Difference between revisions of "User:Denis Maksudov"

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==Introduction==
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[http://cantorsattic.info/User_blog:Denis_Maksudov/Ordinal_functions_collapsing_the_least_weakly_Mahlo_cardinal_and_a_system_of_fundamental_sequences My blogpost in Cantors Attic: Ordinal  functions collapsing the least weakly Mahlo cardinal; a system of fundamental sequences]
  
There are well-known hierarchies of ordinal-indexed functions, such as slow-growing hierarchy, Hardy hierarchy, fast-growing hierarchy and other hierarchies whose definition require assignation of fundamental sequences <math>(\alpha[n])_{n<\omega}</math> for countable limit ordinals. The growth of each of those hierarchies and the limit of its definition depends of choice of system of fundamental sequences. There is well-known and rather simple system formulated by S. Wainer for limit ordinals written in the Cantor normal form up to the first epsilon number. Ordinal notations, which are stronger than the Cantor normal form, require, respectively, stronger systems of fundamental sequences and those systems have much more complicated definition.
+
[http://googology.wikia.com/wiki/User:Denis_Maksudov My account in googology.wikia]
  
Saying about ordinal notations we should mark the contribution of  S. Feferman, W. Buchholz, K. Schütte, G. Jäger and  M. Rathjen.  
+
I’m Denis Maksudov from [https://en.wikipedia.org/wiki/Ufa_State_Aviation_Technical_University Ufa State Aviation Technical University], I’m not a mathematician, but when in May of 2016 I occasionally have found the site [https://googology.wikia.com/wiki/User:Denis_Maksudov 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 [https://googology.wikia.com/wiki/User:Denis_Maksudov googology.wikia] and [http://cantorsattic.info/User_blog:Denis_Maksudov/Ordinal_functions_collapsing_the_least_weakly_Mahlo_cardinal;_a_system_of_fundamental_sequences Cantor’s Attic]:
 +
* Extended arrows (December 2016)
 +
* The extended Wilfried Buchholz's functions (April 2017)
 +
* Fundamental sequences for the functions collapsing <math>\alpha</math>-weakly inaccessible cardinals (August 2017)
 +
* Two notations based on a weakly Mahlo cardinal (May 2018)
 +
* Extension of "illion"-family of number names (May 2018)
  
Based on S. Feferman’s theta-functions, W. Buchholz introduced a hierarchy of ordinal psi-functions, which allows to express large countable ordinals using regular cardinals. G. Jäger developed an extension of this this approach by using of <math>\alpha</math>-weakly inaccessible cardinals. Much more powerful extension, based on using of the least weakly Mahlo cardinal <math>M</math>, was proposed by M. RathjenM. 
+
==General notions==
  
On base of Rathjen’s approach we define a simplified version of functions  <math>\psi_\pi: M\rightarrow \pi</math>  that allows to reduce number of rules for system of fundamental sequences. Later we assign the cofinality and a fundamental sequence for each non-zero ordinal less than <math>\psi_{\chi(0,0)}(\chi(\varepsilon_{M+1},0))</math>
+
Small Greek letters <math>\alpha, \beta, \gamma, \delta, \eta, \xi, \nu, \mu</math> denote ordinals. Each ordinal <math>\alpha</math> is identified with the set of its predecessors <math>\alpha=\{\beta|\beta<\alpha\}</math>. The least ordinal is zero and it is identified with the empty set.  
 
+
 
+
==Basic notions==
+
 
+
Small Greek letters denote ordinals. Each ordinal <math>\alpha</math> is identified with the set of its predecessors <math>\alpha=\{\beta|\beta<\alpha\}</math>.  
+
  
 
<math>\omega</math> is the first transfinite ordinal and the set of all natural numbers.
 
<math>\omega</math> is the first transfinite ordinal and the set of all natural numbers.
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An ordinal <math>\alpha</math> is an additive principal number if <math>\alpha>0</math> and <math>\xi+\eta<\alpha</math> for all <math>\xi,\eta<\alpha</math>.  
 
An ordinal <math>\alpha</math> is an additive principal number if <math>\alpha>0</math> and <math>\xi+\eta<\alpha</math> for all <math>\xi,\eta<\alpha</math>.  
  
<math>P</math> denotes the set of all additive principal numbers.
+
<math>P=\{\alpha>0|\forall\beta,\gamma<\alpha(\beta+\gamma<\alpha)\}</math> is the set of additive principal numbers.
  
For every ordinal <math>\alpha\notin P</math> there exist unique <math>\alpha_1,\alpha_2,..., \alpha_n</math> such that
+
For every ordinal <math>\alpha\notin P\cup\{0\}</math> there exist unique <math>\alpha_1,..., \alpha_n\in P</math> such that
<math>\alpha=\alpha_1+\alpha_2+\cdots+\alpha_n</math> and <math>\alpha>\alpha _{1}\geq \cdots \geq \alpha _{n}</math>
+
<math>\alpha=\alpha_1+\cdots+\alpha_n</math> and <math>\alpha>\alpha _{1}\geq \cdots \geq \alpha _{n}</math>
  
 
<math>\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</math>
 
<math>\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</math>
  
Cofinality <math>\text{cof}(\alpha)</math> of an ordinal <math>\alpha</math> is the least <math>\beta</math> such that there exists a function <math>f:\beta\rightarrow\alpha</math> with <math>\text{sup}\{f(\xi )|\xi <\beta \}=\alpha</math>.  
+
The cofinality of a limit ordinal <math>\alpha</math> is the least length of increasing sequence such that the limit of this sequence is the ordinal <math>\alpha</math>.  
  
The fundamental sequence for an ordinal number <math>\alpha</math> with cofinality <math>\text{cof}(\alpha)=\beta</math> is a strictly increasing sequence <math>(\alpha[\eta])_{\eta<\beta}</math> with length <math>\beta</math> and with limit <math>\alpha</math>, where <math>\alpha[\eta]</math> is the <math>\eta</math>-th element of this sequence. For each ordinal <math>\alpha</math> can be assigned a lot of different fundamental sequences.
+
<math>\text{cof}(\alpha)</math> denotes the cofinality of an ordinal <math>\alpha</math>.
  
'''Properties'''
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An ordinal <math>\alpha</math> is uncountable regular cardinal if it is a limit ordinal larger than <math>\omega</math> and <math>\text{cof}(\alpha)=\alpha</math>.
  
