Buchholz's function

Buchholz's psi functions are a hierarchy of single-argument ordinal collapsing functions \(\psi_\nu(\alpha)\) introduced by German mathematician Wilfried Buchholz in 1986. These functions are a simplified version of Feferman's \(\theta\) functions, but nevertheless have the same strength as those.

Definition
Buchholz defined his functions as follows:


 * \(C_\nu^0(\alpha) = \Omega_\nu\),
 * \(C_\nu^{n+1}(\alpha) = C_\nu^n(\alpha) \cup \{\gamma | P(\gamma) \subseteq C_\nu^n(\alpha)\} \cup \{\psi_\mu(\xi) | \xi \in \alpha \cap C_\nu^n(\alpha) \wedge \xi \in C_\mu(\xi) \wedge \mu \leq \omega\}\),
 * \(C_\nu(\alpha) = \bigcup_{n < \omega} C_\nu^n (\alpha)\),
 * \(\psi_\nu(\alpha) = \min\{\gamma | \gamma \not\in C_\nu(\alpha)\}\),

where

\(\Omega_\nu=\left\{\begin{array}{lcr}1\text{ if }\nu=0\\\aleph_\nu\text{ if }\nu>0\\\end{array}\right.\)

and \(P(\gamma)=\{\gamma_1,\cdots,\gamma_k\}\) is the set of additive principal numbers in form \(\omega^\xi\),

\(P=\{\alpha\in \textrm{On}: 0<\alpha \wedge \forall \xi, \eta < \alpha (\xi+\eta < \alpha)\}=\{\omega^\xi: \xi \in \textrm{On}\}\),

the sum of which gives this ordinal \(\gamma\):

\(\gamma=\alpha_1+\alpha_2+\cdots+\alpha_k\) where \(\alpha_1\geq\alpha_2\geq\cdots\geq\alpha_k\) and \(\alpha_1,\alpha_2,\cdots,\alpha_k \in P(\gamma)\).

Note: Greek letters always denotes ordinals. \(\textrm{On}\) denotes the class of all ordinals.

The limit of this notation is Takeuti-Feferman-Buchholz ordinal.

Properties
Buchholz showed following properties of those functions:


 * \(\psi_\nu(0)=\Omega_\nu\),
 * \(\psi_\nu(\alpha)\in P\),
 * \(\psi_\nu(\alpha+1)=\text{min}\{\gamma\in P: \psi_\nu(\alpha)<\gamma\}\text{ if }\alpha\in C_\nu(\alpha)\),
 * \(\Omega_\nu\le\psi_\nu(\alpha)<\Omega_{\nu+1}\),
 * \(\alpha\le\beta\Rightarrow\psi_\nu(\alpha)\le\psi_\nu(\beta)\),
 * \(\psi_0(\alpha)=\omega^\alpha \text{ if }\alpha<\varepsilon_0\),
 * \(\psi_\nu(\alpha)=\omega^{\Omega_\nu+\alpha} \text{ if }\alpha<\varepsilon_{\Omega_\nu+1} \text{ and } \nu\neq 0\),
 * \(\theta(\varepsilon_{\Omega_\nu+1},0)=\psi_0(\varepsilon_{\Omega_\nu+1})\) for \(0<\nu\le\omega\).

Explanation
Buchholz is working in Zermelo–Fraenkel set theory, that means every ordinal \(\alpha\) is equal to set \(\{\beta|\beta<\alpha\}\). Then condition \(C_\nu^0(\alpha)=\Omega_\nu\) means that set \(C_\nu^0(\alpha)\) includes all ordinals less than \(\Omega_\nu\) in other words \(C_\nu^0(\alpha)=\{\beta|\beta<\Omega_\nu\}\).

The condition \(C_\nu^{n+1}(\alpha) = C_\nu^n(\alpha) \cup \{\gamma | P(\gamma) \subseteq C_\nu^n(\alpha)\} \cup \{\psi_\mu(\xi) | \xi \in \alpha \cap C_\nu^n(\alpha) \wedge \mu \leq \omega\}\) means that set \(C_\nu^{n+1}(\alpha)\) includes:


 * 1) all ordinals from previous set \(C_\nu^n(\alpha)\),
 * 2) all ordinals that can be obtained by summation the additively principal ordinals from previous set \(C_\nu^n(\alpha)\),
 * 3) all ordinals that can be obtained by applying ordinals less than \(\alpha\) from the previous set \(C_\nu^n(\alpha)\) as arguments of functions \(\psi_\mu\), where \(\mu\le\omega\).

