Buchholz's function

Buchholz's psi functions are ordinal collapsing functions created by German mathematician Wilfried Buchholz. Although there are many Buchholz's psi functions, we explain the most famous one in this community, which is a hierarchy of single-argument functions \(\psi_\nu(\alpha)\) introduced in 1986, because the other ones are not currently used in this community. 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 \(\omega\) is the smallest infinite ordinal,

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

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 convention \(\Omega_0\) depends on the context even if we are focusing on Buchholz's function. Actually, Buchholz himself uses \(\Omega_0\) as both of shorthands of \(1\) and \(\omega\), however for the ordinal notation to correspond to Buchholz's function as intended, of the two conventions \(\Omega_0=1\) must be used. Also note that \(P(\omega+\omega)=\{\omega\}\) although the sum of all (distinct) elements of \(P(\omega+\omega)\) isn't equal to \(\omega+\omega\).

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 there is no \(\eta<0\) and 0+0=0).

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

\(C_0(1)\) includes \(\psi_0(0)=1\) and all possible sums of 1s, i.e. the natural numbers. It also includes other ordinals like \(\psi_\nu(0)\) for \(\nu\le\omega\) and sums of those:

\(C_0(1)=\{0,1,2,...,\text{googol},\cdots,\text{TREE(googol)},\cdots,\psi_1(0),\cdots,\psi_1(0)+\psi_1(0),\cdots,\psi_2(0),\cdots\}\).

Then \(\psi_0(1)=\omega\) - first transfinite ordinal, which is greater than all natural numbers by its definition. Since \(\omega\) is the least ordinal not in \(C_0(1)\), we don't need to consider its transfinite elements in this case.

\(C_0(2)\) includes \(\psi_0(0)=1, \psi_0(1)=\omega\) (as well as some other uncountable ordinals) and all possible sums of them.

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

For \(C_0(\omega)\) we have set \(C_0(\omega)=\{0,\psi(0)=1,...,\psi(1)=\omega,...,\psi(2)=\omega^2,...,\psi(3)=\omega^3,...,\psi_1(0),\cdots\}\).

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,

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

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

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

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

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

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

\((\psi_1)^4(0)=\Omega^{\Omega^\Omega}\), where the superscript denotes function iteration,

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

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 \(0, \psi_1(0), \psi_1(\psi_1(0)), \psi_1^3(0),...\)

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 \((n,m) \in \mathbb{N}^2\) with \(1 \leq n+m\), \(a < b\) 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(\varepsilon_{\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 recursive 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 \(\Pi_1^1-\textrm{CA}_0\).

Normal form
The normal form for Buchholz's function can be defined by the pull-back of standard form for the ordinal notation associated to it by \(o\). Namely, the set \(\textrm{NF}\) of predicates on ordinals in \(C_0(\varepsilon_{\Omega_{\omega}+1})\) is defined in the following way: It is also useful to replace the third case by the following one obtained by combining the second condition:
 * 1) The predicate \(\alpha =_{\textrm{NF}} 0\) on an ordinal \(\alpha\) in \(C_0(\varepsilon_{\Omega_{\omega}+1})\) defined as \(\alpha = 0\) belongs to \(\textrm{NF}\).
 * 2) The predicate \(\alpha_0 =_{\textrm{NF}} \psi_{k_1}(\alpha_1)\) on ordinals \(\alpha_0, \alpha_1\) in \(C_0(\varepsilon_{\Omega_{\omega}+1})\) with \(k_1 \in \omega+1\) defined as \(\alpha_0 = \psi_{k_1}(\alpha_1)\) and \(\alpha_1 \in C_{k_1}(\alpha_1)\) belongs to \(\textrm{NF}\).
 * 3) The predicate \(\alpha_0 =_{\textrm{NF}} \alpha_1 + \cdots + \alpha_n\) on ordinals \(\alpha_0, \ldots, \alpha_n\) in \(C_0(\varepsilon_{\Omega_{\omega}+1})\) with an integer \(n > 1\) defined as \(\alpha_0 = \alpha_1 + \cdots + \alpha_n\), \(\alpha_1 \geq \cdots \geq \alpha_n\), and \(\alpha_1,\ldots,\alpha_n \in \textrm{AP}\) belongs to \(\textrm{NF}\). Here \(+\) is a function symbol rather than an actual addition, and \(\textrm{AP}\) denotes the class of additie princpal numbers.
 * 3. The predicate \(\alpha_0 =_{\textrm{NF}} \psi_{k_1}(\alpha_1) + \cdots + \psi_{k_n}(\alpha_n)\) on ordinals \(\alpha_0, \ldots, \alpha_n\) in \(C_0(\varepsilon_{\Omega_{\omega}+1})\) with an integer \(n > 1\) and \(k_1,\ldots,k_n \in \omega+1\) defined as \(\alpha_0 = \psi_{k_1}(\alpha_1) + \cdots + \psi_{k_n}(\alpha_n)\), \(\psi_{k_1}(\alpha_1) \geq \cdots \geq \psi_{k_n}(\alpha_n)\), and \(\alpha_1 \in C_{k_1}(\alpha_1), \ldots, \alpha_n \in C_{k_n}(\alpha_n)\) belongs to \(\textrm{NF}\).

We note that those two formulations are not equivalent. For example, \(\psi_0(\Omega) + 1 =_{\textrm{NF}} \psi_0(\zeta_1) + \psi_0(0)\) is a valid formula which is false with respect to the second formulation because of \(\zeta_1 \notin C_0(\zeta_1)\), while it is a valid formula which is true with respect to the first formulation because of \(\psi_0(\Omega) + 1 = \psi_0(\zeta_1) + \psi_0(0)\), \(\psi_0(\zeta_1), \psi_0(0) \in \textrm{AP}\), and \(\psi_0(\zeta_1) \geq \psi_0(0)\). In this way, the notion of normal form heavily depends on the context.

In both formulations, every ordinal in \(C_0(\varepsilon_{\Omega_{\omega+1}})\) can be uniquely expressed in normal form for Buchholz's function.

Fundamental sequences
Buchholz essentially defined a recursive system of fundamental sequences in the first paper on Buchholz's function and also defined another recursive system of fundamental sequences in the other paper, but the other recursive system of fundamental sequences defined by Googology Wiki user Denis Maksudov is more commonly used in googology. See for more details.

Extension
We introduce the extension of Buchholz's definition by Googology Wiki user Denis Maksudov 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 \(C_n\).

The ordinal notation \((OT,<)\) associated to Buchholz's function is extended to the ordinal notation associated to Extended Buchholz's function by a Japanese Googology Wiki user p進大好きbot in a natural way, and is further extended to a \(3\)-ary notation by a Japanese Googology Wiki user kanrokoti. Also, the notions of normal form and fundamental sequences are extended by Denis Maksudov.

Common misconceptions
Here is a list of some common misconceptions regarding Buchholz's function, which appear in many introductory articles on Buchholz's function: