User blog:Denis Maksudov/The extension of Buchholz's function

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

Definition

\(C_\nu^0(\alpha) = \Omega_\nu\),

\(C_\nu^{n+1}(\alpha) = C_\nu^n(\alpha) \cup \{\gamma | P(\gamma) \subseteq C_\nu^n(\alpha)\} \cup \{\psi_\mu(\xi) | \xi \in \alpha \cap C_\nu^n(\alpha) \wedge \xi \in C_mu(\xi) \wedge \mu \leq \omega\}\),

\(C_\nu(\alpha) = \bigcup_{n < \omega} C_\nu^n (\alpha)\),

\( \psi_\nu(\alpha) = \min\{\gamma | \gamma \not\in C_\nu(\alpha)\}\),

where

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

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

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

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

\(\gamma=\gamma_1+\cdots+\gamma_k\) and \(\gamma_1\geq\cdots\geq\gamma_k\).

Thus \(C_\nu(\alpha)\) denotes the set of all ordinals which can be generated from ordinals \(<\aleph_\nu\) by the functions + (addition) and \(\psi_{\mu\le\omega}(\xi<\alpha)\).

Properties

Buchholz showed following properties of this functions:

\(\psi_\nu(0)=\Omega_\nu\),

\(\psi_\nu(\alpha)\in P\),

\(\psi_\nu(\alpha+1)=\text{min}\{\gamma\in P: \psi_\nu(\alpha)<\gamma\}\),

\(\Omega_\nu\le\psi_\nu(\alpha)<\Omega_{\nu+1} \),

\(\psi_0(\alpha)=\omega^\alpha \text{ if }\alpha<\varepsilon_0\),

\(\psi_\nu(\alpha)=\omega^{\Omega_\nu+\alpha} \text{ if }\alpha<\varepsilon_{\Omega_\nu+1}\),

\(\theta(\varepsilon_{\Omega_\nu+1},0)=\psi(\varepsilon_{\Omega_\nu+1})\) for \(0<\nu\le\omega\).

Extension

Let me 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

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

There is only one little detail difference with original Buchholz definition: ordinal \(\mu\) is not limited by \(\omega\), now ordinal \(\mu\) belong 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 is omega fixed point \(\psi(\Omega_{\Omega_{\Omega_{...}}})\).

Explanation

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

\(C_0(0)=\{0\}\) (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 \(\psi_0(0)=1, \psi_0(0)=\omega\) 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 \(\psi_0(\Omega)=\varepsilon_0\).

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

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

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

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

\(\psi_0(\Omega^\Omega)=\Gamma_0=\theta(\Omega,0)\),

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

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

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

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

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

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

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

\(\psi_1^5(0)=\Omega^{\Omega^\Omega}\),

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

and so on.

Normal form

The ordinal \(\alpha\) is an additive principal number (\(\alpha\in \text{P}\)) if \(\beta+\gamma<\alpha\) for all \(\beta,\gamma<\alpha\)

If \(\alpha\notin \text{P}\) (i.e. \(\alpha\) is not additive principal number) then normal form for \(\alpha:\)

\(\alpha_1+\cdots+\alpha_k\) where \(\alpha_1,...,\alpha_k\in \text{P}\) and \(\alpha_1\geq\cdots\geq\alpha_k\).

If \(\alpha\in \text{P}\) and \(\alpha=\Omega_\nu^\beta \gamma\) then normal form for \(\alpha:\)

\(\alpha=\Omega_\nu^\beta \gamma\) where \(\beta, \gamma<\alpha\) and \(\text{cof}(\beta)\le \Omega_\nu\) and \(\text{cof}(\gamma)< \Omega_\nu\).

Now let's define for this line and all lines below \(\Omega_0=\omega\). Then \(\text{cof}(\Omega_\nu)= \Omega_\nu\) and \(\text{cof}(s)= 1\), where \(s\) is a successor ordinal.