That is why we can rewrite this condition as:

\(C_\nu^{n+1}(\alpha) = \{\beta+\gamma,\psi_\mu(\eta)|\beta, \gamma,\eta\in C_{\nu}^n(\alpha)\wedge\eta<\alpha \wedge \mu \leq \omega\}\).

Thus union of all sets \(C_\nu^n (\alpha)\) with \(n<\omega\) i.e. \(C_\nu(\alpha) = \bigcup_{n < \omega} C_\nu^n (\alpha)\) denotes the set of all ordinals which can be generated from ordinals \(<\aleph_\nu\) by the functions + (addition) and \(\psi_{\mu}(\xi)\), where \(\mu\le\omega\) and \(\xi<\alpha\).

Then \(\psi_\nu(\alpha) = \min\{\gamma | \gamma \not\in C_\nu(\alpha)\}\) is the smallest ordinal that does not belong to this set.

Examples
Consider the following examples:

\(C_0^0(\alpha)=\{0\} =\{\beta:\beta<1\}\),

\(C_0(0)=\{0\}\) (since no functions \(\psi(\eta<0)\) and 0+0=0).

Then \(\psi_0(0)=1\).

\(C_0(1)\) includes \(\psi_0(0)=1\) and all possible sums of natural numbers:

\(C_0(1)=\{0,1,2,...,\text{googol}, ...,\text{TREE(googol)},...\}\).

Then \(\psi_0(1)=\omega\) - first transfinite ordinal, which is greater than all natural numbers by its definition.

\(C_0(2)\) includes \(\psi_0(0)=1, \psi_0(1)=\omega\) and all possible sums of them.

Then \(\psi_0(2)=\omega^2\).

For \(C_0(\omega)\) we have set \(C_0(\omega)=\{0,\psi(0)=1,...,\psi(1)=\omega,...,\psi(2)=\omega^2,...,\psi(3)=\omega^3,...\}\).

Then \(\psi_0(\omega)=\omega^\omega\).

For \(C_0(\Omega)\) we have set \(C_0(\Omega)=\{0,\psi(0)=1,...,\psi(1)=\omega,...,\psi(\omega)=\omega^\omega,...,\psi(\omega^\omega)=\omega^{\omega^\omega},...\}\).

Then \begin{eqnarray*} & & \psi_0(\varepsilon_0)=\psi_0(\varepsilon_0+1) \\ & = & \cdots=\psi_0(\textrm{insert any countable ordinal above } \varepsilon_0 \textrm{which you like very much}) \\ & = & \cdots = \psi_0(\Omega)=\varepsilon_0. \end{eqnarray*} For \(C_0(\Omega+1)\) we have set \(C_0(\Omega)=\{0,1,...,\psi_0(\Omega)=\varepsilon_0,...,\varepsilon_0+\varepsilon_0,...\psi_1(0)=\Omega,...\}\).

Then \(\psi_0(\Omega+1)=\varepsilon_0\omega=\omega^{\varepsilon_0+1}\).

\(\psi_0(\Omega2)=\varepsilon_1\),

\(\psi_0(\Omega^2)=\zeta_0\),

\(\psi_0(\Omega^\alpha(1+\beta)) = \varphi(\alpha,\beta)\),

\(\psi_0(\Omega^\Omega)=\Gamma_0=\theta(\Omega,0)\), using Feferman theta-function,

\(\psi_0(\Omega^{\Omega^\Omega})\) is large Veblen ordinal,

\(\psi_0(\Omega\uparrow\uparrow\omega)=\psi_0(\varepsilon_{\Omega+1})=\theta(\varepsilon_{\Omega+1},0)\).

Now let's research how \(\psi_1\) works:

\(C_1^0(\alpha)=\{\beta:\beta<\Omega_1\}=\{0,\psi(0)=1,2,...\text{googol},...,\psi_0(1)=\omega,...,\psi_0(\Omega)=\varepsilon_0,...\)

\(...,\psi_0(\Omega^\Omega)=\Gamma_0,...,\psi(\Omega^{\Omega^\Omega+\Omega^2}),...\}\) i.e. includes all countable ordinals.

\(C_1(\alpha)\) includes all possible sums of all countable ordinals. Then

\(\psi_1(0)=\Omega_1\) first uncountable ordinal which is greater than all countable ordinal by its definition i.e. smallest number with cardinality \(\aleph_1\).

