{\alpha}\) the enumeration of the closure of \(\{0\} \cup \textrm{Mah}\) in \(\textrm{On}\).
For each \((\alpha,\beta) \in \textrm{On}^2\), hyp cos simultaneously defined \(C(\alpha,\beta) \subset \textrm{On}\), \(\chi_{\beta}(\alpha) \in \textrm{Reg}\), and \(\psi_{\beta}(\alpha)\) in the following way:
\(C(\alpha,\beta) = \bigcup_{n \in \omega} C^n(\alpha,\beta)\), where \(C^n(\alpha,\beta) \subset \textrm{On}\) is defined in the following inductive way on \(n \in \omega\):
If \(n = 0\), then \(C^0(\alpha,\beta)\) is the union of \(\beta\) and \(\{0\}\).
If \(n \neq 0\), then \(C^0(\alpha,\beta)\) is the union of \(\{\gamma + \delta \mid (\gamma,\delta) \in C^{n-1}(\alpha,\beta)^2\}\), \(\{M_{\gamma} \mid \gamma \in C^{n-1}(\alpha,\beta)\}\), \(\{\chi_{\pi}(\gamma) \mid (\pi,\gamma) \in (C^{n-1}(\alpha,\beta) \cap \textrm{Mah}) \times (C^{n-1}(\alpha,\beta) \cap \alpha)\}\), and \(\{\psi_{\pi}(\gamma) \mid (\pi,\gamma) \in (C^{n-1}(\alpha,\beta) \cap \textrm{Reg}) \times (C^{n-1}(\alpha,\beta) \cap \alpha)\}\).
\(\chi_{\beta}(\alpha) = \min \{\pi \in \beta \cap \textrm{Reg} \mid C(\alpha,\pi) \cap \beta \subset \pi\}\)
\(\psi_{\beta}(\alpha) = \min \{\gamma \in \beta \mid C(\alpha,\gamma) \cap \beta \subset \gamma\}\)

I note that I replaced the symbols for variables so that the application of the simultaneous induction becomes clearer, but I have not changed the definition itself. I have pointed out that the whole system is not well-defined, as hyp cos unintentionally extended the first variables of \(\chi\) and \(\psi\) to \(\textrm{On}\), and hence later hyp cos restricted to the first variables. Namely, the domain of \(\chi\) has been replaced by \(\textrm{Mah} \times \textrm{On}\), and the domain of \(\psi\) has been replaced by \(\textrm{Reg} \times \textrm{On}\). See the branch 121 in #Is my simplification cooler than the original one? for a related failure.

The OCF was believed to be much stronger than Rathjen's \(\psi\) by old members in this community without proofs, because it collapses larger Mahlo cardinals and many other members praised it. Is it correct if we consider the definition after the correction of the issue above?

The answer is no. First, we do not know whether \(\chi\) and \(\psi\) are well-defined for any imputs in the restricted domains, because nobody has proved that the issue in the branch 121 in #Is my simplification cooler than the original one? does not occur in the corrected definition. Therefore we assume the well-definedness here. Then Scorcher007 and C7X pointed out that the OCF is much weaker than Rathjen's \(\psi\), as they proved that \(\psi\) always returns \(0\). As a conclusion, the forklore "An OCF collapsing larger cardinal is stronger" in this community is wrong.

You might continue to say "we assume that an OCF successfully collapses higher cardinals!", but what does "successfully" mean? If you are just assuming that it collapses higher cardinals so that it beats Rathjen's \(\psi\), it is just a circular logic. You might continue to say "I can easily fix the issue!", but then could you verify both the well-definedness and the strength? Otherwise, other errors will be perhaps found in the future, and the history will be repeated. Anyway, what we can learn from this story is that belief by old members in this community based on intuition is not reliable. When we state something, we should sufficiently formalise it, and try to prove it instead of believing it by saying "it is obviously true, as old members said so."


Does \(M\) in UNOCF work in the completely same way as \(M\) in Rathjen's \(\psi\)?

No. UNOCF is completely ill-defined. See User blog:p進大好きbot/Historical Background of the Ill-definedness of UNOCF.

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