User:Denis Maksudov

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My blogpost in Cantors Attic: Ordinal functions collapsing the least weakly Mahlo cardinal; a system of fundamental sequences

My account in googology.wikia

I’m Denis Maksudov from Ufa State Aviation Technical University, I’m not a mathematician, but when in May of 2016 I occasionally have found the site googology.wikia I was so fascinated by large numbers and especially by fast-growing hierarchy, that this hobby led me to the studying of notations and systems of fundamental sequences for ordinals. Later I have started to write my own ordinal notations and systems of FS for them. Below you can see some my works previously published in my blogposts on the sites googology.wikia and Cantor’s Attic:

  • Extended arrows (December 2016)
  • The extended Wilfried Buchholz's functions (April 2017)
  • Fundamental sequences for the functions collapsing \(\alpha\)-weakly inaccessible cardinals (August 2017)
  • Two notations based on a weakly Mahlo cardinal (May 2018)
  • Extension of "illion"-family of number names (May 2018)

General notions

Small Greek letters \(\alpha, \beta, \gamma, \delta, \eta, \xi, \nu, \mu\) denote ordinals. Each ordinal \(\alpha\) is identified with the set of its predecessors \(\alpha=\{\beta|\beta<\alpha\}\). The least ordinal is zero and it is identified with the empty set.

\(\omega\) is the first transfinite ordinal and the set of all natural numbers.

Every ordinal \(\alpha\) is either zero, or a successor (if \(\alpha=\beta+1\)), or a limit.

An ordinal \(\alpha\) is a limit ordinal if for all \(\beta<\alpha\) there exists an ordinal \(\gamma\) such that \(\beta<\gamma<\alpha\)

\(S\) denotes the set of all successor ordinals and \(L\) denotes the set of all limit ordinals.

An ordinal \(\alpha\) is an additive principal number if \(\alpha>0\) and \(\xi+\eta<\alpha\) for all \(\xi,\eta<\alpha\).

\(P=\{\alpha>0|\forall\beta,\gamma<\alpha(\beta+\gamma<\alpha)\}\) is the set of additive principal numbers.

For every ordinal \(\alpha\notin P\cup\{0\}\) there exist unique \(\alpha_1,..., \alpha_n\in P\) such that \(\alpha=\alpha_1+\cdots+\alpha_n\) and \(\alpha>\alpha _{1}\geq \cdots \geq \alpha _{n}\)

\(\alpha=_{NF}\alpha _{1}+\cdots +\alpha _{n}:\Leftrightarrow \alpha =\alpha _{1}+\cdots +\alpha _{n}\wedge \alpha>\alpha _{1}\geq \cdots \geq \alpha _{n}\wedge \alpha _{1},... ,\alpha _{n}\in P\)

The cofinality of a limit ordinal \(\alpha\) is the least length of increasing sequence such that the limit of this sequence is the ordinal \(\alpha\).

\(\text{cof}(\alpha)\) denotes the cofinality of an ordinal \(\alpha\).

An ordinal \(\alpha\) is uncountable regular cardinal if it is a limit ordinal larger than \(\omega\) and \(\text{cof}(\alpha)=\alpha\).

\(R=\{\alpha\in L|\alpha>\omega\wedge\text{cof}(\alpha)=\alpha\}\) is the set of all uncountable regular cardinals.

The fundamental sequence for a limit 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.

  • If \(\alpha\) is a limit ordinal then \(\alpha\geq\text{cof}(\alpha)\geq\omega\) and \(\alpha=\sup\{\alpha[\eta]|\eta<\text{cof}(\alpha)\}\).
  • If \(\alpha\) is a successor ordinal then \(\text{cof}(\alpha)=1\) and the fundamental sequence has only one element \(\alpha[0]=\alpha-1\).
  • If \(\alpha=0\) then \(\text{cof}(\alpha)=0\) and \(\alpha\) has not fundamental sequence.

Section I. Extended arrows

We can extend Knuth's up-arrow notation to transfinite ordinals. Let’s define for positive integers \(u, q\) and for ordinal number \(\alpha\):

1) \(u\uparrow^0 q=u\times q\)

2) \(u\uparrow^{\alpha+1}1=u\)

3) \(u\uparrow^{\alpha+1}(q+1)=u\uparrow^{\alpha}(u\uparrow^{\alpha+1}q)\)

4) \(u\uparrow^{\alpha}q=u\uparrow^{\alpha[q]}u\) iff \(\alpha\) is a limit ordinal

where \(\alpha [q]\) denotes the \(q\)-th element of the fundamental sequence assigned to the limit ordinal \(\alpha\).

Let’s also define \(u_{\alpha}^q = u\uparrow^{\alpha}q\)

Hence:

1) \(u_0^q=u\times q\)

2) \(u_ {\alpha+1}^1=u\)

3) \(u_{\alpha+1}^{q+1}=u_{\alpha}^{u_{\alpha+1}^q}\)

4) \(u_{\alpha}^q=u_{\alpha[q]}^u\) iff \(\alpha\) is a limit ordinal.