\(C_1(1)=\{0,...,\psi_0(0)=\omega,...,\psi_1(0)=\Omega,...,\Omega+\omega,...,\Omega+\Omega,...\}\)

Then \(\psi_1(1)=\Omega\omega=\omega^{\Omega+1}\).

Then \(\psi_1(2)=\Omega\omega^2=\omega^{\Omega+2}\),

\(\psi_1(\psi_0(\Omega))=\Omega\varepsilon_0=\omega^{\Omega+\varepsilon_0}\),

\(\psi_1(\psi_0(\Omega^\Omega))=\Omega\Gamma_0=\omega^{\Omega+\Gamma_0}\),

\(\psi_1(\psi_1(0))=\psi_1(\Omega)=\Omega^2=\omega^{\Omega+\Omega}\),

\(\psi_1(\psi_1(\psi_1(0)))=\omega^{\Omega+\omega^{\Omega+\Omega}}=\omega^{\Omega\cdot\Omega}=(\omega^{\Omega})^\Omega=\Omega^\Omega\),

\(\psi_1^4(0)=\Omega^{\Omega^\Omega}\),

\(\psi_1(\Omega_2)=\psi_1^\omega(0)=\Omega\uparrow\uparrow\omega=\varepsilon_{\Omega+1}\).

For case \(\psi(\Omega_2)\) the set \(C_0(\Omega_2)\) includes functions \(\psi_0\) with all arguments less than \(\Omega_2\) i.e. such arguments as \(0, \psi_1(0), \psi_1(\psi_1(0)), \psi_1^3(0),..., \psi_1^\omega(0)\)

and then \(\psi_0(\Omega_2)=\psi_0(\psi_1(\Omega_2))=\psi_0(\varepsilon_{\Omega+1})\).

In general case: \(\psi_0(\Omega_{\nu+1})=\psi_0(\psi_\nu(\Omega_{\nu+1}))=\psi_0(\varepsilon_{\Omega_\nu+1})=\theta(\varepsilon_{\Omega_\nu+1},0)\).

We also can write:

\(\theta(\Omega_\nu,0)=\psi_0(\Omega_\nu^{\Omega_\nu})\) ( for \( 1\le\nu<\omega\)).

Ordinal notation
Buchholz defined an ordinal notation \((OT,<)\) associated to \(\psi\) as an array notation. We explain the original definition of \((OT,<)\).

We simultaneously define the sets \(T\) and \(PT\) of formal strings consisting of \(0\), \(D_v\) indexed by an \(v \in \omega+1\), braces, and commas in the following recursive way: Namely, \(T\) and \(PT\) are the smallest sets satisfying the conditions above. An element of \(T\) is called a term, and an element of \(PT\) is called a principal term. By the definition, \(T\) is a recursive set, \(PT\) is a recursive subset of \(T\), and every term is \(0\), a principal term, or an array of principal terms of length \(\geq 2\) in braces.
 * 1) \(0 \in T\) and \(0 \in PT\).
 * 2) For any \((v,a) \in (\omega+1) \times T\), \(D_va \in T\) and \(D_va \in PT\).
 * 3) For any \((a_i)_{i=0}^{k} \in PT^{k+1}\) with \(k \in \mathbb{N} \setminus \{0\}\), \((a_0,\ldots,a_k) \in T\). More precisely, this condition can be formulated without using ellipses in the following way:
 * 4) For any \((a_0,a_1) \in PT^2\), \((a_0,a_1) \in T\).
 * 5) For any \((a_0,a_1) \in T \times PT\), if there exists formal strings \(s\) such that \(a_0 = (s)\), then \((s,a_1) \in T\).

We also denote an \(a \in PT\) by \((a)\). Since the clause 3 in the definition of \(T\) and \(PT\) is applicable only to arrays of length \(\geq 2\), the additional convention does not cause serious ambiguity. By the convention, every term can be uniquely expressed as either \(0\) or a non-empty array of principal terms in braces.

