Extended Buchholz's function

Extended Buchholz's function is an extension of Buchholz's function by Googology Wiki user Denis Maksudov. The countable limit of Extended Buchholz's function is expressed as \(\psi_0(\Omega_{\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}}}) = \psi_0(\psi_{\psi_{\psi_{\cdots}(0)}(0)}(0)) = \psi_0(\Lambda)\), where \(\Lambda\) denotes the least omega fixed point, and is called Extended Buchholz's ordinal or EBO in Japanese Googology.
== Definition ==
Denis Maksudov defined his functions as follows:
* \(C_\nu^0(\alpha) = \{\beta|\beta<\Omega_\nu\}\),
* \(C_\nu^{n+1}(\alpha) = \{\beta+\gamma,\psi_\mu(\eta)|\mu,\beta, \gamma,\eta\in C_{\nu}^n(\alpha)\wedge\eta<\alpha\}\),
* \(C_\nu(\alpha) = \bigcup_{n < \omega} C_\nu^n (\alpha)\),
* \(\psi_\nu(\alpha) = \min\{\gamma | \gamma \not\in C_\nu(\alpha)\}\),
where
\(\Omega_\nu=\left\{\begin{array}{lcr}
1\text{ if }\nu=0 \\
\text{smallest ordinal with cardinality }\aleph_\nu \text{ if }\nu>0 \\
\end{array}\right.\)
There is only one little detail difference with original Buchholz definition: ordinal \(\mu\) is not limited by \(\omega\), now ordinal \(\mu\) belongs to previous set \(C_{\nu}^n\).
For example if \(C_0^0(1)=\{0\}\) then \(C_0^1(1)=\{0,\psi_0(0)=1\}\) and \(C_0^2(1)=\{0,...,\psi_1(0)=\Omega\}\) and \(C_0^3(1)=\{0,...,\psi_\Omega(0)=\Omega_\Omega\}\) and so on.
== Ordinal notation ==
Since an ordinal collapsing function itself is not a computable function, we need to create an ordinal notation \((OT,<)\) associated to Extended Buchholz's function, i.e. a recursive interpretation of the comparison and the system of fundamental sequences of ordinals using formal expressions, in order to create a computable large number.
Indeed, explicit algorithms to compute them and the associated fast-growing hierarchy extending those for Buchholz's function are given by a Japanese Googology Wiki user p進大好きbot.p進大好きbot, Ordinal Notation Associated to Extended Buchholz's OCF, Googology Wiki user blog. (This is an English translation of a Japanese blog post :ja:ユーザーブログ:P進大好きbot/拡張Buchholz_OCFに伴う順序数表記.) See #External Links for a computation program by koteitan, which essentially supports p進大好きbot's ordinal notation because it deals with its extension KumaKuma ψ function explained in #Extensions section.
Later, 降下段階配列表記 is created by a Japanese Googology Wiki user mrna as an equivalent notation extending 段階配列表記. It also has an extension 多変数段階配列表記 explained in #Extensions section.
== Normal form ==
Googology Wiki user Denis Maksudov also extended the second normal form explained in Buchholz's function#Normal form to ordinals in \(C_0(\Lambda)\) in the following way:
Informally speaking, the normal form for \(0\) is \(0\). If \(\alpha\) is a nonzero ordinal number \(\alpha<\Lambda=\text{min}\{\beta|\psi_\beta(0)=\beta\}\) then the normal form for \(\alpha\) is \(\alpha=\psi_{\nu_1}(\beta_1)+\psi_{\nu_2}(\beta_2)+\cdots+\psi_{\nu_k}(\beta_k)\) where \(k\) is a positive integer and \(\psi_{\nu_1}(\beta_1)\geq\psi_{\nu_2}(\beta_2)\geq\cdots\geq\psi_{\nu_k}(\beta_k)\) and each \(\nu_i\), \(\beta_i\) are ordinals satisfying \(\beta_i \in C_{\nu_i}(\beta_i)\). We note that since \(\beta_i\)'s are also ordinals in \(C_0(\Lambda)\), it is possible to express them in normal form. It roughly means that every ordinal in \(C_0(\Lambda)\) is expressed in "iterated" normal form consisting of \(0\), \(+\), and \(\psi\).
More formally, the set \(\textrm{NF}\) of predicates on ordinals in \(C_0(\Lambda)\) is defined in the following way:
# The predicate \(\alpha =_{\textrm{NF}} 0\) on an ordinal \(\alpha\) in \(C_0(\Lambda)\) defined as \(\alpha = 0\) belongs to \(\textrm{NF}\).