1) <math>\text{cof}(0)=0</math>  
+
<math>R=\{\alpha\in L|\alpha>\omega\wedge\text{cof}(\alpha)=\alpha\}</math> is the set of all uncountable regular cardinals.
  
2) <math>\alpha>0 \Rightarrow \alpha\geq \text{cof}(\alpha) \wedge \alpha=\sup\{\alpha[\eta]|\eta<\text{cof}(\alpha)\}</math>  
+
The fundamental sequence for a limit ordinal number <math>\alpha</math> with cofinality <math>\text{cof}(\alpha)=\beta</math> is a strictly increasing sequence <math>(\alpha[\eta])_{\eta<\beta}</math> with length <math>\beta</math> and with limit <math>\alpha</math>, where <math>\alpha[\eta]</math> is the <math>\eta</math>-th element of this sequence.
 +
* If <math>\alpha</math> is a limit ordinal then <math>\alpha\geq\text{cof}(\alpha)\geq\omega</math> and <math>\alpha=\sup\{\alpha[\eta]|\eta<\text{cof}(\alpha)\}</math>.
 +
* If <math>\alpha</math> is a successor ordinal then <math>\text{cof}(\alpha)=1</math> and the fundamental sequence has only one element <math>\alpha[0]=\alpha-1</math>. 
 +
* If <math>\alpha=0</math> then <math>\text{cof}(\alpha)=0</math> and <math>\alpha</math> has not fundamental sequence.
  
3) <math>\alpha\in S\Leftrightarrow \text{cof}(\alpha)=1</math>.  
+
==Section I. Extended arrows==
  
4) <math>\alpha\in L\Leftrightarrow \text{cof}(\alpha)\geq\omega</math>.
+
We can extend Knuth's up-arrow notation to transfinite ordinals. Let’s define for positive integers <math>u, q</math> and for ordinal number <math>\alpha</math>:
  
5) <math>\beta>\gamma\geq0 \Rightarrow \alpha[\beta]>\alpha[\gamma]</math>
+
1) <math>u\uparrow^0 q=u\times q</math>
  
An ordinal <math>\alpha</math> is uncountable regular cardinal if it is a limit ordinal larger than <math>\omega</math> and <math>\text{cof}(\alpha)=\alpha</math> where <math>\alpha[\eta]</math> denotes the <math>\eta</math>th element of the fundamental sequence assigned to the ordinal <math>\alpha</math>.
+
2) <math>u\uparrow^{\alpha+1}1=u</math>
  
<math>R</math> is the set of uncountable regular cardinals
+
3) <math>u\uparrow^{\alpha+1}(q+1)=u\uparrow^{\alpha}(u\uparrow^{\alpha+1}q)</math>
  
<math>R=\{\alpha\in L|\alpha>\omega\wedge\text{cof}(\alpha)=\alpha\}</math>
+
4) <math>u\uparrow^{\alpha}q=u\uparrow^{\alpha[q]}u</math> iff <math>\alpha</math> is a limit ordinal
  
<math>\kappa</math> is weakly Mahlo iff <math>\kappa</math> is a cardinal such that for every function <math>f: \kappa\rightarrow\kappa</math> there
+
where <math>\alpha [q]</math> denotes the <math>q</math>-th element of the fundamental sequence assigned to the limit ordinal <math>\alpha</math>.
exists a regular cardinal <math>\pi < \kappa</math> such that <math>\forall\alpha<\pi(f(\alpha)< \pi)</math>.
+
  
<math>M</math> is  the least Mahlo cardinal and <math>\varepsilon_{M+1}=\min\{\alpha>M|\alpha=\omega^\alpha\}</math>  
+
Let’s also define <math>u_{\alpha}^q = u\uparrow^{\alpha}q</math>
 +
 
 +
Hence:
 +
 
 +
1) <math>u_0^q=u\times q</math>
 +
 
 +
2) <math>u_ {\alpha+1}^1=u</math>
 +
 
 +
3) <math>u_{\alpha+1}^{q+1}=u_{\alpha}^{u_{\alpha+1}^q}</math>
 +
 
 +
4) <math>u_{\alpha}^q=u_{\alpha[q]}^u</math> iff <math>\alpha</math> is a limit ordinal.
 +
 
 +
==Section II. Extension of "illion"-family of number names==
 +
 
 +
An ordinal <math>\alpha</math> + a Latin prefix denoting the natural number <math>y</math> + "illion" = <math>10_{\alpha}^ {3\times(y+1)}=10\uparrow^{\alpha}(3\times(y+1))</math>
 +
 
 +
For example:
 +
 
 +
2-billion <math>=10_2^9=10\uparrow^2 9</math>
 +
 
 +
2-trillion <math>=10_2^{12}=10\uparrow^2 12</math>
 +
 +
and so on up to 2-centillion <math>=10_2^{303}=10\uparrow^2 303</math>
 +
 
 +
3-billion <math>=10_3^9=10\uparrow^3 9</math>
 +
 
 +
3-trillion <math>=10_3^{12}=10\uparrow^3 12</math>
 +
 +
and so on up to 3-centillion <math>=10_3^{303}=10\uparrow^3 303</math>
 +
 
 +
'''Examples with ordinals''' <math>\alpha\le\varepsilon_0=\min\{\xi|\xi=\omega^\xi\}</math>
 +
 
 +
Every nonzero ordinal <math>\alpha<\varepsilon_0</math> can be represented in a unique Cantor normal form <math>\alpha=\omega^{\beta_{1}}+ \omega^{\beta_{2}}+\cdots+\omega^{\beta_{k-1}}+\omega^{\beta_{k}}</math> where <math>\alpha>\beta_1\geq\beta_2\geq\cdots\geq\beta_{k-1}\geq\beta_k</math>. If <math>\beta_k>0</math> then <math>\alpha</math> is a limit and we can assign to it a fundamental sequence as follows
 +
 
 +
<math>\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.</math>
 +
 
 +
If <math>\alpha=\varepsilon_0</math> then <math>\alpha[0]=0</math> and <math>\alpha[n+1]=\omega^{\alpha[n]}</math>
 +
 
 +
Examples of applying of fundamental sequences
 +
 
 +
<math>\omega[n]=\omega^1[n]= \omega^0 n=n</math> since <math>\omega^0=1</math>
 +
 
 +
<math>10\uparrow^{\omega} 3 = 10\uparrow^{3} 10</math>
 +
 
 +
<math>10\uparrow^{\omega+1} 3 = 10\uparrow^{\omega}(10\uparrow^{\omega}10)=10 \uparrow^{10 \uparrow^{10}10}10</math>
 +
 