We define a binary relation \(a < b\) on \(T\) in the following recursive way: By the definition, \(<\) is a recursive strict total ordering on \(T\). We abbreviate \((a < b) \lor (a = b)\) to \(a \leq b\). Although \(\leq\) itself is not a well-ordering, its restriction to a recursive subset \(OT \subset T\), which will be introduced later, forms a well-ordering.
 * 1) If \(b = 0\), then \(a < b\) is false.
 * 2) If \(a = 0\), then \(a < b\) is equivalent to \(b \neq 0\).
 * 3) Suppose \(a \neq 0\) and \(b \neq 0\).
 * 4) If \(a = D_ua'\) and \(b = D_vb'\) for some \(((u,a'),(v,b')) \in ((\omega+1) \times T)^2\), \(a < b\) is equivalent to that either one of the following holds:
 * 5) \(u < v\).
 * 6) \(u = v\) and \(a' < b'\).
 * 7) If \(a = (a_0,\ldots,a_n)\) and \(b = (b_0,\ldots,b_m)\) for some \((n,m) \in (\omega+1)^2\) with \(1 \leq \min\{n,m\}\), \(a < b\) is equivalent to that either one of the following holds:
 * 8) \(n < m\) and \(a_i = b_i \) for any \(i \in \mathbb{N}\) with \(i \leq n\)
 * 9) There exists some \(k \in \mathbb{N}\) such that \(k \leq \min\{n,m\}\), \(a_k < b_k\), and \(a_i = b_i\) for any \(i \in \mathbb{N}\) with \(i < k\).

In order to define \(OT\), we define a subset \(G_ua \subset T\) for each \((u,a) \in (\omega+1) \times T\) in the following recursive way: By the definition, \(b \in G_ua\) is a recursive relation on \((u,a,b) \in (\omega+1) \times T^2\).
 * 1) If \(a = 0\), then \(G_ua = \emptyset\).
 * 2) Suppose that \(a = D_va'\) for some \((v,a') \in (\omega+1) \times T\).
 * 3) If \(u \leq v\), then \(G_ua = \{a'\} \cup G_ua'\).
 * 4) If \(u > v\), then \(G_ua = \emptyset\).
 * 5) If \(a = (a_0,\ldots,a_k)\) for some \((a_i)_{i=0}^{k} \in PT^{k+1}\) with \(k \in \mathbb{N} \setminus \{0\}\),\(G_ua = \bigcup_{i=0}^{k} G_ua_i\).

Finally, we define a subset \(OT \subset T\) in the following recursive way: By the definition, \(OT\) is a recursive subset of \(T\). An element of \(OT\) is called an ordinal term.
 * 1) \(0 \in OT\).
 * 2) For any \((v,a) \in (\omega+1) \times T\), \(D_va \in OT\) is equivalent to \(a \in OT\) and \(a' < a\) for any \(a' \in G_va\).
 * 3) For any \((a_i)_{i=0}^{k} \in PT^{k+1}\), \((a_0,\ldots,a_k) \in OT\) is equivalent to \((a_i)_{i=0}^{k} \in OT^{k+1}\) and \(a_k \leq \cdots \leq a_0\).

We define a map \begin{eqnarray*} o \colon OT & \to & C_0(\epsilon_{\Omega_{\omega}+1}) \\ a & \mapsto & o(a) \end{eqnarray*} in the following inductive way: Buchholz verified that the map \(o\) satisfies the following:
 * 1) If \(a = 0\), then \(o(a) = 0\).
 * 2) If \(a = D_va'\) for some \((v,a') \in (\omega+1) \times OT\), then \(o(a) = \psi_v(o(a'))\).
 * 3) If \(a = (a_0,\ldots,a_k)\) for some \((a_i)_{i=0}^{k} \in (OT \cap PT)^{k+1}\) with \(k \in \mathbb{N} \setminus \{0\}\), then \(o(a) = o(a_0) \# \cdots \# o(a_k)\), where \(\#\) denotes the descending sum of ordinals, which coincides with the usual addition by the definition of \(OT\).
 * The map \(o\) is an order-preserving bijective map with respect to \(<\) and \(\in\). In particular, \(<\) is a recursive strict well-ordering on \(OT\).
 * For any \(a \in OT\) satisfying \(a < D_10\), \(o(a)\) coincides with the ordinal type of \(<\) restricted to the countable subset \(\{x \in OT \mid x < a\}\).
 * The ordinal \(\psi_0(\varepsilon_{\Omega_{\omega}+1})\) coincides with the ordinal type of \(<\) restricted to the rcursive subset \(\{x \in OT \mid x < D_10\}\). In particular, \((\{x \in OT \mid x < D_10\},<)\) is an ordinal notation equivalent to \(\psi_0(\varepsilon_{\Omega_{\omega}+1})\), and hence \((OT,<)\) is an ordinal notation whose ordinal type is strictly greater than the countable limit of \(\psi\).
 * For any \(v \in \mathbb{N} \setminus \{0\}\), the well-foundedness of \(<\) restricted to the recursive subset \(\{x \in OT \mid x < D_0D_{v+1}0\}\) in the sense of the non-existence of a primitive recursive infinite descending sequence is not provable under \(\textrm{ID}_v\).
 * The well-foundedness of \(<\) restricted to the recursive subset \(\{x \in OT \mid x < D_0D_{\omega}0\}\) in the same sense above is not provable under \(\Pi_1^1-\textrm{CA}_0\).