Section II. Extension of "illion"-family of number names

An ordinal \(\alpha\) + a Latin prefix denoting the natural number \(y\) + "illion" = \(10_{\alpha}^ {3\times(y+1)}=10\uparrow^{\alpha}(3\times(y+1))\)

For example:

2-billion \(=10_2^9=10\uparrow^2 9\)

2-trillion \(=10_2^{12}=10\uparrow^2 12\)

and so on up to 2-centillion \(=10_2^{303}=10\uparrow^2 303\)

3-billion \(=10_3^9=10\uparrow^3 9\)

3-trillion \(=10_3^{12}=10\uparrow^3 12\)

and so on up to 3-centillion \(=10_3^{303}=10\uparrow^3 303\)

Examples with ordinals \(\alpha\le\varepsilon_0=\min\{\xi|\xi=\omega^\xi\}\)

Every nonzero ordinal \(\alpha<\varepsilon_0\) can be represented in a unique Cantor normal form \(\alpha=\omega^{\beta_{1}}+ \omega^{\beta_{2}}+\cdots+\omega^{\beta_{k-1}}+\omega^{\beta_{k}}\) where \(\alpha>\beta_1\geq\beta_2\geq\cdots\geq\beta_{k-1}\geq\beta_k\). If \(\beta_k>0\) then \(\alpha\) is a limit and we can assign to it a fundamental sequence as follows

\(\alpha[n]=\omega^{\beta_{1}}+ \omega^{\beta_{2}}+\cdots+\omega^{\beta_{k-1}}+\left\{\begin{array}{lcr} \omega^\gamma n \text{ if } \beta_k=\gamma+1\\ \omega^{\beta_k[n]} \text{ if } \beta_k \text{ is a limit}\\ \end{array}\right.\)

If \(\alpha=\varepsilon_0\) then \(\alpha[0]=0\) and \(\alpha[n+1]=\omega^{\alpha[n]}\)

Examples of applying of fundamental sequences

\(\omega[n]=\omega^1[n]= \omega^0 n=n\) since \(\omega^0=1\)

\(10\uparrow^{\omega} 3 = 10\uparrow^{3} 10\)

\(10\uparrow^{\omega+1} 3 = 10\uparrow^{\omega}(10\uparrow^{\omega}10)=10 \uparrow^{10 \uparrow^{10}10}10\)

Examples of numbers

\(\omega\)-billion \(=10_\omega ^9=10\uparrow^\omega 9\)

\(\omega\)-trillion \(=10_\omega ^{12}=10\uparrow^\omega 12\)

and so on up to \(\omega\)-centillion \(=10_\omega ^{303}=10\uparrow^\omega 303\)

\(\varepsilon_0\)-billion \(=10_{\varepsilon_0}^9=10\uparrow^{\varepsilon_0}9\)

\(\varepsilon_0\)-trillion \(=10_{\varepsilon_0}^{12}=10\uparrow^{\varepsilon_0}12\)

and so on up to \(\varepsilon_0\)-centillion \(=10_{\varepsilon_0}^{303}=10\uparrow^{\varepsilon_0}303\)

Section III. The extended Wilfried Buchholz's functions

Definition of the extended Wilfried Buchholz's functions

We 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 \(\mu\) belongs to previous set \(C_n\). Limit of this notation must be omega fixed point \(\psi_0(\Omega_{\Omega_{\Omega_{...}}})=\psi_0(\psi_{\psi_{...}(0)}(0))\)

Normal form for the extended Wilfried Buchholz's functions

The normal form for 0 is 0. If \(\alpha\) is a nonzero ordinal number \(\alpha<\Xi=\text{min}\{\beta|\psi_\beta(0)=\beta\}\) 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 \(k\) is a positive integer and \(\psi_{\nu_1}(\beta_1)\geq\psi_{\nu_2}(\beta_2)\geq\cdots\geq\psi_{\nu_k}(\beta_k)\) and each \(\nu_i, \beta_i\) are also written in normal form.

Fundamental sequences for the extended Wilfried Buchholz's functions

For nonzero ordinals \(\alpha<\Xi\), written in normal form, fundamental sequences are defined as follows:

  • If \(\alpha=\psi_{\nu_1}(\beta_1)+\psi_{\nu_2}(\beta_2)+\cdots+\psi_{\nu_k}(\beta_k)\) where \(k\geq2\) then \(\text{cof}(\alpha)=\text{cof}(\psi_{\nu_k}(\beta_k))\) and \(\alpha[\eta]=\psi_{\nu_1}(\beta_1)+\cdots+\psi_{\nu_{k-1}}(\beta_{k-1})+(\psi_{\nu_k}(\beta_k)[\eta])\)
  • If \(\alpha=\psi_{0}(0)=1\), then \(\text{cof}(\alpha)=1\) and \(\alpha[0]=0\)
  • If \(\alpha=\psi_{\nu+1}(0)\), then \(\text{cof}(\alpha)=\Omega_{\nu+1}\) and \(\alpha[\eta]=\Omega_{\nu+1}[\eta]=\eta\)
  • If \(\alpha=\psi_{\nu}(0)\) and \(\text{cof}(\nu)\in\{\omega\}\cup\{\Omega_{\mu+1}|\mu\geq 0\}\), then \(\text{cof}(\alpha)=\text{cof}(\nu)\) and \(\alpha[\eta]=\psi_{\nu[\eta]}(0)=\Omega_{\nu[\eta]}\)
  • If \(\alpha=\psi_{\nu}(\beta+1)\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[\eta]=\psi_{\nu}(\beta)\cdot \eta\) (and note \(\psi_\nu(0)=\Omega_\nu\))
  • If \(\alpha=\psi_{\nu}(\beta)\) and \(\text{cof}(\beta)\in\{\omega\}\cup\{\Omega_{\mu+1}|\mu<\nu\}\) then \(\text{cof}(\alpha)=\text{cof}(\beta)\) and \(\alpha[\eta]=\psi_{\nu}(\beta[\eta])\)
  • If \(\alpha=\psi_{\nu}(\beta)\) and \(\text{cof}(\beta)\in\{\Omega_{\mu+1}|\mu\geq\nu\}\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[\eta]=\psi_{\nu}(\beta[\gamma[\eta]])\) where \(\left\{\begin{array}{lcr} \gamma[0]=\Omega_\mu \\ \gamma[\eta+1]=\psi_\mu(\beta[\gamma[\eta]])\\ \end{array}\right.\)

