Buchholz psi functions

Buchholz's psi-functions are a hierarchy of single-argument ordinal functions $$\psi_\nu(\alpha)$$ introduced by German mathematician Wilfried Buchholz in 1986. These functions are a simplified version of the $$\theta$$-functions, but nevertheless have the same strength as those. Later on this approach was extended by Jäger and Schütte.

Definition
Buchholz defined his functions as follows. Define: The functions &psi;v(&alpha;) for &alpha; an ordinal, v an ordinal at most &omega;, are defined by induction on &alpha; as follows: where Cv(&alpha;) is the smallest set such that
 * &Omega;&xi; = &omega;&xi; if &xi; > 0, &Omega;0 = 1
 * &psi;v(&alpha;) is the smallest ordinal not in Cv(&alpha;)
 * Cv(&alpha;) contains all ordinals less than &Omega;v
 * Cv(&alpha;) is closed under ordinal addition
 * Cv(&alpha;) is closed under the functions &psi;u (for u&le;&omega;) applied to arguments less than &alpha;.

The limit of this notation is the Takeuti–Feferman–Buchholz ordinal.

Properties
Let $$P$$ be the class of additively principal ordinals. Buchholz showed following properties of this functions:


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

Normal form
The normal form for 0 is 0. If $$\alpha$$ is a nonzero ordinal number $$\alpha<\Omega_\omega$$ 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 $$\nu_i\in\mathbb N_0, k\in\mathbb N_{>0}, \beta_i\in C_{\nu_i}(\beta_i)$$ and $$\psi_{\nu_1}(\beta_1)\geq\psi_{\nu_2}(\beta_2)\geq\cdots\geq\psi_{\nu_k}(\beta_k)$$ and each $$ \beta_i$$ is also written in normal form.

Fundamental sequences
The fundamental sequence for an ordinal number $$\alpha$$ with cofinality $$\operatorname{cof}(\alpha)=\beta$$ 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 $$\operatorname{cof}(\alpha)=1$$ and the fundamental sequence has only one element $$\alpha[0]=\alpha-1$$. If $$\alpha$$ is a limit ordinal then $$\operatorname{cof}(\alpha)\in\{\omega\} \cup \{\Omega_{\mu+1}\mid\mu\geq 0\}$$.

For nonzero ordinals $$\alpha<\Omega_\omega$$, 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 $$\operatorname{cof}(\alpha)=\operatorname{cof}(\psi_{\nu_k}(\beta_k))$$ 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 $$\operatorname{cof}(\alpha)=1$$ and $$\alpha[0]=0$$,
 * 3) If $$\alpha=\psi_{\nu+1}(0)$$, then $$\operatorname{cof}(\alpha)=\Omega_{\nu+1}$$ and $$\alpha[\eta]=\Omega_{\nu+1}[\eta]=\eta$$,
 * 4) If $$\alpha=\psi_{\nu}(\beta+1)$$ then $$\operatorname{cof}(\alpha)=\omega$$ and $$\alpha[\eta]=\psi_{\nu}(\beta)\cdot \eta$$ (and note: $$\psi_\nu(0)=\Omega_\nu$$),
 * 5) If $$\alpha=\psi_{\nu}(\beta)$$ and $$\operatorname{cof}(\beta)\in\{\omega\}\cup\{\Omega_{\mu+1}\mid\mu<\nu\}$$ then $$\operatorname{cof}(\alpha)=\operatorname{cof}(\beta)$$ and $$\alpha[\eta]=\psi_{\nu}(\beta[\eta])$$,
 * 6) If $$\alpha=\psi_{\nu}(\beta)$$ and $$\operatorname{cof}(\beta)\in\{\Omega_{\mu+1}\mid\mu\geq\nu\}$$ then $$\operatorname{cof}(\alpha)=\omega$$ and $$\alpha[\eta]=\psi_\nu(\beta[\gamma[\eta]])$$ where $$\left\{\begin{array}{lcr} \gamma[0]=\Omega_\mu \\ \gamma[\eta+1]=\psi_\mu(\beta[\gamma[\eta]])\\ \end{array}\right.$$.

Explanation
Buchholz is working in Zermelo–Fraenkel set theory, that means every ordinal $$\alpha$$ is equal to set $$\{\beta\mid\beta<\alpha\}$$. 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 $$C_\nu^0(\alpha)=\{\beta\mid\beta<\Omega_\nu\}$$.