 +
Examples of numbers
 +
 
 +
<math>\omega</math>-billion <math>=10_\omega ^9=10\uparrow^\omega 9</math>
 +
 
 +
<math>\omega</math>-trillion <math>=10_\omega ^{12}=10\uparrow^\omega 12</math>
 +
 +
and so on up to <math>\omega</math>-centillion <math>=10_\omega ^{303}=10\uparrow^\omega 303</math>
 +
 
 +
<math>\varepsilon_0</math>-billion <math>=10_{\varepsilon_0}^9=10\uparrow^{\varepsilon_0}9</math>
 +
 
 +
<math>\varepsilon_0</math>-trillion <math>=10_{\varepsilon_0}^{12}=10\uparrow^{\varepsilon_0}12</math>
 +
 +
and so on up to <math>\varepsilon_0</math>-centillion <math>=10_{\varepsilon_0}^{303}=10\uparrow^{\varepsilon_0}303</math>
 +
 
 +
==Section III. The extended Wilfried Buchholz's functions==
 +
 
 +
===Definition of the extended Wilfried Buchholz's functions===
 +
 
 +
We rewrite [https://core.ac.uk/download/pdf/12164720.pdf Buchholz's definition] as follows:
 +
 
 +
* <math>C_\nu^0(\alpha) = \{\beta|\beta<\Omega_\nu\}</math>
 +
* <math>C_\nu^{n+1}(\alpha) = \{\beta+\gamma,\psi_\mu(\eta)|\mu,\beta, \gamma,\eta\in C_{\nu}^n(\alpha)\wedge\eta<\alpha\}</math>
 +
* <math>C_\nu(\alpha) = \bigcup_{n < \omega} C_\nu^n (\alpha)</math>
 +
* <math>\psi_\nu(\alpha) = \min\{\gamma | \gamma \not\in C_\nu(\alpha)\}</math>
 +
 
 +
where
 +
 
 +
<math>\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.</math>
 +
 
 +
There is only one little detail difference with original Buchholz definition: ordinal <math>\mu</math> is not limited by <math>\omega</math>, now ordinal <math>\mu</math> belongs to previous set <math>C_n</math>. Limit of this notation must be omega fixed point <math>\psi_0(\Omega_{\Omega_{\Omega_{...}}})=\psi_0(\psi_{\psi_{...}(0)}(0))</math>
 +
 
 +
===Normal form for the extended Wilfried Buchholz's functions===
 +
 
 +
The normal form for 0 is 0. If <math>\alpha</math> is a nonzero ordinal number <math>\alpha<\Xi=\text{min}\{\beta|\psi_\beta(0)=\beta\}</math> then the normal form for <math>\alpha</math> is <math>\alpha=\psi_{\nu_1}(\beta_1)+\psi_{\nu_2}(\beta_2)+\cdots+\psi_{\nu_k}(\beta_k)</math> where <math>k</math> is a positive integer and <math>\psi_{\nu_1}(\beta_1)\geq\psi_{\nu_2}(\beta_2)\geq\cdots\geq\psi_{\nu_k}(\beta_k)</math> and each <math>\nu_i, \beta_i</math> are also written in normal form.
 +
 
 +
===Fundamental sequences for the extended Wilfried Buchholz's functions===
 +
 
 +
For nonzero ordinals <math>\alpha<\Xi</math>, written in normal form, fundamental sequences are defined as follows:
 +
 
 +
* If <math>\alpha=\psi_{\nu_1}(\beta_1)+\psi_{\nu_2}(\beta_2)+\cdots+\psi_{\nu_k}(\beta_k)</math> where <math>k\geq2</math> then <math>\text{cof}(\alpha)=\text{cof}(\psi_{\nu_k}(\beta_k))</math> and <math>\alpha[\eta]=\psi_{\nu_1}(\beta_1)+\cdots+\psi_{\nu_{k-1}}(\beta_{k-1})+(\psi_{\nu_k}(\beta_k)[\eta])</math>
 +
* If <math>\alpha=\psi_{0}(0)=1</math>, then <math>\text{cof}(\alpha)=1</math> and <math>\alpha[0]=0</math>
 +
* If <math>\alpha=\psi_{\nu+1}(0)</math>, then <math>\text{cof}(\alpha)=\Omega_{\nu+1}</math> and <math>\alpha[\eta]=\Omega_{\nu+1}[\eta]=\eta</math>
 +
* If <math>\alpha=\psi_{\nu}(0)</math> and <math>\text{cof}(\nu)\in\{\omega\}\cup\{\Omega_{\mu+1}|\mu\geq 0\}</math>, then <math>\text{cof}(\alpha)=\text{cof}(\nu)</math> and <math>\alpha[\eta]=\psi_{\nu[\eta]}(0)=\Omega_{\nu[\eta]}</math>
 +
* If <math>\alpha=\psi_{\nu}(\beta+1)</math> then <math>\text{cof}(\alpha)=\omega</math> and <math>\alpha[\eta]=\psi_{\nu}(\beta)\cdot \eta</math> (and note <math>\psi_\nu(0)=\Omega_\nu</math>)
 +
* If <math>\alpha=\psi_{\nu}(\beta)</math> and <math>\text{cof}(\beta)\in\{\omega\}\cup\{\Omega_{\mu+1}|\mu<\nu\}</math>  then <math>\text{cof}(\alpha)=\text{cof}(\beta)</math> and <math>\alpha[\eta]=\psi_{\nu}(\beta[\eta])</math>
 +
* If <math>\alpha=\psi_{\nu}(\beta)</math> and <math>\text{cof}(\beta)\in\{\Omega_{\mu+1}|\mu\geq\nu\}</math> then <math>\text{cof}(\alpha)=\omega</math> and <math>\alpha[\eta]=\psi_{\nu}(\beta[\gamma[\eta]])</math> where <math>\left\{\begin{array}{lcr} \gamma[0]=\Omega_\mu \\ \gamma[\eta+1]=\psi_\mu(\beta[\gamma[\eta]])\\ \end{array}\right.</math>
 +
 
 +
Below in the text the small Greek letter <math>\upsilon</math> denotes the largest countable limit ordinal such that the ruleset in this section allows to define fundamental sequences for all  limit ordinals less than <math>\upsilon</math>. Then <math>\text{cof}(\upsilon)=\omega</math> and <math>\upsilon[\eta]=\psi_0(\alpha[\eta])</math> where <math>\alpha[0]=0</math> and <math>\alpha[z+1]=\psi_{\alpha[z]}(0)=\Omega_{\alpha[z]}</math> for all integers <math>z \geq 0</math>
 +
 