Below in the text the small Greek letter \(\upsilon\) denotes the largest countable limit ordinal such that the ruleset in this section allows to define fundamental sequences for all limit ordinals less than \(\upsilon\). Then \(\text{cof}(\upsilon)=\omega\) and \(\upsilon[\eta]=\psi_0(\alpha[\eta])\) where \(\alpha[0]=0\) and \(\alpha[z+1]=\psi_{\alpha[z]}(0)=\Omega_{\alpha[z]}\) for all integers \(z \geq 0\)

Examples of numbers

\(\upsilon \)-billion \(=10_\upsilon ^9=10\uparrow^\upsilon 9\)

\(\upsilon \)-trillion \(=10_\upsilon ^{12}=10\uparrow^\upsilon 12\)

and so on up to \(\upsilon \)-centillion \(=10_\upsilon ^{303}=10\uparrow^\upsilon 303\)

Section IV. Fundamental sequences for the functions collapsing \(\alpha\)-weakly inaccessible cardinals

Definition of the functions collapsing \(\alpha\)-weakly inaccessible cardinals

An ordinal is \(\alpha\)-weakly inaccessible if it's an uncountable regular cardinal and it's a limit of \(\gamma\)-weakly inaccessible cardinals for all \(\gamma<\alpha\)

Let \(I(\alpha, 0)\) be the first \(\alpha\)-weakly inaccessible cardinal, \(I(\alpha, \beta+1)\) be the next \(\alpha\)-weakly inaccessible cardinal after \(I(\alpha,\beta)\), and \(I(\alpha,\beta)=\sup\{I(\alpha,\gamma)|\gamma<\beta\}\) for limit ordinal \(\beta\)

In this section the variables \(\rho\), \(\pi\) are reserved for uncountable regular cardinals of the form \(I(\alpha,0)\) or \(I(\alpha,\beta+1)\)

Then,

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

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

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

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

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

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

Properties

  • \(I(0,\alpha)=\Omega_{1+\alpha}=\aleph_{1+\alpha}\)
  • \(I(1,\alpha)=I_{1+\alpha}\)
  • \(\psi_{I(0,0)}(\alpha)=\omega^\alpha\) for \(\alpha<\varepsilon_0\)
  • \(\psi_{I(0,\alpha+1)}(\beta)=\omega^{I(0,\alpha)+1+\beta}\) for \(\beta<\varepsilon_{I(0,\alpha)+1}\)

Standard form for ordinals \(\alpha<\psi_{I(1,0,0)}(0)=\text{min}\{\xi|I(\xi,0)=\xi\}\)

  • The standard form for 0 is 0
  • If \(\alpha\) is of the form \(I(\beta,\gamma)\), then the standard form for \(\alpha\) is \(\alpha=I(\beta,\gamma)\) where \(\beta, \gamma<\alpha\) and \(\beta, \gamma\) are expressed in standard form
  • If \(\alpha\) is not additively principal and \(\alpha>0\), then the standard form for \(\alpha\) is \(\alpha=\alpha_1+\alpha_2+\cdots+\alpha_n\), where the \(\alpha_i\) are principal ordinals with \(\alpha_1\geq\alpha_2\geq\cdots\geq\alpha_n\), and the \(\alpha_i\) are expressed in standard form
  • If \(\alpha\) is an additively principal ordinal but not of the form \(I(\beta,\gamma)\), then \(\alpha\) is expressible in the form \(\psi_\pi(\delta)\). Then the standard form for \(\alpha\) is \(\alpha=\psi_\pi(\delta)\) where \(\pi\) and \(\delta\) are expressed in standard form

Fundamental sequences

For non-zero ordinals \(\alpha<\psi_{I(1,0,0)}(0)\) written in standard form fundamental sequences are defined as follows:

  • If \(\alpha=\alpha_1+\alpha_2+\cdots+\alpha_n\) with \(n\geq 2\) then \(\text{cof}(\alpha)=\text{cof}(\alpha_n)\) and \(\alpha[\eta]=\alpha_1+\alpha_2+\cdots+(\alpha_n[\eta])\)
  • If \(\alpha=\psi_{I(0,0)}(0)\) then \(\alpha=\text{cof}(\alpha)=1\) and \(\alpha[0]=0\)
  • If \(\alpha=\psi_{I(0,\beta+1)}(0)\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[\eta]=I(0,\beta)\cdot\eta\)
  • If \(\alpha=\psi_{I(0,\beta)}(\gamma+1)\) and \(\beta\in\{0\}\cup S\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[\eta]=\psi_{I(0,\beta)}(\gamma)\cdot\eta\)
  • If \(\alpha=\psi_{I(\beta+1,0)}(0)\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[0]=0\) and \(\alpha[\eta+1]=I(\beta,\alpha[\eta])\)
  • If \(\alpha=\psi_{I(\beta+1,\gamma+1)}(0)\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[0]=I(\beta+1,\gamma)+1\) and \(\alpha[\eta+1]=I(\beta,\alpha[\eta])\)
  • If \(\alpha=\psi_{I(\beta+1,\gamma)}(\delta+1)\) and \(\gamma\in\{0\}\cup S\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[0]=\psi_{I(\beta+1,\gamma)}(\delta)+1\) and \(\alpha[\eta+1]=I(\beta,\alpha[\eta])\)
  • If \(\alpha=\psi_{I(\beta,0)}(0)\) and \(\beta\in L\) then \(\text{cof}(\alpha)=\text{cof}(\beta)\) and \(\alpha[\eta]=I(\beta[\eta],0)\)
  • If \(\alpha=\psi_{I(\beta,\gamma+1)}(0)\) and \(\beta\in L\) then \(\text{cof}(\alpha)=\text{cof}(\beta)\) and \(\alpha[\eta]=I(\beta[\eta],I(\beta,\gamma)+1)\)
  • If \(\alpha=\psi_{I(\beta,\gamma)}(\delta+1)\) and \(\beta\in L\) and \(\gamma\in \{0\}\cup S\) then \(\text{cof}(\alpha)=\text{cof}(\beta)\) and \(\alpha[\eta]=I(\beta[\eta],\psi_{I(\beta,\gamma)}(\delta)+1)\)
  • If \(\alpha=\pi\) then \(\text{cof}(\alpha)=\pi\) and \(\alpha[\eta]=\eta\)
  • If \(\alpha=I(\beta,\gamma)\) and \(\gamma\in L\) then \(\text{cof}(\alpha)=\text{cof}(\gamma)\) and \(\alpha[\eta]=I(\beta,\gamma[\eta])\)
  • If \(\alpha=\psi_\pi(\beta)\) and \(\omega\le\text{cof}(\beta)<\pi\) then \(\text{cof}(\alpha)=\text{cof}(\beta)\) and \(\alpha[\eta]=\psi_\pi(\beta[\eta])\)
  • If \(\alpha=\psi_\pi(\beta)\) and \(\text{cof}(\beta)=\rho\geq\pi\) then \(\text{cof}(\alpha)=\omega\) and \(\alpha[\eta]=\psi_\pi(\beta[\gamma[\eta]])\) with \(\gamma[0]=1\) and \(\gamma[\eta+1]=\psi_{\rho}(\beta[\gamma[\eta]])\)

Below \(\sigma\) denotes the ordinal \(\psi_{I(0,0)}(I(\omega,0))\). Then \(\text{cof}(\sigma)=\omega\) and \(\sigma[\eta]=\psi_{I(0,0)}(I(\eta,0))\)

Examples of numbers

\(\sigma\)-billion \(=10_{\sigma}^9=10\uparrow^{\sigma} 9\)

\(\sigma\)-trillion \(=10_{\sigma}^{12}=10\uparrow^{\sigma}12\)

and so on up to \(\sigma\)-centillion \(=10_{\sigma}^{303}=10\uparrow^{\sigma}303\)

Below in the text the small Greek letter \(\tau\) denotes the largest countable limit ordinal such that the ruleset in this section allows to define fundamental sequences for all limit ordinals less than \(\tau\). Then \(\text{cof}(\tau)=\omega\) and \(\tau[\eta]=\psi_{I(0,0)}(\alpha[\eta])\) where \(\alpha[0]=0\) and \(\alpha[z+1]=I(\alpha[z],0)\) for all integers \(z \geq 0\)

Examples of numbers

\(\tau\)-billion \(=10_{\tau}^9=10\uparrow^{\tau} 9\)

\(\tau\)-trillion \(=10_{\tau}^{12}=10\uparrow^{\tau}12\)

and so on up to \(\tau\)-centillion \(=10_{\tau}^{303}=10\uparrow^{\tau}303\)

Section V. The first notation based on the least weakly Mahlo cardinal

Basic notions

\(\kappa\) is weakly Mahlo iff \(\kappa\) is a cardinal such that for every function \(f: \kappa\rightarrow\kappa\) there exists a regular cardinal \(\pi < \kappa\) such that \(\forall\alpha<\pi(f(\alpha)< \pi)\).

\(M\) is the least weakly Mahlo cardinal and \(\varepsilon_{M+1}=\min\{\alpha>M|\alpha=\omega^\alpha\}\)

\(\alpha=_{NF}M^\beta\gamma\Leftrightarrow\alpha=M^\beta\gamma\wedge\gamma<M\)

In this section the variables \(\pi\), \(\rho\), \(\kappa\) are reserved for regular uncountable cardinals less than \(M\).

Enumeration function \(F\) of class of ordinals \(X\) is the unique increasing function such that \(X=\{F(\alpha)|\alpha\in\text{dom}(F)\}\) where domain of \(F\), \(\text{dom}(F)\) is an ordinal number. We use \(\text{Enum}(X)\) to denote \(F\).