# The predicate \(\alpha_0 =_{\textrm{NF}} \psi_{\nu_1}(\alpha_1)\) on ordinals \(\alpha_0, \alpha_1, \nu_1\) in \(C_0(\Lambda)\) defined as \(\alpha_0 = \psi_{\nu_1}(\alpha_1)\) and \(\alpha_1 \in C_{\nu_1}(\alpha_1)\) belongs to \(\textrm{NF}\).
# The predicate \(\alpha_0 =_{\textrm{NF}} \psi_{\nu_1}(\alpha_1) + \cdots + \psi_{\nu_n}(\alpha_n)\) on ordinals \(\alpha_0, \ldots, \alpha_n, \nu_1, \ldots, \nu_k\) in \(C_0(\Lambda)\) with an integer \(k > 1\) defined as \(\alpha_0 = \psi_{\nu_1}(\alpha_1) + \cdots + \psi_{\nu_k}(\alpha_k)\), \(\psi_{\nu_1}(\alpha_1) \geq \cdots \geq \psi_{\nu_k}(\alpha_k)\), and \(\alpha_1 \in C_{\nu_1}(\alpha_1), \ldots, \alpha_k \in C_{\nu_k}(\alpha_k)\) belongs to \(\textrm{NF}\).
Moreover, the normality of an expression can be described in a recursive way with respect to the corresponding ordinal notation system extending the original ordinal notation system \((OT,<)\) explained above.
== Fundamental sequences ==
The fundamental sequence for an ordinal number \(\alpha\) with cofinality \(\text{cof}(\alpha)=\beta\) is a strictly increasing sequence \((\alpha[\eta])_{\eta<\beta}\) with length \(\beta\) and with limit \(\alpha\), where \(\alpha[\eta]\) is the \(\eta\)-th element of this sequence. If \(\alpha\) is a successor ordinal then \(\text{cof}(\alpha)=1\) and the fundamental sequence has only one element \(\alpha[0]\) satisfying \(\alpha=\alpha[0]+1\). If \(\alpha\) is a limit ordinal then \(\text{cof}(\alpha)\in\{\omega\}\cup\{\Omega_{\mu+1}|\mu\geq 0\}\).
Although a system of fundamental sequences is not unique, there is a canonical choice of fundamental sequences in this community given by Denis. For nonzero ordinals \(\alpha<\Lambda\), written in normal form, fundamental sequences are defined as follows:
#If \(\alpha=_{\textrm{NF}}\psi_{\nu_1}(\beta_1)+\psi_{\nu_2}(\beta_2)+\cdots+\psi_{\nu_k}(\beta_k)\) where \(k\geq2\) then \(\text{cof}(\alpha)=\text{cof}(\psi_{\nu_k}(\beta_k))\) and \(\alpha[\eta]=\psi_{\nu_1}(\beta_1)+\cdots+\psi_{\nu_{k-1}}(\beta_{k-1})+(\psi_{\nu_k}(\beta_k)[\eta])\),
#If \(\alpha=_{\textrm{NF}}\psi_{0}(0)=1\), then \(\text{cof}(\alpha)=1\) and \(\alpha[0]=0\),
#If \(\alpha=_{\textrm{NF}}\psi_{\nu+1}(0)\), then \(\text{cof}(\alpha)=\Omega_{\nu+1}\) and \(\alpha[\eta]=\Omega_{\nu+1}[\eta]=\eta\),
#If \(\alpha=_{\textrm{NF}}\psi_{\nu}(0)\) and \(\text{cof}(\nu)\in\{\omega\}\cup\{\Omega_{\mu+1}|\mu\geq 0\}\), then \(\text{cof}(\alpha)=\text{cof}(\nu)\) and \(\alpha[\eta]=\psi_{\nu[\eta]}(0)=\Omega_{\nu[\eta]}\),
#If \(\alpha=_{\textrm{NF}}\psi_{\nu}(\beta+1)\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[\eta]=\psi_{\nu}(\beta)\cdot \eta\) (and note: \(\psi_\nu(0)=\Omega_\nu\)),
#If \(\alpha=_{\textrm{NF}}\psi_{\nu}(\beta)\) and \(\text{cof}(\beta)\in\{\omega\}\cup\{\Omega_{\mu+1}|\mu<\nu\}\) then \(\text{cof}(\alpha)=\text{cof}(\beta)\) and \(\alpha[\eta]=\psi_{\nu}(\beta[\eta])\),
#If \(\alpha=_{\textrm{NF}}\psi_{\nu}(\beta)\) and \(\text{cof}(\beta) = \Omega_{\mu+1}\) for a \(\mu\geq\nu\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[\eta]=\psi_{\nu}(\beta[\gamma[\eta]])\) where \(\left\{\begin{array}{lcr} \gamma[0]=\Omega_\mu \\ \gamma[\eta+1]=\psi_\mu(\beta[\gamma[\eta]])\\ \end{array}\right.\).
It is an extension of the system of fundamental sequences up to \(\psi_0(\varepsilon_{\Omega_{\omega}+1})\) in Buchholz hierarchy given by modifying the rule ([].5) (ii) in recursive definition of the \(\textrm{dom}\) function and \([]\) in Buchholz's original paperW. Buchholz, A New System of Proof-Theoretic Ordinal Functions, Annals of Pure and Applied Logic, vol. 32 (1986), pp. 195--207. by the rule 6 in the definition of \([]\) in p.6 in Buchholz's another paperW. Buchholz, Relating ordinals to proofs in a perspicuous way, Reflections on the foundations of mathematics (Stanford, CA, 1998) 15 (2017): 37--59. applied to the convention \(\Omega_0 = 1\) except for the minor differences related to the difference \(\omega[n] = n+1\) in the original definition and \(\omega[n] = n\) in the definition here. (Please remember that \(\Omega_0\) is defined as \(1\) in the original paper, while it is defined as \(\omega\) in the other paper.) More precisely, the fundamental sequence of \(\psi_0(2) = \omega \times \omega\) is given as \(\omega \times \omega [n] = \omega \times (n+1)\) in the original definition while we have \(\omega \times \omega[n] = \omega \times n\) in the definition here, and the fundamental sequence of \(\psi_{\omega}(0) = \Omega_{\omega}\) is given as \(\Omega_{\omega}[n] = \Omega_{n+1}\) while we have \(\Omega_{\omega}[n] = \Omega_n\) in the definition here.
If \(\alpha=\Lambda\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[0]=0\) and \(\alpha[\eta+1]=\psi_{\alpha[\eta]}(0)=\Omega_{\alpha[\eta]}\).
== Variants ==
Since the structure of Extended Buchholz's function is quite sophisticated and is deeply studied in Japanese googology, there are many variants of Extended Buchholz's function or the ordinal notation associated to it. Namely, it plays a role of the base of other notations, i.e. many variants are invented, in Japanese googology.
=== Extension ===
The pioneer of those extensions is the \(3\)-ary function notation by a Japanese Googology Wiki user kanrokoti.kanrokoti, 3 variables ψ which is larger than EBOCF, Googology Wiki user blog. Later, mrna extended 降下段階配列表記 to 多変数段階配列表記. There are several trials in Japanese googology to even extend those extensions.
=== Simplification ===
Extended Buchholz's function is defined by using the closure with respect to the addition, while Extended Weak Buchholz's function is defined in the same way except that it does not use the closure with respect to the addition. Therefore the definitions directly imply that Extended Weak Buchholz's function is bounded by Extended Buchholz's function. One non-trivial expectation is that the two functions have the same countable limit, but it has not been proved yet.
== Common misconceptions ==
Since there are so many misconceptions about Buchholz's function and Extended Buchholz's function, readers are strongly recommended to check Buchholz's function#Common misconceptions before talking about them.
== External Links ==
* Buchholz ψ analyzer by Fish
* Online calculator for fast-growing hierarchy with Buchholz function by Denis Maksudov
* Online calculator for fast-growing hierarchy with Extended Buchholz function by Denis Maksudov
* ordex: Ordinal Expander in JavaScript by koteitan (Koteitan's online calculator for kanrokoti's \(3\)-ary extension of p進大好きbot's ordinal notation associated to Denis' extension of Buchholz's function. Therefore it essentially supports p進大好きbot's ordinal notation. A detailed explanation is available here.)
== Sources ==
ja:拡張ブーフホルツのψ関数
== See also ==
* Buchholz's function
* Extended Weak Buchholz's function
Category:Buchholz's psi-functions
Category:Functions
Category:Transfinite ordinals
Category:Eponymous functions
Category:Ordinal collapsing functions
Category:Ordinal notations