 +
Examples of numbers
 +
 
 +
<math>\upsilon </math>-billion <math>=10_\upsilon ^9=10\uparrow^\upsilon 9</math>
 +
 
 +
<math>\upsilon </math>-trillion <math>=10_\upsilon ^{12}=10\uparrow^\upsilon 12</math>
 +
 +
and so on up to <math>\upsilon </math>-centillion <math>=10_\upsilon ^{303}=10\uparrow^\upsilon 303</math>
 +
 
 +
==Section IV. Fundamental sequences for the functions collapsing <math>\alpha</math>-weakly inaccessible cardinals==
 +
 
 +
===Definition of the functions collapsing <math>\alpha</math>-weakly inaccessible cardinals===
 +
 
 +
An ordinal is <math>\alpha</math>-weakly inaccessible if it's an uncountable regular cardinal and it's a limit of <math>\gamma</math>-weakly inaccessible cardinals for all <math>\gamma<\alpha</math>
 +
 
 +
Let <math>I(\alpha, 0)</math> be the first <math>\alpha</math>-weakly inaccessible cardinal, <math>I(\alpha, \beta+1)</math> be the next <math>\alpha</math>-weakly inaccessible cardinal after <math>I(\alpha,\beta)</math>, and <math>I(\alpha,\beta)=\sup\{I(\alpha,\gamma)|\gamma<\beta\}</math> for limit ordinal <math>\beta</math>
 +
 
 +
In this section the variables <math>\rho</math>, <math>\pi</math> are reserved for uncountable regular cardinals of the form <math>I(\alpha,0)</math> or <math>I(\alpha,\beta+1)</math>
 +
 
 +
Then,
 +
 
 +
<math>C_0(\alpha,\beta) = \beta\cup\{0\}</math>
 +
 
 +
<math>C_{n+1}(\alpha,\beta) = \{\gamma+\delta|\gamma,\delta\in C_n(\alpha,\beta)\}</math>
 +
 
 +
<math>\cup \{I(\gamma,\delta)|\gamma,\delta\in C_n(\alpha,\beta)\}</math>
 +
 
 +
<math>\cup \{\psi_\pi(\gamma)|\pi,\gamma\in C_n(\alpha,\beta)\wedge\gamma<\alpha\}</math>
 +
 
 +
<math>C(\alpha,\beta) = \bigcup_{n<\omega} C_n(\alpha,\beta)</math>
 +
 
 +
<math>\psi_\pi(\alpha) = \min\{\beta<\pi|C(\alpha,\beta)\cap\pi\subseteq\beta\}</math>
 +
 
 +
===Properties===
 +
 
 +
* <math>I(0,\alpha)=\Omega_{1+\alpha}=\aleph_{1+\alpha}</math>
 +
* <math>I(1,\alpha)=I_{1+\alpha}</math>
 +
* <math>\psi_{I(0,0)}(\alpha)=\omega^\alpha</math> for <math>\alpha<\varepsilon_0</math>
 +
* <math>\psi_{I(0,\alpha+1)}(\beta)=\omega^{I(0,\alpha)+1+\beta}</math> for <math>\beta<\varepsilon_{I(0,\alpha)+1}</math>
 +
 
 +
===Standard form for ordinals <math>\alpha<\psi_{I(1,0,0)}(0)=\text{min}\{\xi|I(\xi,0)=\xi\}</math>===
 +
 
 +
* The standard form for 0 is 0
 +
* If <math>\alpha</math> is of the form <math>I(\beta,\gamma)</math>, then the standard form for <math>\alpha</math> is <math>\alpha=I(\beta,\gamma)</math> where <math>\beta, \gamma<\alpha</math> and <math>\beta, \gamma</math> are expressed in standard form
 +
* If <math>\alpha</math> is not additively principal and <math>\alpha>0</math>, then the standard form for <math>\alpha</math> is <math>\alpha=\alpha_1+\alpha_2+\cdots+\alpha_n</math>, where the <math>\alpha_i</math> are principal ordinals with <math>\alpha_1\geq\alpha_2\geq\cdots\geq\alpha_n</math>, and the <math>\alpha_i</math> are expressed in standard form
 +
* If <math>\alpha</math> is an additively principal ordinal but not of the form <math>I(\beta,\gamma)</math>, then <math>\alpha</math> is expressible in the form <math>\psi_\pi(\delta)</math>. Then the standard form for <math>\alpha</math> is <math>\alpha=\psi_\pi(\delta)</math> where <math>\pi</math> and <math>\delta</math> are expressed in standard form
 +
 
 +
===Fundamental sequences===
 +
 
 +
For non-zero ordinals <math>\alpha<\psi_{I(1,0,0)}(0)</math> written in standard form fundamental sequences are defined as follows:
 +
 