\(cl(X) \) is closure of \(X\)

\(cl_M(X)=X\cup\{\alpha<M|\alpha=\sup(X\cap\alpha)\} \)

Definition of Veblen function

\(\varphi_\alpha=\text{Enum}(\{\beta\in P|\forall\gamma<\alpha(\varphi_\gamma(\beta)=\beta)\})\)

Below we write \(\varphi(\alpha,\beta)\) for \(\varphi_\alpha(\beta)\)

\(\alpha=_{NF}\varphi(\beta,\gamma)\Leftrightarrow\alpha=\varphi(\beta,\gamma)\wedge\beta,\gamma<\alpha\)

Let \(M^{\Gamma}=\min\{\alpha>M|\alpha=\varphi(\alpha,0)\}\)

Definition of functions \(\chi_\alpha(\beta) \) and \(\psi_\pi(\gamma) \)

Inductive Definition of functions \(\chi_\alpha: M\rightarrow M\) for \(\alpha <M^{\Gamma}\) (Rathjen, 1990)

1) \(\{0,M\}\cup\beta\subseteq B^n(\alpha, \beta)\)

2) \(\gamma=_{NF}\gamma_1+\cdots+\gamma_k\wedge\gamma_1,...,\gamma_k\in B^n(\alpha, \beta)\Rightarrow\gamma\in B^{n+1}(\alpha, \beta)\)

3) \(\gamma=\chi_\eta(\xi)\wedge\eta,\xi\in B^n(\alpha, \beta)\wedge\eta<\alpha\wedge\xi<M\Rightarrow\gamma\in B^{n+1}(\alpha, \beta)\)

4) \(\gamma=_{NF}\varphi(\delta,\eta) \wedge\delta,\eta\in B^n(\alpha, \beta)\Rightarrow\gamma\in B^{n+1}(\alpha, \beta)\)

5) \(\gamma<\pi\wedge\pi\in B^n(\alpha, \beta)\Rightarrow\gamma\in B^{n+1}(\alpha, \beta)\)

6) \(B(\alpha,\beta)=\bigcup_{n<\omega}B^{n}(\alpha, \beta)\)

7) \(\chi_\alpha=\text{Enum}(cl_M(\{\kappa|\kappa\notin B(\alpha,\kappa)\wedge\alpha\in B(\alpha,\kappa)\}))\)

Below we write \(\chi(\alpha,\beta)\) for \(\chi_\alpha(\beta)\)

Properties of \(\chi\)-functions:

1) \(\chi(\alpha,\beta)<M\)

2) \(\beta>\gamma\geq 0 \Rightarrow \chi(\alpha,\beta)>\chi(\alpha,\gamma)\)

3) \(\alpha>\gamma\geq 0 \Rightarrow \chi(\alpha,\beta)=\chi(\gamma,\chi(\alpha,\beta))\)

4) \(\chi(\alpha,0),\chi(\alpha,\beta+1) \in R\)

5) \(\chi(0,\alpha)=\aleph_{1+\alpha}\)

6) \(\chi(\alpha,\beta)=I(\alpha,\beta)\) for all \(\alpha<\gamma\) where \(\gamma=\sup\{\delta(n)|n<\omega\}\) with \(\delta(0)=0\) and \(\delta(n+1)=\chi(\delta(n),0)\)

Definition: \(\alpha=_{NF}\chi(\beta,\gamma) \Leftrightarrow\alpha=\chi(\beta,\gamma)\wedge\gamma<\alpha\)

Let \(\Pi\) be the set of uncountable regular cardinals of the form \(\chi(\alpha,0)\) or \(\chi(\alpha,\beta+1)\)

\(\Pi=\{\chi(\alpha,0)|\alpha<\varepsilon_{M+1}\}\cup\{\chi(\alpha,\beta+1)|\alpha<\varepsilon_{M+1}\wedge\beta<M\}\)

On base of Rathjen’s approach we define a simplified version of functions \(\psi_\pi: M\rightarrow \pi\) that allows to reduce number of rules for system of fundamental sequences, and after this we get set of 20 rules.

Inductive Definition of functions \(\psi_\pi: M\rightarrow \pi\) for \(\pi\in \Pi\)

1) \(C^0(\alpha, \beta)=\{0,M\}\cup\beta\)

2) \(C^{n+1}(\alpha, \beta)=\{\gamma+\delta,\chi(\gamma,\delta), \omega^{M+\gamma}, \psi_\kappa(\eta)|\gamma,\delta,\eta,\kappa\in C^{n}(\alpha, \beta)\wedge\eta<\alpha\wedge\kappa\in\Pi\}\)

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

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

Properties of \(\psi\)-functions:

1) \(\psi_{\chi(0,0)}(0)=1\)

2) \(\alpha>\beta\geq 0 \Rightarrow \psi_\pi(\beta)<\psi_ \pi(\alpha)<\pi\)

3) \(\psi_\pi(\alpha)\in P\)

Definition: \(\alpha=_{NF}\psi_\pi(\beta)\Leftrightarrow\alpha=\psi_\pi(\beta) \wedge\beta\in C(\beta, \psi_\pi(\beta))\)

A system of fundamental sequences

Inductive definition of \(T\)

1) \(0 \in T\)

2) \(\alpha=_{NF}\alpha_1+\alpha_2+\cdots+\alpha_n\wedge\alpha_1,\alpha_2,...,\alpha_n\in T\Rightarrow\alpha\in T\)

3) \(\alpha=_{NF}\chi(\beta,\gamma)\wedge\beta,\gamma\in T\Rightarrow\alpha\in T\)

4) \(\alpha=_{NF}\psi_\pi(\beta)\wedge\pi,\beta\in T\Rightarrow\alpha\in T\)

5) \(\alpha=_{NF}M^\beta\gamma\wedge\beta,\gamma\in T\Rightarrow\alpha\in T\)

Definition of fundamental sequences for non-zero ordinals \(\alpha\in T\):

1) \(\alpha=\alpha_1+\alpha_2+\cdots+\alpha_n \wedge \alpha_1\geq\alpha_2\geq\cdots\geq\alpha_n \Rightarrow \text{cof} (\alpha)= \text{cof} (\alpha_n) \wedge \alpha[\eta]=\alpha_1+\alpha_2+\cdots+(\alpha_n[\eta])\)