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


 * all ordinals from previous set $$C_\nu^n(\alpha)$$,
 * all ordinals that can be obtained by summation the additively principal ordinals from previous set $$C_\nu^n(\alpha)$$,
 * 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:


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

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 $$\psi_{\mu}(\eta)$$, where $$\mu\le\omega$$ and $$\eta<\alpha$$.

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

Examples

Consider the following examples:


 * $$C_0^0(\alpha)=\{0\} =\{\beta\mid\beta<1\},$$


 * $$C_0(0)=\{0\}$$ (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 and therefore $$\psi_0(1)=\omega$$ – 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 and therefore $$\psi_0(2)=\omega^2$$.

If $$\alpha=\omega$$ then $$C_0(\alpha)=\{0,\psi(0)=1,\ldots,\psi(1)=\omega,\ldots,\psi(2)=\omega^2,\ldots,\psi(3)=\omega^3, \ldots\}$$ and $$\psi_0(\omega)=\omega^\omega$$.

If $$\alpha=\Omega$$ then $$C_0(\alpha)=\{0,\psi(0) = 1, \ldots, \psi(1) = \omega, \ldots, \psi(\omega) = \omega^\omega, \ldots, \psi(\omega^\omega) = \omega^{\omega^\omega},\ldots\}$$ and $$\psi_0(\Omega)=\varepsilon_0$$ – the smallest epsilon number i.e. first fixed point of $$\alpha=\omega^\alpha$$.

If $$\alpha=\Omega+1$$ then $$C_0(\alpha)=\{0,1,\ldots,\psi_0(\Omega)=\varepsilon_0,\ldots,\varepsilon_0+\varepsilon_0,\ldots,\psi_1(0)=\Omega,\ldots\}$$ and $$\psi_0(\Omega+1)=\varepsilon_0\omega=\omega^{\varepsilon_0+1}$$.

$$\psi_0(\Omega2)=\varepsilon_1$$ the second epsilon number,


 * $$\psi_0(\Omega^2) = \varepsilon_{\varepsilon_\cdots}=\zeta_0$$ i.e. first fixed point of $$\alpha=\varepsilon_\alpha$$,

$$\varphi(\alpha,\beta)=\psi_0(\Omega^\alpha(1+\beta))$$, where $$\varphi$$ denotes the Veblen function,

$$\psi_0(\Omega^\Omega)=\Gamma_0=\varphi(1,0,0)=\theta(\Omega,0)$$, where $$\theta$$ denotes the Feferman's function and $$\Gamma_0$$ is the Feferman–Schütte ordinal,


 * $$\psi_0(\Omega^{\Omega^2})=\varphi(1,0,0,0)$$ is the Ackermann ordinal,


 * $$\psi_0(\Omega^{\Omega^\omega})$$ is the small Veblen ordinal,


 * $$\psi_0(\Omega^{\Omega^\Omega})$$ is the 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:


 * $$C_1^0(0)=\{\beta\mid\beta<\Omega_1\}=\{0,\psi(0) = 1,2, \ldots \text{googol}, \ldots, \psi_0(1)=\omega, \ldots, \psi_0(\Omega) =\varepsilon_0,\ldots$$

$$\ldots,\psi_0(\Omega^\Omega)=\Gamma_0,\ldots,\psi(\Omega^{\Omega^\Omega+\Omega^2}),\ldots\}$$ i.e. includes all countable ordinals. And therefore $$C_1(0)$$ includes all possible sums of all countable ordinals and $$\psi_1(0)=\Omega_1$$ first uncountable ordinal which is greater than all countable ordinal by its definition i.e. smallest number with cardinality $$\aleph_1$$.

If $$\alpha=1$$ then $$C_1(\alpha)=\{0,\ldots,\psi_0(0) = \omega, \ldots, \psi_1(0) = \Omega,\ldots,\Omega+\omega,\ldots,\Omega+\Omega,\ldots\}$$ and $$\psi_1(1)=\Omega\omega=\omega^{\Omega+1}$$.