 +
* If <math>\alpha=\alpha_1+\alpha_2+\cdots+\alpha_n</math> with <math>n\geq 2</math> then <math>\text{cof}(\alpha)=\text{cof}(\alpha_n)</math> and <math>\alpha[\eta]=\alpha_1+\alpha_2+\cdots+(\alpha_n[\eta])</math>
 +
*  If <math>\alpha=\psi_{I(0,0)}(0)</math> then <math>\alpha=\text{cof}(\alpha)=1</math> and <math>\alpha[0]=0</math>
 +
*  If <math>\alpha=\psi_{I(0,\beta+1)}(0)</math> then <math>\text{cof}(\alpha)=\omega</math> and <math>\alpha[\eta]=I(0,\beta)\cdot\eta</math>
 +
*  If <math>\alpha=\psi_{I(0,\beta)}(\gamma+1)</math> and <math>\beta\in\{0\}\cup S</math> then <math>\text{cof}(\alpha)=\omega</math> and <math>\alpha[\eta]=\psi_{I(0,\beta)}(\gamma)\cdot\eta</math>
 +
*  If <math>\alpha=\psi_{I(\beta+1,0)}(0)</math> then <math>\text{cof}(\alpha)=\omega</math> and <math>\alpha[0]=0</math> and <math>\alpha[\eta+1]=I(\beta,\alpha[\eta])</math>
 +
*  If <math>\alpha=\psi_{I(\beta+1,\gamma+1)}(0)</math> then <math>\text{cof}(\alpha)=\omega</math> and <math>\alpha[0]=I(\beta+1,\gamma)+1</math> and <math>\alpha[\eta+1]=I(\beta,\alpha[\eta])</math>
 +
*  If <math>\alpha=\psi_{I(\beta+1,\gamma)}(\delta+1)</math> and <math>\gamma\in\{0\}\cup S</math> then <math>\text{cof}(\alpha)=\omega</math> and <math>\alpha[0]=\psi_{I(\beta+1,\gamma)}(\delta)+1</math> and <math>\alpha[\eta+1]=I(\beta,\alpha[\eta])</math>
 +
*  If <math>\alpha=\psi_{I(\beta,0)}(0)</math> and <math>\beta\in L</math> then <math>\text{cof}(\alpha)=\text{cof}(\beta)</math> and <math>\alpha[\eta]=I(\beta[\eta],0)</math>
 +
*  If <math>\alpha=\psi_{I(\beta,\gamma+1)}(0)</math> and <math>\beta\in L</math> then <math>\text{cof}(\alpha)=\text{cof}(\beta)</math> and <math>\alpha[\eta]=I(\beta[\eta],I(\beta,\gamma)+1)</math>
 +
*  If <math>\alpha=\psi_{I(\beta,\gamma)}(\delta+1)</math> and <math>\beta\in L</math> and <math>\gamma\in \{0\}\cup S</math> then <math>\text{cof}(\alpha)=\text{cof}(\beta)</math> and <math>\alpha[\eta]=I(\beta[\eta],\psi_{I(\beta,\gamma)}(\delta)+1)</math>
 +
*  If <math>\alpha=\pi</math> then <math>\text{cof}(\alpha)=\pi</math> and <math>\alpha[\eta]=\eta</math>
 +
*  If <math>\alpha=I(\beta,\gamma)</math> and <math>\gamma\in L</math> then <math>\text{cof}(\alpha)=\text{cof}(\gamma)</math> and <math>\alpha[\eta]=I(\beta,\gamma[\eta])</math>
 +
*  If <math>\alpha=\psi_\pi(\beta)</math> and <math>\omega\le\text{cof}(\beta)<\pi</math> then <math>\text{cof}(\alpha)=\text{cof}(\beta)</math> and <math>\alpha[\eta]=\psi_\pi(\beta[\eta])</math>
 +
*  If <math>\alpha=\psi_\pi(\beta)</math> and <math>\text{cof}(\beta)=\rho\geq\pi</math> then <math>\text{cof}(\alpha)=\omega</math> and <math>\alpha[\eta]=\psi_\pi(\beta[\gamma[\eta]])</math> with <math>\gamma[0]=1</math> and <math>\gamma[\eta+1]=\psi_{\rho}(\beta[\gamma[\eta]])</math>
 +
 
 +
Below <math>\sigma</math> denotes the ordinal  <math>\psi_{I(0,0)}(I(\omega,0))</math>. Then <math>\text{cof}(\sigma)=\omega</math> and <math>\sigma[\eta]=\psi_{I(0,0)}(I(\eta,0))</math>
 +
 
 +
Examples of numbers
 +
 
 +
<math>\sigma</math>-billion <math>=10_{\sigma}^9=10\uparrow^{\sigma} 9</math>
 +
 
 +
<math>\sigma</math>-trillion <math>=10_{\sigma}^{12}=10\uparrow^{\sigma}12</math>
 +
 +
and so on up to <math>\sigma</math>-centillion <math>=10_{\sigma}^{303}=10\uparrow^{\sigma}303</math>
 +
 
 +
Below in the text the small Greek letter <math>\tau</math> denotes the largest countable limit ordinal such that the ruleset in this section allows to define fundamental sequences for all  limit ordinals less than <math>\tau</math>.  Then <math>\text{cof}(\tau)=\omega</math> and <math>\tau[\eta]=\psi_{I(0,0)}(\alpha[\eta])</math> where <math>\alpha[0]=0</math> and <math>\alpha[z+1]=I(\alpha[z],0)</math> for all integers <math>z \geq 0</math>
 +
 
 +
Examples of numbers
 +
 
 +
<math>\tau</math>-billion <math>=10_{\tau}^9=10\uparrow^{\tau} 9</math>
 +
 
 +
<math>\tau</math>-trillion <math>=10_{\tau}^{12}=10\uparrow^{\tau}12</math>
 +
 +
and so on up to <math>\tau</math>-centillion <math>=10_{\tau}^{303}=10\uparrow^{\tau}303</math>
 +
 
 +
==Section V. The first notation based on the least weakly Mahlo cardinal==
 +
 
 +
===Basic notions===
 +
 
 +
<math>\kappa</math> is weakly Mahlo iff <math>\kappa</math> is a cardinal such that for every function <math>f: \kappa\rightarrow\kappa</math> there exists a regular cardinal <math>\pi < \kappa</math> such that <math>\forall\alpha<\pi(f(\alpha)< \pi)</math>.
 +
 
 +
<math>M</math> is  the least weakly Mahlo cardinal and <math>\varepsilon_{M+1}=\min\{\alpha>M|\alpha=\omega^\alpha\}</math>  
  
 
<math>\alpha=_{NF}M^\beta\gamma\Leftrightarrow\alpha=M^\beta\gamma\wedge\gamma<M</math>
 
<math>\alpha=_{NF}M^\beta\gamma\Leftrightarrow\alpha=M^\beta\gamma\wedge\gamma<M</math>
  
The variables <math>\pi</math>, <math>\mu</math>, <math>\kappa</math> are reserved for regular uncountable cardinals less than <math>M</math>.
+
In this section the variables <math>\pi</math>, <math>\rho</math>, <math>\kappa</math> are reserved for regular uncountable cardinals less than <math>M</math>.
  
Enumeration function <math>F</math> of class of ordinals <math>X</math> is the unique increasing function such that <math>X=\{F(\alpha)|\alpha\in\text{dom}(F)\}</math> where domain of <math>F</math>, <math>\text{dom}(F)</math> is an ordinal number. We use <math>\text{Enum}(X)</math> to donate <math>F</math>.
+
Enumeration function <math>F</math> of class of ordinals <math>X</math> is the unique increasing function such that <math>X=\{F(\alpha)|\alpha\in\text{dom}(F)\}</math> where domain of <math>F</math>, <math>\text{dom}(F)</math> is an ordinal number. We use <math>\text{Enum}(X)</math> to denote <math>F</math>.
  