2) \(\alpha=0\Rightarrow\text{cof}(\alpha)=0\)

3) \(\alpha=\psi_{\chi(0,0)}(0)=1 \vee \alpha=\chi(\beta,0) \vee \alpha=\chi(\beta,\gamma+1) \vee \alpha=M\Rightarrow \text{cof} (\alpha)=\alpha \wedge \alpha[\eta]=\eta\)

4) \(\alpha=\psi _{\chi(0,\beta+1)}(0) \Rightarrow \text{cof}(\alpha)=\omega \wedge \alpha[n]=\chi(0,\beta)\times n\)

5) \(\alpha=\psi_{ \chi(0,\beta)}(\gamma+1) \Rightarrow \text{cof}(\alpha)=\omega \wedge \alpha[n]=\psi_{\chi(0,\beta)}(\gamma)\times n\)

6) \(\alpha=\psi _{\chi(\beta+1,0)}(0) \Rightarrow \text{cof}(\alpha)=\omega \wedge \alpha[0]=0 \wedge \alpha[n+1]=\chi(\beta,\alpha[n])\)

7) \(\alpha=\psi _{\chi(\beta+1,\gamma+1)}(0) \Rightarrow \text{cof}(\alpha)=\omega \wedge \alpha[0]=\chi(\beta+1,\gamma)+1 \wedge \alpha[n+1]=\chi(\beta,\alpha[n])\)

8) \(\alpha=\psi_{\chi(\beta+1,\gamma)}(\delta+1) \Rightarrow \text{cof}(\alpha)=\omega \wedge \alpha[0]= \psi_{\chi(\beta+1,\gamma)}(\delta)+1 \wedge \alpha[n+1]=\chi(\beta,\alpha[n])\)

9) \(\alpha=\psi _{\chi(\beta,0)}(0) \wedge M>\text{cof}(\beta)\geq\omega \Rightarrow \text{cof} (\alpha)= \text{cof} (\beta) \wedge \alpha[\eta]=\chi(\beta[\eta],0)\)

10) \(\alpha=\psi_{ \chi(\beta,\gamma+1)}(0) \wedge M>\text{cof}(\beta)\geq\omega \Rightarrow \text{cof}(\alpha)=\text{cof}(\beta)\wedge \alpha[\eta]=\chi(\beta[\eta],\chi(\beta,\gamma)+1)\)

11) \(\alpha=\psi_{ \chi(\beta,\gamma)}(\delta+1) \wedge M>\text{cof} (\beta)\geq\omega \Rightarrow \text{cof}(\alpha)=\text{cof}(\beta) \wedge \alpha[\eta]=\chi(\beta[\eta],\psi_{\chi(\beta,\gamma)}(\delta)+1)\)

12) \(\alpha=\psi_{\chi(\beta,0)}(0) \wedge \text{cof}(\beta)=M\Rightarrow \text{cof}(\alpha)= \omega \wedge \alpha[0]=1 \wedge \alpha[n+1]=\chi(\beta[\alpha[n]],0)\)

13) \(\alpha=\psi_{ \chi(\beta,\gamma+1)}(0) \wedge \text{cof} (\beta)=M \Rightarrow \text{cof} (\alpha)= \omega \wedge \alpha[0]=\chi(\beta,\gamma)+1 \wedge \alpha[n+1]=\chi(\beta[\alpha[n]],0)\)

14) \(\alpha=\psi_{\chi(\beta,\gamma)}(\delta+1) \wedge \text{cof} (\beta)=M \Rightarrow \text{cof} (\alpha)= \omega \wedge \alpha[0]= \psi_{ \chi(\beta,\gamma)}(\delta)+1 \wedge \alpha[n+1]=\chi(\beta[\alpha[n]],0)\)

15) \(\alpha=M^{\beta}\times\gamma \wedge \gamma<M \wedge \text{cof} (\gamma)\geq\omega \Rightarrow \text{cof} (\alpha)= \text{cof}(\gamma)\wedge\alpha[\eta]=M^{\beta}\times(\gamma[\eta])\)

16) \(\alpha=M^{\beta+1}\times(\gamma+1) \wedge \gamma<M \Rightarrow \text{cof} (\alpha)=M \wedge\alpha[\eta]=M^{\beta+1}\times\gamma+M^\beta\times\eta\)

17) \(\alpha=M^\beta\times(\gamma+1) \wedge \gamma<M \wedge\text{cof}(\beta)\geq\omega \Rightarrow \text{cof}(\alpha)= \text{cof}(\beta) \wedge \alpha[\eta]=M^\beta\times\gamma+M^{\beta[\eta]}\)

18) \(\alpha=\chi(\beta,\gamma) \wedge \text{cof}(\gamma)\geq\omega \Rightarrow \text{cof} (\alpha)=\text{cof}(\gamma)\wedge \alpha[\eta]=\chi(\beta,\gamma[\eta])\)

19) \(\alpha=\psi_\pi(\beta) \wedge \pi>\text{cof}(\beta)\geq\omega \Rightarrow \text{cof} (\alpha)= \text{cof}(\beta) \wedge \alpha[\eta]=\psi_\pi(\beta[\eta])\)

20) \(\alpha=\psi_\pi(\beta) \wedge \text{cof}(\beta)=\rho\geq\pi \Rightarrow \text{cof} (\alpha)=\omega \wedge \alpha[n]=\psi _\pi(\beta[\gamma[n]])\) where \(\gamma[0]=1\) and \(\gamma[k+1]=\psi_\rho(\beta[\gamma[k]])\)

