User blog:Hyp cos/Degrees of Recursive Inaccessibility and other OCFs

Degrees of Recursive Inaccessibility
Degrees of Recursive Inaccessibility is an OCF created by Dmytro Taranovsky, earlier than other ordinal notations. It seems to be the simplest OCF and yield a simplest ordinal notation among this kind.

In this OCF, the a in C(a,b,c) means "admissibility degree"; b means degree and is related to how "rare" this ordinal is; c is related to the absolute size of this ordinal. It is addition-free and yield \(C(0,a,b)=b+\omega^a\) iff \(a\omega\land\beta\) is admissible \(\land\forall\gamma<\beta\exists\beta'(\gamma<\beta'<\beta\land\beta'\in I^\alpha)\}\), and \(I^\alpha=\{\beta|\forall\alpha'<\alpha\forall\gamma<\beta\exists\beta'(\gamma<\beta'<\beta\land\beta'\in I^\alpha')\}\) for limit ordinal \(\alpha\). So \(I^{\alpha+1}\) is the class of recursively \(\alpha\)-inaccessible ordinals.

Let \(A^\alpha\) be the closure of \(I^\alpha\). Ordinals in \(A^\alpha\) are called of admissibility degree \(\alpha\). Alternatively we could define: every ordinal is of admissibility degree 0, while 0 is only of admissibility degree 0; an ordinal is of admissibility degree \(\alpha+1\) iff it is recursively \(\alpha\)-inaccessible, or in the closure of them; for limit \(\alpha\), having admissibility degree \(\alpha\) is the same as having every admissibility degree \(<\alpha\).

\begin{eqnarray*} H_0(\alpha,\beta) &=& \beta\\ H_{i+1}(\alpha,\beta) &=& \{C(\xi,\gamma,\delta)|\xi,\gamma,\delta\in H_i(\alpha,\beta)\land\gamma<\alpha\}\\ H(\alpha,\beta) &=& \cup_{i<\omega}H_i(\alpha,\beta)\\ C(\alpha,\beta,\gamma) &=& \min\{\delta|\delta\in I^\alpha\land\delta\notin H(\beta,\delta)\land\delta>\gamma\} \end{eqnarray*}

Syntax
As an ordinal notation, terms are constructed using a constant: 0, and a ternary function symbol: C.

Comparison
Terms are compared in this way, resulting "<", ">" or "=".
 * 1) \(0<C(a,b,c)\)
 * 2) \(C(a,b,c)<C(d,e,f)\leftrightarrow C(a,b,c)\le f\lor(c<C(d,e,f)\land(a<d\lor(a=d\land b<e)))\)

Standard Terms
Some terms are standard. One standard term means one ordinal, and different standard terms mean different ordinals. The ordering of ordinals is defined to be exactly the ordering of standard terms.
 * 0 is standard
 * C(a,b,c) is standard if all the following are true
 * a,b,c are all standard
 * If c=C(d,e,f), then \(a<d\lor(a=d\land b\le e)\)
 * \(b\in H(b,C(a,b,c))\), where
 * \(x\in H(y,z)\leftrightarrow x<z\lor(x=C(x_1,x_2,x_3)\land x_1,x_2,x_3\in H(y,z)\land x_2<y)\)

\(\psi\)
The \(\psi\) is aimed for an OCF beginning with these values: Here, this OCF makes the "\(\Omega\)" also a result from OCF, while in the ordinal notation "\(\Omega\)" is a shorthand.
 * \(\psi(0)=1\)
 * \(\psi(1)=\omega\)
 * \(\psi(2)=\omega^2\)
 * \(\psi(\omega)=\omega^\omega\)
 * \(\psi(\Omega)=\varepsilon_0\)
 * \(\psi(\Omega+1)=\varepsilon_0\cdot\omega\)
 * \(\psi(\Omega+\psi(\Omega))=\varepsilon_0^2\)
 * \(\psi(\Omega+\psi(\Omega+1))=\varepsilon_0^\omega\)
 * \(\psi(\Omega\cdot2)=\varepsilon_1\)
 * \(\psi(\Omega\cdot\omega)=\varepsilon_\omega\)
 * \(\psi(\Omega\cdot\psi(\Omega))=\varepsilon_{\varepsilon_0}\)
 * \(\psi(\Omega^2)=\zeta_0\)
 * \(\psi(\Omega^\omega)=\varphi(\omega,0)\)
 * \(\psi(\Omega^{\psi(\Omega)})=\varphi(\varepsilon_0,0)\)
 * \(\psi(\Omega^\Omega)=\Gamma_0\)
 * \(\psi(\Omega^\Omega+1)=\Gamma_0\cdot\omega\)
 * \(\psi(\Omega^\Omega+\Omega)=\varepsilon_{\Gamma_0+1}\)
 * \(\psi(\Omega^\Omega+\Omega^{\psi(\Omega^\Omega)})=\varphi(\Gamma_0,1)\)
 * \(\psi(\Omega^\Omega\cdot2)=\Gamma_1\)
 * \(\psi(\Omega^{\Omega+1})=\varphi(1,1,0)\)
 * \(\psi(\Omega^{\Omega^2})=\varphi(1,0,0,0)\)
 * \(\psi(\Omega^{\Omega^n\cdot\alpha_n+\cdots+\Omega\cdot\alpha_1+\alpha_0}\cdot(1+\beta))=\varphi(\alpha_n,\cdots,\alpha_1,\alpha_0,\beta)\)
 * \(\psi(\Omega^{\Omega^\omega})=\)SVO
 * \(\psi(\Omega^{\Omega^\Omega})=\)LVO