 
<math>cl(X) </math> is closure of <math>X</math>  
 
<math>cl(X) </math> is closure of <math>X</math>  
  
<math>cl_M(X)=X\cup\{\lambda<M|\lambda=\sup(X\cap\lambda)\} </math>
+
<math>cl_M(X)=X\cup\{\alpha<M|\alpha=\sup(X\cap\alpha)\} </math>
  
 
'''Definition''' of Veblen function
 
'''Definition''' of Veblen function
Line 78: Line 266:
 
Let <math>M^{\Gamma}=\min\{\alpha>M|\alpha=\varphi(\alpha,0)\}</math>
 
Let <math>M^{\Gamma}=\min\{\alpha>M|\alpha=\varphi(\alpha,0)\}</math>
  
'''Definition''' of Jäger's function <math>I_\alpha:M\rightarrow M</math> which enumerate the <math>\alpha</math>-inaccessible ordinals less than <math>M</math> and their limits
+
===Definition of functions <math>\chi_\alpha(\beta) </math> and <math>\psi_\pi(\gamma) </math>===
 
+
<math>I_\alpha=\text{Enum}(\{\beta\in R|\forall\gamma<\alpha(I_\gamma(\beta)=\beta)\}) </math>
+
 
+
Below we write <math>I(\alpha,\beta)</math> for <math>I_\alpha(\beta)</math>
+
 
+
 
+
==Definition of functions <math>\chi_\alpha(\beta) </math> and <math>\psi_\pi(\gamma) </math>==
+
  
 
'''Inductive Definition''' of  functions <math>\chi_\alpha: M\rightarrow M</math> for <math>\alpha <M^{\Gamma}</math> (Rathjen, 1990)
 
'''Inductive Definition''' of  functions <math>\chi_\alpha: M\rightarrow M</math> for <math>\alpha <M^{\Gamma}</math> (Rathjen, 1990)
  
1) <math>\{0,M\}\cup\beta\subset B^n(\alpha, \beta)</math>
+
1) <math>\{0,M\}\cup\beta\subseteq B^n(\alpha, \beta)</math>
  
 
2) <math>\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)</math>
 
2) <math>\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)</math>
Line 99: Line 280:
 
5) <math>\gamma<\pi\wedge\pi\in B^n(\alpha, \beta)\Rightarrow\gamma\in B^{n+1}(\alpha, \beta)</math>
 
5) <math>\gamma<\pi\wedge\pi\in B^n(\alpha, \beta)\Rightarrow\gamma\in B^{n+1}(\alpha, \beta)</math>
  
6) <math>B(\alpha,\beta)=\cup_{n<\omega}B^{n}(\alpha, \beta)</math>
+
6) <math>B(\alpha,\beta)=\bigcup_{n<\omega}B^{n}(\alpha, \beta)</math>
  
 
7) <math>\chi_\alpha=\text{Enum}(cl_M(\{\kappa|\kappa\notin B(\alpha,\kappa)\wedge\alpha\in B(\alpha,\kappa)\}))</math>
 
7) <math>\chi_\alpha=\text{Enum}(cl_M(\{\kappa|\kappa\notin B(\alpha,\kappa)\wedge\alpha\in B(\alpha,\kappa)\}))</math>
 
Note: as was said <math>\kappa</math> and <math>\pi </math> are reserved for uncountable regular cardinals less than <math>M</math>.
 
  
 
Below we write <math>\chi(\alpha,\beta)</math> for <math>\chi_\alpha(\beta)</math>
 
Below we write <math>\chi(\alpha,\beta)</math> for <math>\chi_\alpha(\beta)</math>
Line 119: Line 298:
 
5) <math>\chi(0,\alpha)=\aleph_{1+\alpha}</math>
 
5) <math>\chi(0,\alpha)=\aleph_{1+\alpha}</math>
  
6) <math>\chi(\alpha,\beta)=I(\alpha,\beta)</math> for all <math>\alpha<\lambda</math> where <math>\lambda=\sup\{\gamma(n)|n<\omega\}</math> with <math>\gamma(0)=0</math> and <math>\gamma(n+1)=\chi(\gamma(n),0)</math>
+
6) <math>\chi(\alpha,\beta)=I(\alpha,\beta)</math> for all <math>\alpha<\gamma</math> where <math>\gamma=\sup\{\delta(n)|n<\omega\}</math> with <math>\delta(0)=0</math> and <math>\delta(n+1)=\chi(\delta(n),0)</math>
  
 
'''Definition:''' <math>\alpha=_{NF}\chi(\beta,\gamma) \Leftrightarrow\alpha=\chi(\beta,\gamma)\wedge\gamma<\alpha</math>
 
'''Definition:''' <math>\alpha=_{NF}\chi(\beta,\gamma) \Leftrightarrow\alpha=\chi(\beta,\gamma)\wedge\gamma<\alpha</math>
 
  
 
Let <math>\Pi</math>  be the set of uncountable regular cardinals of the form <math>\chi(\alpha,0)</math> or <math>\chi(\alpha,\beta+1)</math>  
 
Let <math>\Pi</math>  be the set of uncountable regular cardinals of the form <math>\chi(\alpha,0)</math> or <math>\chi(\alpha,\beta+1)</math>  
  
 
<math>\Pi=\{\chi(\alpha,0)|\alpha<\varepsilon_{M+1}\}\cup\{\chi(\alpha,\beta+1)|\alpha<\varepsilon_{M+1}\wedge\beta<M\}</math>
 
<math>\Pi=\{\chi(\alpha,0)|\alpha<\varepsilon_{M+1}\}\cup\{\chi(\alpha,\beta+1)|\alpha<\varepsilon_{M+1}\wedge\beta<M\}</math>
 +
 +
On base of [https://www1.maths.leeds.ac.uk/~rathjen/Ord_Notation_Weakly_Mahlo.pdf Rathjen’s approach] we define a simplified version of functions  <math>\psi_\pi: M\rightarrow \pi</math> 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 <math>\psi_\pi: M\rightarrow \pi</math> for <math>\pi\in \Pi</math>  
 
'''Inductive Definition''' of  functions <math>\psi_\pi: M\rightarrow \pi</math> for <math>\pi\in \Pi</math>  
Line 132: Line 312:
 