Below in the text the capital Greek letter \(\Lambda\) denotes the largest countable limit ordinal such that the ruleset in this section allows to assign fundamental sequences for all limit ordinals less than \(\Lambda\). Then \(\text{cof}(\Lambda)=\omega \) and \(\Lambda[n]=\psi_{\chi(0,0)}(\chi(\alpha[n],0))\) where \(\alpha[0]=0\) and \(\alpha[z+1]=M^{\alpha[z]}\) for all integers \(z \geq 0\)

Note \(M^0=1\)

Examples of numbers

\(\Lambda \)-billion \(=10_\Lambda^9=10\uparrow^\Lambda 9\)

\(\Lambda \)-trillion \(=10_\Lambda^{12}=10\uparrow^\Lambda 12\)

and so on up to \(\Lambda \)-centillion \(=10_\Lambda ^{303}=10\uparrow^\Lambda 303\)

Section VI. The second notation based on the least weakly Mahlo cardinal

This notation allows to obtain much simpler system of fundamental sequences.

Basic notions

\(M\) is the least weakly Mahlo cardinal.

Normal form. \(\alpha=_{NF}M^\beta\gamma\Leftrightarrow\alpha=M^\beta\gamma\wedge\gamma<M\)

\(\varepsilon_{M+1}=\min\{\alpha>M|\alpha=\omega^\alpha\}\) is the least epsilon number greater than \(M\).

In this section:

  • \(\alpha\in R\Leftrightarrow\alpha=\chi(\beta)\vee\alpha=M\),
  • the variables \(\pi, \rho\) are reserved for uncountable regular cardinals less than \(M\).

Definition of the function \(\chi:\varepsilon_{M+1}\rightarrow M\)

1) \(B_0(\alpha,\beta)=\beta\cup\{0\}\)

2) \(\gamma=_{NF}\gamma_1+\cdots+\gamma_k\wedge\gamma_1,...,\gamma_k\in B_n(\alpha,\beta)\Rightarrow\gamma\in B_{n+1}(\alpha,\beta)\)

3) \(\gamma=\omega^{M+\delta}\wedge\delta\in B_n(\alpha,\beta)\Rightarrow\gamma\in B_{n+1}(\alpha,\beta)\)

4) \(\gamma=\chi(\eta)\wedge\eta\in B_n(\alpha,\beta)\cap\alpha \Rightarrow\gamma\in B_{n+1}(\alpha,\beta)\)

5) \(\gamma<\pi\wedge\pi\in B_n(\alpha,\beta) \Rightarrow\gamma\in B_{n+1}(\alpha,\beta)\)

6) \(B(\alpha,\beta)=\bigcup_{n<\omega}B_n(\alpha,\beta)\)

7) \(\chi(\alpha)=\min\{\beta<M|B(\alpha,\beta)\cap M\subseteq\beta\wedge\beta\in R\}\)

Normal form. \(\alpha=_{NF}\chi(\beta)\Leftrightarrow\alpha=\chi(\beta)\wedge\beta\in B(\beta,\chi(\beta))\)

Definition of functions \(\psi_\pi:M\rightarrow \pi\)

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

2) \(\gamma=_{NF}\gamma_1+\cdots+\gamma_k\wedge\gamma_1,...,\gamma_k \in C_n(\alpha,\beta)\Rightarrow\gamma\in C_{n+1}(\alpha,\beta)\)

3) \(\gamma=\omega^{M+\delta}\wedge\delta\in C_n(\alpha,\beta)\Rightarrow\gamma\in C_{n+1}(\alpha,\beta)\)

4) \(\gamma=_{NF}\chi(\eta)\wedge\eta\in C_n(\alpha,\beta) \Rightarrow\gamma\in C_{n+1}(\alpha,\beta)\)

5) \(\gamma=\psi_\pi(\eta)\wedge\eta<\alpha\wedge\pi,\eta\in C_n(\alpha,\beta)\Rightarrow\gamma\in C_{n+1}(\alpha,\beta)\)

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

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

Normal form. \(\alpha=_{NF}\psi_\pi(\beta)\Leftrightarrow\alpha=\psi_\pi(\beta)\wedge\beta\in C(\beta,\psi_\pi(\beta))\)

A system of fundamental sequences

Definition of the set \(T\) of ordinals which can be generated from the ordinals \(0\) and \(M\) using addition, multiplication, exponentiation and the functions \(\chi,\psi_\pi\)

1) \(0\in T\)

2) \(\alpha=_{NF}\alpha_1+\cdots+\alpha_k\wedge\alpha_1,...,\alpha_k\in T\Rightarrow\alpha\in T\)

3) \(\alpha=_{NF}M^\beta\gamma\wedge\beta,\gamma\in T\Rightarrow\alpha\in T\)

4) \(\alpha=_{NF}\psi_\pi(\beta)\wedge\pi,\beta\in T\Rightarrow\alpha\in T\)

5) \(\alpha=_{NF}\chi(\beta)\wedge\beta\in T\Rightarrow\alpha\in T\)

Definition of fundamental sequences for non-zero ordinals \(\alpha\in T\):

1) \(\alpha=\alpha_1+\cdots+\alpha_k\Rightarrow\text{cof}(\alpha)=\text{cof}(\alpha_k)\wedge\alpha[\eta]=\alpha_1+\cdots+(\alpha_k[\eta])\)

2) \(\alpha=\psi_{\chi(\beta+1)}(0)\Rightarrow\text{cof}(\alpha)=\omega\wedge\alpha[\eta]=\chi(\beta)\times \eta\)