The \(\xi\) in \(\psi^\xi_\nu(\alpha)\) means "admissibility degree"; \(\alpha\) is related to how "rare" this ordinal is; \(\nu\) is related to the absolute size of this ordinal.

OCF definition
\(A^\alpha\) is defined as before, \(B^1=A^1\cup\{1\}\) and \(B^\alpha=A^\alpha\) for \(\alpha\neq1\). \(\text{enum}\) is the enumeration function, so \(\text{enum}(X)(\alpha)\) is the \(1+\alpha\)th ordinal in class \(X\).

\begin{eqnarray*} H_0(\alpha,\beta)&=&\beta\\ H_{i+1}(\alpha,\beta)&=&\{\gamma+\delta|\gamma,\delta\in H_i(\alpha,\beta)\}\\ &\cup&\{\psi^\xi_\nu(\gamma)|\gamma,\nu,\xi\in H_i(\alpha,\beta)\land\gamma<\alpha\}\\ H(\alpha,\beta)&=&\bigcup_{i<\omega}H_i(\alpha,\beta)\\ \psi^\xi_\nu(\alpha)&=&\min(A^\xi\backslash H(\alpha,\text{enum}(B^{\xi+1})(\nu))) \end{eqnarray*}

As we see, \(\psi\) uses addition in its definition, making it more complicated.

\(\theta\)
The \(\theta\) is aimed for an OCF beginning with these values: Here, this OCF makes the "\(\Omega\)" also a result from OCF, while in the ordinal notation "\(\Omega\)" is a shorthand.
 * \(\theta(0,\beta)=\omega^\beta\)
 * \(\theta(1,\beta)=\varepsilon_\beta\)
 * \(\theta(2,\beta)=\zeta_\beta\)
 * \(\theta(\omega,\beta)=\varphi(\omega,\beta)\)
 * \(\theta(\Omega,\beta)=\Gamma_\beta\)
 * \(\theta(\Omega+1,0)=\varphi(1,1,0)\)
 * \(\theta(\Omega^2,0)=\varphi(1,0,0,0)\)
 * \(\theta(\Omega^n\cdot\alpha_n+\cdots+\Omega\cdot\alpha_1+\alpha_0,\beta)=\varphi(\alpha_n,\cdots,\alpha_1,\alpha_0,\beta)\)
 * \(\theta(\Omega^\omega,0)=\)SVO
 * \(\theta(\Omega^\Omega,0)=\)LVO

The \(\xi\) in \(\theta^\xi(\alpha,\beta)\) means "admissibility degree"; \(\alpha\) is related to how "rare" this ordinal is; \(\beta\) is related to the absolute size of this ordinal.

OCF definition
\(A^\alpha\) is defined as before. \(\text{enum}\) is the enumeration function, so \(\text{enum}(X)(\alpha)\) is the \(1+\alpha\)th ordinal in class \(X\).

\begin{eqnarray*} H_0(\alpha,\beta)&=&\beta\cup\{0\}\\ H_{i+1}(\alpha,\beta)&=&\{\gamma+\delta|\gamma,\delta\in H_i(\alpha,\beta)\}\\ &\cup&\{\theta^\xi(\gamma,\delta)|\gamma,\delta,\xi\in H_i(\alpha,\beta)\land\gamma<\alpha\}\\ H(\alpha,\beta)&=&\bigcup_{i<\omega}H_i(\alpha,\beta)\\ B^\xi(\beta)&=&\{\gamma\in A^\xi|\gamma\notin H(\beta,\gamma)\}\\ \theta^\xi(\alpha,\beta)&=&\text{enum}(B^\xi(\alpha))(\beta) \end{eqnarray*}

\(\theta\) also uses addition in its definition.