Extension
We rewrite Buchholz's definition 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

\(\Omega_\nu=\left\{\begin{array}{lcr} 1\text{ if }\nu=0 \\ \text{smallest ordinal with cardinality }\aleph_\nu \text{ if }\nu>0 \\ \end{array}\right.\)

and \(\omega\) is the smallest infinite ordinal.

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 \(C_n\).

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.

Limit of this notation must be omega fixed point \(\psi_0(\Omega_{\Omega_{\Omega_{\cdots}}})=\psi_0(\psi_{\psi_{\cdots}(0)}(0)) = \psi_0(\Lambda)\) (see the next section).

Normal form
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 \(\beta_i \in C_{\nu_i}(\beta_i)\) also written in normal form. More precisely, the normality of an expression of an ordinal can be described in a primitive 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 \(\alpha[0]=\alpha-1\). If \(\alpha\) is a limit ordinal then \(\text{cof}(\alpha)\in\{\omega\}\cup\{\Omega_{\mu+1}|\mu\geq 0\}\).

For nonzero ordinals \(\alpha<\Lambda\), written in normal form, fundamental sequences are defined as follows:


 * 1) If \(\alpha=\psi_{\nu_1}(\beta_1)+\psi_{\nu_2}(\beta_2)+\cdots+\psi_{\nu_k}(\beta_k)\) where \(k\geq2\) then \(\text{cof}(\alpha)=\text{cof}(\psi_{\nu_k}(\beta_k))\) and \(\alpha[\eta]=\psi_{\nu_1}(\beta_1)+\cdots+\psi_{\nu_{k-1}}(\beta_{k-1})+(\psi_{\nu_k}(\beta_k)[\eta])\),
 * 2) If \(\alpha=\psi_{0}(0)=1\), then \(\text{cof}(\alpha)=1\) and \(\alpha[0]=0\),
 * 3) If \(\alpha=\psi_{\nu+1}(0)\), then \(\text{cof}(\alpha)=\Omega_{\nu+1}\) and \(\alpha[\eta]=\Omega_{\nu+1}[\eta]=\eta\),
 * 4) If \(\alpha=\psi_{\nu}(0)\) and \(\text{cof}(\nu)\in\{\omega\}\cup\{\Omega_{\mu+1}|\mu\geq 0\}\), then \(\text{cof}(\alpha)=\text{cof}(\nu)\) and \(\alpha[\eta]=\psi_{\nu[\eta]}(0)=\Omega_{\nu[\eta]}\),
 * 5) If \(\alpha=\psi_{\nu}(\beta+1)\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[\eta]=\psi_{\nu}(\beta)\cdot \eta\) (and note: \(\psi_\nu(0)=\Omega_\nu\)),
 * 6) If \(\alpha=\psi_{\nu}(\beta)\) and \(\text{cof}(\beta)\in\{\omega\}\cup\{\Omega_{\mu+1}|\mu<\nu\}\) then \(\text{cof}(\alpha)=\text{cof}(\beta)\) and \(\alpha[\eta]=\psi_{\nu}(\beta[\eta])\),
 * 7) If \(\alpha=\psi_{\nu}(\beta)\) 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 original system of fundamental sequences up to \(\psi_0(\varepsilon_{\Omega_{\omega}+1})\) in Buchholz hierarchy. If \(\alpha=\Lambda\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[0]=0\) and \(\alpha[\eta+1]=\psi_{\alpha[\eta]}(0)=\Omega_{\alpha[\eta]}\).

For comparison of ordinals written in normal form use the following property:

if \(\alpha<\beta\) and \(1\le\eta<\omega\) then \(\left\{\begin{array}{lcr} 0<\psi_\alpha(\gamma)\cdot\eta <\psi_\beta(\delta)\\ 0<\psi_\gamma(\alpha)\cdot\eta<\psi_\gamma(\beta)\\ \end{array}\right.\)

The comparison of expressions of ordinals can be described in a primitive recursive way with respect to the corresponding ordinal notation system extending the original ordinal notation system \(OT\). In particular, the system of fundamental sequences above induce a primitive recursive system of fundamental sequences on the corresponding ordinal notation, and hence the fast-growing hierarchy associated to it is recursive.