User blog:Denis Maksudov/Canonical fundamental sequences for Hypcos's notation with the first weakly inaccessible cardinal.

Canonical fundamental sequences for Hypcos's notation with the first weakly inaccessible  cardinal.

Defenition:

\(\rho\) and \(\pi\) are always regular cardinals i.e. \(\rho,\pi\in\{\Omega_{\nu+1}\}\cup\{ I\}\) i.e. \(\text{cof}(\rho)=\rho\) and \(\text{cof}(\pi)=\pi\).

\( I\) is the first weakly inaccessible cardinal, \(\Omega_\nu\) with \(\nu>0\) is the \(\alpha\)-th uncountable cardinal and for this notation \(\Omega_0=0\).

Then,

\(C_0(\alpha,\beta) = \beta\cup\{0,I\} \)

\(C_{n+1}(\alpha,\beta) = \{\gamma+\delta|\gamma,\delta\in C_n(\alpha,\beta)\} \cup \{\Omega_\gamma|\gamma\in C_n(\alpha,\beta)\}  \cup \{\psi_\pi(\gamma)|\pi,\gamma\in C_n(\alpha,\beta)\wedge\gamma<\alpha\} \)

\(C(\alpha,\beta) = \bigcup_{n<\omega} C_n(\alpha,\beta) \)

\(\psi_\pi(\alpha) = \min\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\} \)

Properties

\(\psi_{\Omega_1}(\alpha)=\omega^\alpha\)

\(\psi_{\pi}(0)=1\) for \(\pi\le I\)

\(\psi_{\Omega_{\nu+1}}(\alpha)=\omega^{\Omega_\nu+\alpha}\) for \(\alpha>1\) and \(0<\nu< I\)

\(\psi_{\Omega_{\nu+1}}(\alpha)=\omega^{\Omega_\nu+1+\alpha}\) for \(\nu\geq I\)

Standard form


 * 1) If \(\alpha=0\), then the standard form for \(\alpha\) is \(0\).
 * 2) If \(\alpha=\Omega_\beta\), then the standard form for \(\alpha\) is \(\Omega_\beta\) where \(\beta\) is expressed in standard form.
 * 3) If \(\alpha= I\), then the standard form for \(\alpha\) is \( I\).
 * 4) If \(\alpha=\alpha_1+\alpha_2+\cdots+\alpha_n\) with \(n\geq 2\), then the standard form for \(\alpha\) is \(\alpha=\alpha_1+\alpha_2+\cdots+\alpha_n\wedge\alpha_1\geq\alpha_2\geq\cdots\geq\alpha_n\wedge\alpha_1,\alpha_2,\cdots,\alpha_n\in P\), where \(P\) is the set of additive principal numbers and and the \(\alpha_i\) are expressed in standard form.
 * 5) If \(\alpha=I^\gamma\delta\), then the standard form for \(\alpha\) is \(\alpha= I^\gamma\delta\) where \( \delta< I, \gamma< I^\gamma\) and \(\gamma,\delta\) are expressed in standard form.
 * 6) If \(\alpha=\psi_\pi(\beta)\), then the standard form for \(\alpha\) is \(\alpha=\psi_\pi(\beta)\) where \(\pi\) and \(\beta\) are expressed in standard form.

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.

Let \(\alpha\in S\Leftrightarrow\text{cof}(\alpha)=1\) and \(\alpha\in L\Leftrightarrow\text{cof}(\alpha)\in\{\omega\}\cup\{\Omega_{\nu+1}\}\cup\{ I\}\) where \(S\) denotes the set of successor ordinals and \(L\) denotes the set of limit ordinals.

\text{cof}(\alpha)=\omega \text{ and }\alpha[\eta]=\Omega_\nu\cdot\eta \text{ if }\pi= \Omega_{\nu+1} \text{ where } \nu\geq I\\\end{array}\right.\).
 * 1) If \(\alpha=\alpha_1+\alpha_2+\cdots+\alpha_n\) with \(n\geq 2\) then \(\text{cof}(\alpha)=\text{cof}(\alpha_n)\) and \(\alpha[\eta]=\alpha_1+\alpha_2+\cdots+(\alpha_n[\eta])\).
 * 2) If \(\alpha=I^\gamma\delta\) then \(\left\{\begin{array}{lcr} \text{cof}(\alpha)=I \text{ and }\alpha[\eta]=I^\gamma(\delta-1)+I^{\gamma-1}\cdot\eta \text{ if }\gamma,\delta\in S\\ \text{cof}(\alpha)=\text{cof}(\delta)\text{ and }\alpha[\eta]=I^\gamma(\delta[\eta])\text{ if }\delta\in L\\ \text{cof}(\alpha)=\text{cof}(\gamma)\text{ and }\alpha[\eta]=I^\gamma(\delta-1)+I^{\gamma[\eta]}\text{ if }\delta\in S, \gamma\in L\\\end{array}\right.\).
 * 3) If \(\alpha=\psi_{\pi}(0)\) then \(\left\{\begin{array}{lcr} \alpha=\text{cof}(\alpha)=1 \text{ and }\alpha[0]=0 \text{ if }\pi\le I\\
 * 1) If \(\alpha=\psi_{\Omega_{\nu+1}}(\beta+1)\) and \(\beta\geq 1\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[\eta]=\alpha=\psi_{\Omega_{\nu+1}}(\beta)\cdot\eta\).
 * 2) If \(\alpha=\psi_{ I}(\beta+1)\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[0]=\psi_{ I}(\beta)+1\) and \(\alpha[\eta+1]=\Omega_{\alpha[\eta]}\).
 * 3) If \(\alpha=\pi\) then \(\text{cof}(\alpha)=\pi\) and \(\alpha[\eta]=\eta\).
 * 4) If \(\alpha=\Omega_\nu\) and \(\nu\in L\) then \(\text{cof}(\alpha)=\text{cof}(\nu)\) and \(\alpha[\eta]=\Omega_{\nu[\eta]}\).
 * 5) If \(\alpha=\psi_\pi(\beta)\) and \(\text{cof}(\beta)\in\{\omega\}\cup\{\rho|\rho<\pi\}\) then \(\text{cof}(\alpha)=\text{cof}(\beta)\) and \(\alpha[\eta]=\psi_\pi(\beta[\eta])\).
 * 6) If \(\alpha=\psi_\pi(\beta)\) and \(\text{cof}(\beta)\in\{\rho|\rho\geq\pi\}\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[\eta]=\psi_\pi(\beta[\gamma[\eta]])\) with \(\gamma[0]=1\) and \(\gamma[\eta+1]=\psi_{\rho}(\beta[\gamma[\eta]])\).

Limit of this notation is \(\lambda\). If \(\alpha=\lambda\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[0]=I+1\) and \(\alpha[\eta+1]=\Omega_{\alpha[\eta]}\).