1) <math>C^0(\alpha, \beta)=\{0,M\}\cup\beta</math>
 
1) <math>C^0(\alpha, \beta)=\{0,M\}\cup\beta</math>
  
2) <math>C^{n+1}(\alpha, \beta)=\{\gamma+\delta,\chi(\gamma,\delta), \omega^{M+\gamma}, \psi_\kappa(\eta)|\gamma,\delta,\eta,\xi,\kappa\in C^{n}(\alpha, \beta)\wedge\eta<\alpha\wedge\kappa\in\Pi\}</math>
+
2) <math>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\}</math>
  
3) <math>C(\alpha,\beta)=\cup_{n<\omega}C^{n}(\alpha, \beta)</math>
+
3) <math>C(\alpha,\beta)=\bigcup_{n<\omega}C^{n}(\alpha, \beta)</math>
  
4) <math>\psi_\pi(\alpha)=\min\{\beta<\pi|C(\alpha,\beta)\cap \pi\subset\beta\}</math>
+
4) <math>\psi_\pi(\alpha)=\min\{\beta<\pi|C(\alpha,\beta)\cap \pi\subseteq\beta\}</math>
  
 
'''Properties''' of  <math>\psi</math>-functions:
 
'''Properties''' of  <math>\psi</math>-functions:
Line 145: Line 325:
  
 
3) <math>\psi_\pi(\alpha)\in P</math>
 
3) <math>\psi_\pi(\alpha)\in P</math>
 
We write <math>\psi(\alpha)</math>  for <math>\psi_{\chi(0,0)}(\alpha)</math>
 
  
 
'''Definition:''' <math>\alpha=_{NF}\psi_\pi(\beta)\Leftrightarrow\alpha=\psi_\pi(\beta) \wedge\beta\in C(\beta, \psi_\pi(\beta))</math>
 
'''Definition:''' <math>\alpha=_{NF}\psi_\pi(\beta)\Leftrightarrow\alpha=\psi_\pi(\beta) \wedge\beta\in C(\beta, \psi_\pi(\beta))</math>
  
==A system of fundamental sequences==
+
===A system of fundamental sequences===
  
 
'''Inductive definition''' of <math>T</math>
 
'''Inductive definition''' of <math>T</math>
Line 163: Line 341:
  
 
5) <math>\alpha=_{NF}M^\beta\gamma\wedge\beta,\gamma\in T\Rightarrow\alpha\in T</math>
 
5) <math>\alpha=_{NF}M^\beta\gamma\wedge\beta,\gamma\in T\Rightarrow\alpha\in T</math>
 
For better understanding of collapsing functions <math>\psi_\pi</math> we define for each ordinal  <math>\alpha\in T</math>  its cofinality <math>\text{cof}(\alpha) </math> and  sequence  <math> (\alpha[\eta])_{\eta<\text{cof}(\alpha) }</math>  such that <math>\alpha=\sup\{\alpha[\eta]|\eta<<\text{cof}(\alpha) \}</math>
 
  
 
'''Definition''' of fundamental sequences for non-zero ordinals <math>\alpha\in T</math>:
 
'''Definition''' of fundamental sequences for non-zero ordinals <math>\alpha\in T</math>:
Line 172: Line 348:
 
2) <math>\alpha=0\Rightarrow\text{cof}(\alpha)=0</math>  
 
2) <math>\alpha=0\Rightarrow\text{cof}(\alpha)=0</math>  
  
 
+
3) <math>\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</math>
3) <math>\alpha=\psi _{\chi(0,0)}(0) \vee \alpha=\chi(\beta,0) \vee \alpha=\chi(\beta,\gamma+1) \vee \alpha=M\Rightarrow \text{cof} (\alpha)=\alpha \wedge \alpha[\eta]=\eta</math>
+
  
 
4) <math>\alpha=\psi _{\chi(0,\beta+1)}(0) \Rightarrow \text{cof}(\alpha)=\omega \wedge \alpha[n]=\chi(0,\beta)\times n</math>
 
4) <math>\alpha=\psi _{\chi(0,\beta+1)}(0) \Rightarrow \text{cof}(\alpha)=\omega \wedge \alpha[n]=\chi(0,\beta)\times n</math>
  
 
5) <math>\alpha=\psi_{ \chi(0,\beta)}(\gamma+1) \Rightarrow \text{cof}(\alpha)=\omega \wedge \alpha[n]=\psi_{\chi(0,\beta)}(\gamma)\times n</math>
 
5) <math>\alpha=\psi_{ \chi(0,\beta)}(\gamma+1) \Rightarrow \text{cof}(\alpha)=\omega \wedge \alpha[n]=\psi_{\chi(0,\beta)}(\gamma)\times n</math>
 
 
  
 
6) <math>\alpha=\psi _{\chi(\beta+1,0)}(0) \Rightarrow \text{cof}(\alpha)=\omega \wedge \alpha[0]=0 \wedge \alpha[n+1]=\chi(\beta,\alpha[n])</math>
 
6) <math>\alpha=\psi _{\chi(\beta+1,0)}(0) \Rightarrow \text{cof}(\alpha)=\omega \wedge \alpha[0]=0 \wedge \alpha[n+1]=\chi(\beta,\alpha[n])</math>
Line 186: Line 359:
  
 
8) <math>\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])</math>
 
8) <math>\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])</math>
 
  
 
9) <math>\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)</math>
 
9) <math>\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)</math>
Line 193: Line 365:
  
 
11) <math>\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)</math>
 
11) <math>\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)</math>
 
  
 
12) <math>\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)</math>
 
12) <math>\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)</math>
Line 200: Line 371:
  
 
14) <math>\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)</math>
 
14) <math>\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)</math>
 
  
 
15) <math>\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])</math>
 
15) <math>\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])</math>
  
16) <math>\alpha=M^{\beta+1}\times(\gamma+1) \wedge \gamma+1<M \Rightarrow \text{cof} (\alpha)=M \wedge\alpha[\eta]=M^{\beta+1}\times\gamma+M^\beta\times\eta</math>
+
16) <math>\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</math>
 
+
17) <math>\alpha=M^\beta\times(\gamma+1) \wedge \gamma+1<M \wedge\text{cof}(\beta)\geq\omega \Rightarrow \text{cof}(\alpha)= \text{cof}(\beta) \wedge \alpha[\eta]=M^\beta\times\gamma+M^{\beta[\eta]}</math>
+
  
 +
17) <math>\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]}</math>
  
 
18) <math>\alpha=\chi(\beta,\gamma) \wedge \text{cof}(\gamma)\geq\omega \Rightarrow \text{cof} (\alpha)=\text{cof}(\gamma)\wedge \alpha[\eta]=\chi(\beta,\gamma[\eta])</math>
 