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


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


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


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


 * $$\psi_1^{k+1}(0)=\Omega\uparrow\uparrow k$$ where $$k$$ is a natural number, $$k \geq 2$$,


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

For case $$\psi(\psi_2(0))=\psi(\Omega_2)$$ the set $$C_0(\Omega_2)$$ includes functions $$\psi_0$$ with all arguments less than $$\Omega_2$$ i.e. such arguments as $$0, \psi_1(0), \psi_1(\psi_1(0)), \psi_1^3(0),\ldots, \psi_1^\omega(0)$$

and then


 * $$ \psi_0(\Omega_2) = \psi_0(\psi_1(\Omega_2)) = \psi_0(\varepsilon_{\Omega+1}).$$

In the 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:


 * $$\theta(\Omega_\nu,0)=\psi_0(\Omega_\nu^\Omega) \text{ (for } 1\le\nu)<\omega$$

Ordinal notation
Buchholz defined an ordinal notation $$\mathsf{(OT, <)}$$ associated to the $$\psi$$ function. We simultaneously define the sets $$T$$ and $$PT$$ as formal strings consisting of $$0, D_v$$ indexed by $$v \in \omega + 1$$, braces and commas in the following way:


 * $$0 \in T \and 0 \in PT$$,
 * $$\forall (v, a) \in (\omega + 1) \times T( D_va \in T \and D_va \in PT) $$.
 * $$\forall (a_0, a_1) \in PT^2((a_0, a_1) \in T)$$.
 * $$\exists s (a_0 = s) \and (a_0, a_1) \in T \times PT \rightarrow (s, a_1) \in T$$.

An element of $$T$$ is called a term, and an element of $$PT$$ is called a principal term. By definition, $$T$$ is a recursive set and $$PT$$ is a recursive subset of $$T$$. Every term is either $$0$$, a principal term or a braced array of principal terms of length $$\geq 2$$. We denote $$a \in PT$$ by $$(a)$$. By convention, every term can be uniquely expressed as either $$0$$ or a braced, non-empty array of principal terms. Since clauses 3 and 4 in the definition of $$T$$ and $$PT$$ are applicable only to arrays of length $$\geq 2$$, this convention does not cause serious ambiguity.

We then define a binary relation $$a < b$$ on $$T$$ in the following way:


 * $$b = 0 \rightarrow a < b = \bot$$
 * $$a = 0 \rightarrow (a < b \leftrightarrow b \neq 0)$$
 * If $$a \neq 0 \and b \neq 0$$, then:
 * If $$a = D_ua' \and b = D_vb'$$ for some $$((u, a'), (v, b')) \in ((\omega + 1) \times T)^2$$, then $$a < b$$ is true if either of the following are true:
 * $$u < v$$
 * $$u = v \and a' < b'$$
 * If $$a = (a_0, ..., a_n) \and b = (b_0, ..., b_m)$$ for some $$(n, m) \in \N^2 \and 1 \leq n + m$$, then $$a < b$$ is true if either of the following are true:
 * $$\forall i \in \N \and i \leq n(n < m \and a_i = b_i)$$
 * $$\exists k \in \N\forall i \in \N \and i < k(k \leq min\{n, m\} \and a_k < b_k \and a_i = b_1)$$

$$<$$ is a recursive strict total ordering on $$T$$. We abbreviate $$(a < b) \or (a = b)$$ as $$a \leq b$$. Although $$\leq$$ itself is not a well-ordering, its restriction to a recursive subset $$OT \subset T$$, which will be described later, forms a well-ordering. 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 way:


 * $$a = 0 \rightarrow G_ua = \varnothing$$
 * Suppose that $$a = D_va'$$ for some $$(v, a') \in (\omega + 1) \times T$$, then:
 * $$u \leq v \rightarrow G_ua = \{a'\} \cup G_ua'$$
 * $$u > v \rightarrow G_ua = \varnothing$$


 * If $$a = (a_0, ..., a_k)$$ for some $$(a_i)_{i=0}^k \in PT^{k + 1}$$ for some $$k \in \N \backslash \{0\}, G_ua = \bigcup_{i=0}^k G_ua_i$$.

$$b \in G_ua$$ is a recursive relation on $$(u, a, b) \in (\omega + 1) \times T^2$$. Finally, we define a subset $$OT \subset T$$ in the following way:


 * $$0 \in OT$$
 * For any $$(v, a) \in (\omega + 1) \times T$$, $$D_va \in OT$$ is equivalent to $$a \in OT$$ and, for any $$a' \in G_va$$, $$a' < a$$.
 * For any $$(a_i)_{i=0}^k \in PT^{k + 1}$$, $$(a_0, ..., a_k) \in OT$$ is equivalent to $$(a_i)_{i=0}^k \in OT$$ and $$a_k \leq ... \leq a_0$$.

$$OT$$ is a recursive subset of $$T$$, and an element of $$OT$$ is called an ordinal term. We can then define a map $$o: OT \rightarrow C_0(\varepsilon_{\Omega_\omega+1})$$ in the following way:


 * $$a = 0 \rightarrow o(a) = 0$$
 * If $$a = D_va'$$ for some $$(v, a') \in (\omega + 1) \times OT$$, then $$o(a) = \psi_v(o(a'))$$.
 * If $$a = (a_0, ..., a_k)$$ for some $$(a_i)_{i=0}^k \in (OT \cap PT)^{k+1}$$ with $$k \in \N \backslash \{0\}$$, then $$o(a) = o(a_0) \# ... \# o(a_k)$$, where $$\#$$ denotes the descending sum of ordinals, which coincides with the usual addition by the definition of $$OT$$.