\(\vartheta\)
The \(\vartheta\) is aimed for an OCF beginning with these values: This OCF is very tricky. C, \(\psi\) and \(\theta\) naturally handle "simple limit point of something" and need one entry for admissibility; while \(\vartheta\) naturally handles "of some class, and also limit point of something", or recursive inaccessibility, and needs one entry for recursive Mahloness.
 * \(\vartheta(0)=1\)
 * \(\vartheta(1)=\omega\)
 * \(\vartheta(2)=\omega^2\)
 * \(\vartheta(\omega)=\omega^\omega\)
 * \(\vartheta(\Omega)=\varepsilon_0\)
 * \(\vartheta(\varepsilon_0)=\varepsilon_0\cdot\omega\)
 * \(\vartheta(\varepsilon_0\cdot\omega)=\varepsilon_0^\omega\)
 * \(\vartheta(\Omega+1)=\varepsilon_1\)
 * \(\vartheta(\Omega+\vartheta(\Omega))=\varepsilon_{\varepsilon_0}\)
 * \(\vartheta(\Omega\cdot2)=\zeta_0\)
 * \(\vartheta(\zeta_0)=\zeta_0\cdot\omega\)
 * \(\vartheta(\Omega+\zeta_0)=\varepsilon_{\zeta_0+1}\)
 * \(\vartheta(\Omega\cdot2+1)=\zeta_1\)
 * \(\vartheta(\Omega\cdot3)=\varphi(3,0)\)
 * \(\vartheta(\Omega\cdot\omega)=\varphi(\omega,0)\)
 * \(\vartheta(\Omega^2)=\Gamma_0\)
 * \(\vartheta(\Omega\cdot\Gamma_0)=\varphi(\Gamma_0,1)\)
 * \(\vartheta(\Omega^2+1)=\Gamma_1\)
 * \(\vartheta(\Omega^2+\Omega)=\varphi(1,1,0)\)
 * \(\vartheta(\Omega^3)=\varphi(1,0,0,0)\)
 * \(\vartheta(\Omega^n\cdot\alpha_n+\cdots+\Omega\cdot\alpha_1+\alpha_0,\beta)=\varphi(\alpha_n,\cdots,\alpha_1,\alpha_0)\)
 * \(\vartheta(\Omega^\omega)=\)SVO
 * \(\vartheta(\Omega^\Omega)=\)LVO

The \(\xi\) in \(\vartheta^\xi_\pi(\alpha)\) means "Mahloness degree" (instead of "admissibility degree"); \(\alpha\) is related to how "rare" this ordinal is; \(\pi\) is related to the absolute size of this ordinal.

OCF definition
Ordinals in \(A^\alpha\) are called of Mahloness degree \(\alpha\), defined as Alternatively define \(M^0=\text{Ord}\), \(M^{\alpha+1}\) is the class of such admissible ordinal \(\beta\) that is Mahlo on \(M^\alpha\cap\beta\), and \(M^\alpha=\{\beta|\forall\alpha'<\alpha\forall\gamma<\beta\exists\beta'(\gamma<\beta'<\beta\land\beta'\in M^\alpha')\}\) for limit ordinal \(\alpha\), and \(A^\alpha\) is the closure of \(M^\alpha\).
 * 1) every ordinal is of Mahloness degree 0, while 0 is only of Mahloness degree 0;
 * 2) an ordinal is of Mahloness degree \(\alpha+1\) iff it is recursively \(\alpha\)-Mahlo, or in the closure of them;
 * 3) for limit \(\alpha\), having Mahloness degree \(\alpha\) is the same as having every Mahloness degree \(<\alpha\).

Let \(\Omega(\alpha)=\min A^{1+\alpha}\).

\begin{eqnarray*} H_0(\alpha,\beta)&=&\beta\cup\{0\}\\ H_{i+1}(\alpha,\beta)&=&\{\gamma+\delta|\gamma,\delta\in H_i(\alpha,\beta)\}\\ &\cup&\{\Omega(\gamma)|\gamma\in H_i(\alpha,\beta)\}\\ &\cup&\{\vartheta^\xi_\pi(\gamma)|\gamma,\pi,\xi\in H_i(\alpha,\beta)\land\gamma<\alpha\}\\ H(\alpha,\beta)&=&\bigcup_{i<\omega}H_i(\alpha,\beta)\\ \vartheta_\pi^\xi(\alpha)&=&\min(\{\beta<\pi|H(\alpha,\beta)\cap\pi\subseteq\beta\land\alpha,\pi\in H(\alpha,\beta)\land\beta\in A^\xi\}\cup\{\pi\}) \end{eqnarray*}

\(\pi<\vartheta_\pi^\xi(\alpha)\) does not happen, so we always need a large ordinal and then construct ordinals downwards. As a cost, \(\Omega(\alpha)\) is necessary, making it more complicated than \(\psi\) and \(\theta\).