Countable limit of Extended Buchholz's function

View full site to see MathJax equation

The countable limit \(\psi_0(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}})\) of Extended Buchholz's ordinal collapsing function is a large countable ordinal, where \(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}}\) denotes the least omega fixed point. It is called Extended Buchholz's ordinal or EBO in Japanese googology, and the term was coined by a Japanese googologist 叢武 (Muratake).

Properties

It is the smallest ordinal which is not expressed by using \(0, +, \psi\). It is also the smallest ordinal which is not expressed by the ordinal notation associated to Extended Buchholz's function with respect to the natural correspondence \(o\), but can be expressed as \(\langle \langle (), () \rangle , () \rangle\) or \(\psi_{\psi_0(0)}(0)\) with respect to the order type.

For any ordinal \(\alpha\) greater than or equal to \(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}}\), e.g. \(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}} + 1\), the least weakly inaccessible cardinal \(I\), and the least weakly Mahlo cardinal \(M\), \(\psi_0(\alpha)\) coincides with the countable limit of Extended Buchholz's function. Therefore claims such as "\(\psi_0(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}} + 1) = \psi_0(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}}) \times \omega\) and hence is larger than \(\psi_0(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}})\)" and "\(\psi_0(I)\) collapses into smaller ordinals of the form \(\psi_0(\psi_I(\psi_I(\ldots \psi_I(0) \ldots)))\), and hence is larger than \(\psi_0(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}})\)" are incorrect. See also #Warning for a related issue.

Warning

In this community, it is frequently denoted by \(\psi(\psi_I(0))\), but the expression has a problem. Here \(I\) is the least weakly inaccessible cardinal, and \(\psi\) is an unspecified or undefined ordinal collapsing function, and \(\psi_I(0)\) is intended to be the least omega fixed point.

Since beginners tend to talk about specific values of ordinal collapsing functions without specifying definitions under the wrong assumptions that the difference of the definition does not cause the difference of specific values or that an ordinal collapsing function just gives roles to symbols to diagonalise other functions, the \(\psi\) function is often confounded with Buchholz's ordinal collapsing function, Extended Buchholz's ordinal collapsing function, and Rathjen's ordinal collapsing functions based on a weakly Mahlo cardinal. However, p進大好きbot pointed out that they do not have either the same expression or the required property:

1. Buchholz's function does not have the expression \(\psi_I(0)\).
2. Neither Buchholz's \(\psi_0\) nor Rathjen's \(\psi_{\Omega}\) has an official abbreviation to \(\psi\).
3. Extended Buchholz's function does not satisfy \(\psi_I(0) = \Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}}\).
4. Rathjen's standard ordinal collapsing function based on a weakly Mahlo cardinal does not satisfy \(\psi_I(0) = \Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}}\).
5. Rathjen's simplified ordinal collapsing function based on a weakly Mahlo cardinal does not have the expression \(\psi_I(0)\).
6. Rathjen's simplfied ordinal notation based on a weakly Mahlo cardinal has a function symbol \(\psi\), but it is not an actual function. Although the notation is associated to a simplified ordinal collapsing function based on a weakly Mahlo cardinal, the ordinal collapsing function is unpublished, and hence cannot be a candidate due to the lack of an explicit source.

As the last example shows, it is possible (and actually easy) to formulate an ordinal collapsing function such that \(\psi_I(0)\) is a valid expression and its value precisely coincides with the least omega fixed point. Also, Jäger's function and Jäger-Buchholz function satisfy the property as long as we assume that \(\psi\) is an abbreviation of \(\psi_{\Omega}\), although either of them is rarely used in this community. The significant point of this story is that it clearly tells us that arguments on unspecified or undefined ordinal collapsing functions under the assumptions above actually cause serious errors.

Why do people tell others wrong informations about OCFs? It is because there are so many wrong articles written by those who do not know the precise definitions of OCFs. When beginners want to know OCFs, they will check precise articles, and judge such articles useless because the contents are quite difficult for them. Then they will search "introductory" articles written by those who do not know the precide definitions of OCFs, and praise such articles as useful ones. However, such articles are unsourced, and include many wrong informations. After then, the beginners guess that they have already understood OCFs. They try to write useful "introduction" before correcting their understandings by checking sources, and tell others wrong informations. This is a general story, but the expression \(\psi(\psi_I(0))\) actually frequently experiences the misfortune.

As a result, beginners should be very careful when other googologists talk about the expression "\(\psi(\psi_I(0))\)" without specifying the definition of \(\psi\), and ask the precise definition of the \(\psi\).

See also

• Buchholz's function#Common misconceptions

Ordinals, ordinal analysis, and set theory

Basics: cardinal numbers · ordinal numbers · limit ordinals · fundamental sequence · normal form · transfinite induction · ordinal notation · Absolute infinity
Theories: Robinson arithmetic · Presburger arithmetic · Peano arithmetic · KP · second-order arithmetic · ZFC
Model Theoretic Concepts: structure · elementary embedding
Countable ordinals: \(\omega\) · \(\varepsilon_0\) · \(\zeta_0\) · \(\eta_0\) · \(\Gamma_0\) (Feferman–Schütte ordinal) · \(\varphi(1,0,0,0)\) (Ackermann ordinal) · \(\psi_0(\Omega^{\Omega^\omega})\) (small Veblen ordinal) · \(\psi_0(\Omega^{\Omega^\Omega})\) (large Veblen ordinal) · \(\psi_0(\varepsilon_{\Omega + 1}) = \psi_0(\Omega_2)\) (Bachmann-Howard ordinal) · \(\psi_0(\Omega_\omega)\) with respect to Buchholz's ψ · \(\psi_0(\varepsilon_{\Omega_\omega + 1})\) (Takeuti-Feferman-Buchholz ordinal) · \(\psi_0(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}})\) (countable limit of Extended Buchholz's function)‎ · \(\omega_1^\mathfrak{Ch}\) (Omega one of chess) · \(\omega_1^{\text{CK}}\) (Church-Kleene ordinal) · \(\omega_\alpha^\text{CK}\) (admissible ordinal) · recursively inaccessible ordinal · recursively Mahlo ordinal · reflecting ordinal · stable ordinal · \(\lambda,\gamma,\zeta,\Sigma\) (Infinite time Turing machine ordinals) · gap ordinal · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Hardy hierarchy · Slow-growing hierarchy · Middle-growing hierarchy · N-growing hierarchy
Ordinal functions: enumeration · normal function · derivative · Veblen function · ordinal collapsing function · Weak Buchholz's function · Bachmann's function · Madore's function · Feferman's theta function · Buchholz's function · Extended Weak Buchholz's function · Extended Buchholz's function · Jäger-Buchholz function · Jäger's function · Rathjen's psi function · Rathjen's Psi function · Stegert's Psi function · Arai's psi function
Uncountable cardinals: \(\omega_1\) · omega fixed point · inaccessible cardinal \(I\) · Mahlo cardinal \(M\) · weakly compact cardinal \(K\) · indescribable cardinal · rank-into-rank cardinal · Berkeley cardinal · (Club Berkeley cardinal · limit club Berkeley cardinal)
Classes: \(\textrm{Card}\) · \(\textrm{On}\) · \(V\) · \(L\) · \(\textrm{Lim}\) · \(\textrm{AP}\) · Class (set theory)