Buchholz verified the following properties of $$o$$:


 * The map $$o$$ is an order-preserving bijective map with respect to $$<$$ and $$\in$$. In particular, $$o$$ 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 $$\{x \in OT \; | \; x < a\}$$.
 * The Takeuti-Feferman-Buchholz ordinal coincides with the ordinal type of $$<$$ restricted to the recursive subset $$\{x \in OT \; | \; x < D_10\}$$.
 * The ordinal type of $$(OT, <)$$ is greater than the Takeuti-Feferman-Buchholz ordinal.
 * For any $$v \in \N \; \backslash \; \{0\}$$, the well-foundedness of $$<$$ restricted to the recursive subset $$\{x \in OT \; | \; x < D_0D_{v+1}0\}$$ in the sense of the non-existence of a primitive recursive infinite descending sequence is not provable under $$\mathsf{ID}_v$$.
 * The well-foundedness of $$<$$ restricted to the recursive subset$$\{x \in OT \; | \; x < D_0D_\omega0\}$$ in the same sense as above is not provable under $$\Pi^1_1 - CA_0$$.

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 $$NF$$ of predicates on ordinals in $$C_0(\varepsilon_{\Omega_\omega + 1})$$ is defined in the following way:


 * The predicate $$\alpha = _{NF}0$$ on an ordinal $$\alpha \in C_0(\varepsilon_{\Omega_\omega + 1})$$ defined as $$\alpha = 0$$ belongs to $$NF$$.


 * The predicate $$\alpha_0 = _{NF}\psi_{k_1}(\alpha_1)$$ on ordinals $$\alpha_0, \alpha_1 \in C_0(\varepsilon_{\Omega_\omega + 1})$$ with $$k_1 = \omega + 1$$ defined as $$\alpha_0 = \psi_{k_1}(\alpha_1)$$ and $$\alpha_1 = C_{k_1}(\alpha_1)$$ belongs to $$NF$$.
 * The predicate $$\alpha_0 = _{NF}\alpha_1 + ... + \alpha_n$$ on ordinals $$\alpha_0, \alpha_1, ..., \alpha_n \in C_0(\varepsilon_{\Omega_\omega + 1})$$ with an integer $$n > 1$$ defined as $$\alpha_0 = \alpha_1 + ... + \alpha_n$$, $$\alpha_1 \geq ... \geq \alpha_n$$, and $$\alpha_1, ..., \alpha_n \in AP$$ belongs to $$NF$$. Here $$+$$ is a function symbol rather than an actual addition, and $$AP$$ denotes the class of additive principal numbers.

It is also useful to replace the third case by the following one obtained by combining the second condition:


 * The predicate $$\alpha_0 = _{NF}\psi_{k_1}(\alpha_1) + ... + \psi_{k_n}(\alpha_n)$$ on ordinals $$\alpha_0, \alpha_1, ..., \alpha_n \in C_0(\varepsilon_{\Omega_\omega + 1})$$ with an integer $$n > 1$$ and $$k_1, ..., k_n \in \omega + 1$$ defined as $$\alpha_0 = \psi_{k_1}(\alpha_1) + ... + \psi_{k_n}(\alpha_n)$$, $$\psi_{k_1}(\alpha_1) \geq ... \geq \psi_{k_n}(\alpha_n)$$, and $$\alpha_1 \in C_{k_1}(\alpha_1), ..., \alpha_n \in C_{k_n}(\alpha_n) \in AP$$ belongs to $$NF$$.

We note that those two formulations are not equivalent. For example, $$\psi_0(\Omega) + 1 = _{NF} \psi_0(\zeta_1) + \psi_0(0)$$ is a valid formula which is false with respect to the second formulation because of $$\zeta_1 \neq C_0(\zeta_1)$$, while it is a valid formula which is true with respect to the first formulation because of $$\psi_0(\Omega) + 1 = \psi_0(\zeta_1) + \psi_0(0)$$, $$\psi_0(\zeta_1), \psi_0(0) \in AP$$, 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.