3) \(\alpha=\psi_{\chi(\beta)}(0)\wedge\omega\le\text{cof}(\beta)<M\Rightarrow\text{cof}(\alpha)=\text{cof}(\beta)\wedge\alpha[\eta]=\chi(\beta[\eta])\)

4) \(\alpha=\psi_{\chi(\beta)}(0)\wedge\text{cof}(\beta)=M\Rightarrow\text{cof}(\alpha)=\omega\wedge\alpha[0]=1\wedge\alpha[\eta+1]=\chi(\beta[\alpha[\eta]])\)

5) \(\alpha=\psi_{\chi(\beta)}(\gamma+1)\wedge(\beta=0\vee\beta=\delta+1\vee\omega\le\text{cof}(\beta)<M)\Rightarrow\text{cof}(\alpha)=\omega\wedge\alpha[\eta]=\psi_{\chi(\beta)}(\gamma)\times \eta\)

6) \(\alpha=\psi_{\chi(\beta)}(\gamma+1)\wedge\text{cof}(\beta)=M\Rightarrow\text{cof}(\alpha)=\omega\wedge\alpha[0]=\psi_{\chi(\beta)}(\gamma)+1\wedge\alpha[\eta+1]=\chi(\beta[\alpha[\eta]])\)

7) \(\alpha=\psi_{\chi(0)}(0)=1\vee\alpha=\chi(\beta)\vee\alpha=M\Rightarrow\text{cof}(\alpha)=\alpha\wedge\alpha[\eta]=\eta\)

8) \(\alpha=M^{\beta}\times\gamma \wedge \text{cof} (\gamma)\geq\omega \Rightarrow \text{cof} (\alpha)= \text{cof}(\gamma)\wedge\alpha[\eta]=M^{\beta}\times(\gamma[\eta])\)

9) \(\alpha=M^{\beta+1}\times(\gamma+1) \Rightarrow \text{cof} (\alpha)=M \wedge\alpha[\eta]=M^{\beta+1}\times\gamma+M^\beta\times\eta\)

10) \(\alpha=M^\beta\times(\gamma+1) \wedge\text{cof}(\beta)\geq\omega \Rightarrow \text{cof}(\alpha)= \text{cof}(\beta) \wedge \alpha[\eta]=M^\beta\times\gamma+M^{\beta[\eta]}\)

11) \(\alpha=\psi_\pi(\beta) \wedge \pi>\text{cof}(\beta)\geq\omega \Rightarrow \text{cof} (\alpha)= \text{cof}(\beta) \wedge \alpha[\eta]=\psi_\pi(\beta[\eta])\)

12) \(\alpha=\psi_\pi(\beta) \wedge \text{cof}(\beta)=\rho\geq\pi \Rightarrow \text{cof} (\alpha)=\omega \wedge \alpha[\eta]=\psi _\pi(\beta[\gamma[\eta]])\) where \(\gamma[0]=1\) and \(\gamma[z+1]=\psi_\rho(\beta[\gamma[z]])\) for all integers \(z \geq 0\)

Below in the text the small Greek letter \(\lambda\) denotes the largest countable limit ordinal such that the ruleset in this section allows to assign fundamental sequences for all limit ordinals less than \(\lambda\). Then \( \text{cof} (\lambda)=\omega\) and \(\lambda[\eta]=\psi_{\chi(0)} (\chi(\alpha[\eta]))\) where \(\alpha[0]=0\) and \(\alpha[z+1]=M^{\alpha[z]}\) for all integers \(z \geq 0\)

Note \(M^0=1\)

Examples of numbers

\(\lambda \)-billion \(=10_\lambda ^9=10\uparrow^\lambda 9\)

\(\lambda \)-trillion \(=10_\lambda ^{12}=10\uparrow^\lambda 12\)

and so on up to \(\lambda \)-centillion \(=10_\lambda ^{303}=10\uparrow^\lambda 303\)

Curiously, is there in our possibly infinite physical universe a cosmological object , such that for measure of its parameters, for example linear size in parsecs, requires at least one \(\lambda \)-centillion? Just wishing to somehow apply huge numbers mentioned above we can define for example (under the assumption that our universe is infinite):

the name of a number \(j\) + "er" is the set of all points of the physical space, which are located not further than \(j\) parsecs from the point of the Earth's center.

For example, \(\lambda\)-billioner is the set of all points of the physical space which are located not further than \(\lambda\)-billion parsecs from the point of the Earth's center.

Other examples. \(\omega\)-billioner, \(\varepsilon_0\)-trillioner, \(\upsilon \)-billioner, \(\sigma\)-trillioner, \(\tau\)-centillioner, \(\Lambda \)-billioner


Author: Denis Maksudov (Ufa, Russia)

E-mail: md77@list.ru

References

1. Buchholz, W.; Wainer, S.S (1987). "Provably Computable Functions and the Fast Growing Hierarchy". Logic and Combinatorics, edited by S. Simpson, Contemporary Mathematics, Vol. 65, AMS, 179-198.

2. W.Buchholz (1986). A New System of Proof-Theoretic Ordinal Functions. Annals of Pure and Applied Logic, Vol. 32, 195-207

3. M.Jäger (1984). \(\rho\)-inaccessible ordinals, collapsing functions and a recursive notation system. Arch. Math. Logik Grundlagenforsch, Vol. 24, 49-62

4. M. Rathjen (1990). Ordinal Notations Based on a Weakly Mahlo Cardinal. Arch. Math. Logic, Vol. 29, 249-263