If \(\alpha\notin \text{P}\) is an ordinal, such that \(\Omega_\nu\le\alpha<\Omega_{\nu+1}\) then normal form for \(\alpha\):

\(\Omega_\nu^{\beta_1}\gamma_1+\Omega_\nu^{\beta_2}\gamma_2+\cdots+\Omega_\nu^{\beta_k}\gamma_k+\gamma_{k+1}\), where


 * \(\beta_1 \geq \beta_2 \geq \cdots \geq \beta_k\geq 1\),


 * \(\text{cof}(\beta_m)\le\Omega_\nu\) for \(1\le m \le k\),


 * \(\text{cof}(\gamma_n)<\Omega_\nu\) for \(1\le n \le k+1\).

Fundamental sequences

The fundamental sequence for an ordinal with cofinality \(\Omega_\nu\) is a distinguished strictly increasing sequence with length \(\Omega_\nu\), which has the ordinal as its limit. Let \(\alpha\in s\) denotes \(\alpha\) is a successor ordinal and \(\alpha\in L_\nu\) denotes \(\alpha\) is a limit ordinal with cofinality \(\Omega_\nu\) (with length of fundamental sequence \(\Omega_\nu\)). For example, \(\alpha\in L_0\) denotes a limit countable ordinal with cofnality \(\Omega_0=\omega\).

The fundamental sequences are defined as follows:

1) If \(\alpha=\alpha_1+\alpha_2+\cdots+\alpha_k\), where \(\alpha_1\geq\alpha_2\geq\cdots\geq\alpha_k\) then \(\text{cof}(\alpha)=\text{cof}(\alpha_k)\) and \(\alpha[\eta]=\alpha_1+\alpha_2+\cdots+(\alpha_k[\eta])\),

2) if \(\alpha=\psi_\nu(\beta+1)\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[n]=\psi_\nu(\beta)\cdot n\), and note that \(\psi_0(0)=1\) and \(\psi_\nu(0)=\Omega_\nu\) for \(\nu>0\),

3) if \(\alpha=\psi_\nu(\beta)\) and \(\beta\in L_{\mu\le\nu}\) then \(\text{cof}(\alpha)=\text{cof}(\beta)\) and \(\alpha[\eta]=\psi_\nu(\beta[\eta])\),

4) if \(\alpha=\psi_\nu(\beta)\) and \(\beta\in L_{\mu+1>\nu}\) then \(\text{cof}(\alpha)=\omega\) and \(\left\{\begin{array}{lcr} \alpha[\eta]=\psi_\nu(\beta[\gamma[\eta]])\\ \gamma[0]=\Omega_\mu \text{ if }\mu\geq 1\\ \gamma[0]=0\text{ if }\mu=0\\ \gamma[\eta+1]=\psi_\mu(\beta[\gamma[\eta]])\\ \end{array}\right.,\)

5) if \(\alpha=\Omega_\nu^\beta \gamma\) and \(\beta,\gamma\in s\) then \(\text{cof}(\alpha)=\Omega_\nu\) and

\(\alpha[\eta]=\Omega_\nu^\beta (\gamma -1)+\Omega_\nu^{\beta-1}\cdot \eta\),

6) if \(\alpha=\Omega_\nu^\beta \gamma\) and \(\gamma \in L_{\mu<\nu}\) then \(\text{cof}(\alpha)=\text{cof}(\gamma)\) and \(\alpha[\eta]=\Omega_\nu^\beta (\gamma[\eta])\),

7) if \(\alpha=\Omega_\nu^\beta \gamma\) and \(\gamma\in s, \beta \in L_{\mu\le\nu}\) then \(\text{cof}(\alpha)=\text{cof}(\beta)\) and \(\alpha[\eta]=\Omega_\nu^\beta (\gamma-1)+\Omega_\nu^{\beta[\eta]}\),

8) if \(\alpha=\Omega_{\mu+1}\) then \(\text{cof}(\alpha)=\Omega_{\mu+1}\) and \(\alpha(\eta)=\eta\) (as well as \(\omega[n]=n\)),

9) if \(\alpha=\Omega_{\beta}\) and \(\beta\in L_{\mu}\) then \(\text{cof}(\alpha)=\text{cof}(\beta)\) and \(\alpha(\eta)=\Omega_{\beta[\eta]}\).

Rules 1-9 assign FS for each limit ordinal up to omega fixed point \(\psi(\Omega_{\Omega_{\Omega_{...}}})\).