User blog:Denis Maksudov/FS for Hypcos's notation (with weakly inaccessibles cardinals) up to Ψ(M^2)

This is the continuation of my previous post. I added few new rules in fundamental sequences system to extend up to \(\psi(I_{I_{I...}})=\psi(M^2)\). That allow to define FS for Hypcos's notation with weakly inaccessibles. I publish it to take into account possible critical remarks and after this to add this FS-system in article List of systems of fundamental sequences.

Defenition:

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

\(\Omega_\alpha\) with \(\alpha>0\) is the \(\alpha\)-th uncountable cardinal, \(I_\alpha\) with \(\alpha>0\) is the \(\alpha\)-th weakly inaccessible cardinal and for this notation \(I_0=\Omega_0=0\).

Then,

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

\(C_{n+1}(\alpha,\beta) = \{\gamma+\delta|\gamma,\delta\in C_n(\alpha,\beta)\}\)

\(\cup \{\Omega_\gamma|\gamma\in C_n(\alpha,\beta)\}\)

\(\cup \{I_\gamma|\gamma\in C_n(\alpha,\beta)\}\)

\(\cup \{\psi_\pi(\gamma)|\pi,\gamma\in C_n(\alpha,\beta)\wedge\gamma<\alpha\}\)

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

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

Properties


 * 1) \(\psi_{\pi}(0)=1\)
 * 2) \(\psi_{\Omega_1}(\alpha)=\omega^\alpha\) for \(\alpha<\varepsilon_0\)
 * 3) \(\psi_{\Omega_{\nu+1}}(\alpha)=\omega^{\Omega_\nu+\alpha}\) for \(1\le\alpha<\varepsilon_{\Omega_\nu+1}\) and \(\nu>0\)

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 \(\alpha=I_\beta\), then the standard form for \(\alpha\) is \(I_\beta\) where \(\beta\) is expressed in standard form
 * 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 \(\alpha=\psi_\pi(\beta)\), 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 \(\alpha\in L\Leftrightarrow\text{cof}(\alpha)\in\{\omega\}\cup\{\Omega_{\nu}|\nu>0\}\cup\{ I_{\mu}|\mu>0\}\) where \(S\) denotes the set of successor ordinals and \(L\) denotes the set of limit ordinals.

For non-zero ordinals written in standard form fundamental sequences defined as follows:


 * 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 \(\alpha=\psi_{\pi}(0)\) then \(\alpha=\text{cof}(\alpha)=1\) and \(\alpha[0]=0\)
 * 3) If \(\alpha=\psi_{\Omega_{\nu+1}}(1)\) then \(\text{cof}(\alpha)=\omega\) and \(\left\{\begin{array}{lcr} \alpha[\eta]=\Omega_{\nu}\cdot\eta \text{ if }\nu>0 \\ \alpha[\eta]=\eta \text{ if }\nu=0\\ \end{array}\right.\)
 * 4) If \(\alpha=\psi_{\Omega_{\nu+1}}(\beta+1)\) and \(\beta\geq 1\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[\eta]=\psi_{\Omega_{\nu+1}}(\beta)\cdot\eta\)
 * 5) If \(\alpha=\psi_{ I_{\nu+1}}(1)\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[0]=I_\nu+1\) and \(\alpha[\eta+1]=\Omega_{\alpha[\eta]}\).
 * 6) If \(\alpha=\psi_{ I_{\nu+1}}(\beta+1)\) and \(\beta\geq 1\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[0]=\psi_{ I_{\nu+1}}(\beta)+1\) and \(\alpha[\eta+1]=\Omega_{\alpha[\eta]}\).
 * 7) If \(\alpha=\pi\) then \(\text{cof}(\alpha)=\pi\) and \(\alpha[\eta]=\eta\)
 * 8) If \(\alpha=\Omega_\nu\) and \(\nu\in L\) then \(\text{cof}(\alpha)=\text{cof}(\nu)\) and \(\alpha[\eta]=\Omega_{\nu[\eta]}\)
 * 9) If \(\alpha=I_\nu\) and \(\nu\in L\) then \(\text{cof}(\alpha)=\text{cof}(\nu)\) and \(\alpha[\eta]=I_{\nu[\eta]}\).
 * 10) If \(\alpha=\psi_\pi(\beta)\) and \(\text{cof}(\beta)\in\{\omega\}\cup\{\rho|\rho<\pi\}\) then \(\text{cof}(\alpha)=\text{cof}(\beta)\) and \(\alpha[\eta]=\psi_\pi(\beta[\eta])\)
 * 11) 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 \(\psi(M^2)\) where \(\psi\) is a shorthand for \(\psi_{\Omega_1}\) and \(M\) is the first weakly Mahlo cardinal. If \(\alpha=\psi(M^2)\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[\eta]=\psi(\beta[\eta])\) where \(\beta[0]=1\) and \(\beta[\eta+1]=I_{\beta[\eta]}\).