18) <math>\alpha=\chi(\beta,\gamma) \wedge \text{cof}(\gamma)\geq\omega \Rightarrow \text{cof} (\alpha)=\text{cof}(\gamma)\wedge \alpha[\eta]=\chi(\beta,\gamma[\eta])</math>
Line 213: Line 382:
 
19) <math>\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])</math>
 
19) <math>\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])</math>
  
20) <math>\alpha=\psi_\pi(\beta) \wedge \text{cof}(\beta)=\mu\geq\pi \Rightarrow \text{cof} (\alpha)=\omega \wedge \alpha[n]=\psi _\pi(\beta[\gamma[n]]) \wedge \gamma[0]=1 \wedge \gamma[k+1]=\psi_\mu(\beta[\gamma[k]])</math>
+
20) <math>\alpha=\psi_\pi(\beta) \wedge \text{cof}(\beta)=\rho\geq\pi \Rightarrow \text{cof} (\alpha)=\omega \wedge \alpha[n]=\psi _\pi(\beta[\gamma[n]])</math> where  <math>\gamma[0]=1</math> and  <math>\gamma[k+1]=\psi_\rho(\beta[\gamma[k]])</math>
 +
 
 +
Below in the text the capital Greek letter <math>\Lambda</math> denotes the largest countable limit ordinal such that the ruleset in this section allows to assign fundamental sequences for all  limit ordinals less than <math>\Lambda</math>.  Then <math>\text{cof}(\Lambda)=\omega </math> and <math>\Lambda[n]=\psi_{\chi(0,0)}(\chi(\alpha[n],0))</math> where <math>\alpha[0]=0</math> and  <math>\alpha[z+1]=M^{\alpha[z]}</math> for all integers <math>z \geq 0</math>
 +
 
 +
Note  <math>M^0=1</math>
 +
 
 +
Examples of numbers
 +
 
 +
<math>\Lambda </math>-billion <math>=10_\Lambda^9=10\uparrow^\Lambda 9</math>
 +
 
 +
<math>\Lambda </math>-trillion <math>=10_\Lambda^{12}=10\uparrow^\Lambda 12</math>
 +
 +
and so on up to <math>\Lambda </math>-centillion <math>=10_\Lambda ^{303}=10\uparrow^\Lambda 303</math>
 +
 
 +
==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'''
 +
 
 +
<math>M</math> is the least weakly Mahlo cardinal.
 +
 
 +
Normal form. <math>\alpha=_{NF}M^\beta\gamma\Leftrightarrow\alpha=M^\beta\gamma\wedge\gamma<M</math>
 +
 
 +
<math>\varepsilon_{M+1}=\min\{\alpha>M|\alpha=\omega^\alpha\}</math> is the least epsilon number greater than <math>M</math>.
 +
 
 +
In this section:
 +
* <math>\alpha\in R\Leftrightarrow\alpha=\chi(\beta)\vee\alpha=M</math>,
 +
* the variables <math>\pi, \rho</math> are reserved for uncountable regular cardinals less than <math>M</math>.
 +
 
 +
'''Definition''' of the function <math>\chi:\varepsilon_{M+1}\rightarrow M</math>
 +
 
 +
1) <math>B_0(\alpha,\beta)=\beta\cup\{0\}</math>
 +
 
 +
2) <math>\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)</math>
 +
 
 +
3) <math>\gamma=\omega^{M+\delta}\wedge\delta\in B_n(\alpha,\beta)\Rightarrow\gamma\in B_{n+1}(\alpha,\beta)</math>
 +
 
 +
4) <math>\gamma=\chi(\eta)\wedge\eta\in B_n(\alpha,\beta)\cap\alpha \Rightarrow\gamma\in B_{n+1}(\alpha,\beta)</math>
 +
 
 +
5) <math>\gamma<\pi\wedge\pi\in B_n(\alpha,\beta) \Rightarrow\gamma\in B_{n+1}(\alpha,\beta)</math>
 +
 
 +
6) <math>B(\alpha,\beta)=\bigcup_{n<\omega}B_n(\alpha,\beta)</math>
 +
 
 +
7) <math>\chi(\alpha)=\min\{\beta<M|B(\alpha,\beta)\cap M\subseteq\beta\wedge\beta\in R\}</math>
 +
 
 +
Normal form. <math>\alpha=_{NF}\chi(\beta)\Leftrightarrow\alpha=\chi(\beta)\wedge\beta\in B(\beta,\chi(\beta))</math>
 +
 
 +
'''Definition''' of functions <math>\psi_\pi:M\rightarrow \pi</math>
 +
 
 +
1) <math>C_0(\alpha,\beta)=\beta\cup\{0\}</math>
 +
 
 +
2) <math>\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)</math>
 +
 
 +
3) <math>\gamma=\omega^{M+\delta}\wedge\delta\in C_n(\alpha,\beta)\Rightarrow\gamma\in C_{n+1}(\alpha,\beta)</math>
 +
 
 +
4) <math>\gamma=_{NF}\chi(\eta)\wedge\eta\in C_n(\alpha,\beta) \Rightarrow\gamma\in C_{n+1}(\alpha,\beta)</math>
 +
 
 +
5) <math>\gamma=\psi_\pi(\eta)\wedge\eta<\alpha\wedge\pi,\eta\in C_n(\alpha,\beta)\Rightarrow\gamma\in C_{n+1}(\alpha,\beta)</math>
 +
 
 +
6) <math>C(\alpha,\beta)=\bigcup_{n<\omega}C_n(\alpha,\beta)</math>
 +
 
 +
7) <math>\psi_\pi(\alpha)=\min\{\beta<\pi|C(\alpha,\beta)\cap \pi\subseteq\beta\}</math>
 +
 
 +
Normal form. <math>\alpha=_{NF}\psi_\pi(\beta)\Leftrightarrow\alpha=\psi_\pi(\beta)\wedge\beta\in C(\beta,\psi_\pi(\beta))</math>
 +
 
 +
===A system of fundamental sequences===
 +
 
 +
'''Definition''' of the set <math>T</math> of ordinals which can be generated from the ordinals <math>0</math> and <math>M</math> using addition, multiplication, exponentiation and the functions <math>\chi,\psi_\pi</math>
 +
 
 +
1) <math>0\in T</math>
 +
 
 +
2) <math>\alpha=_{NF}\alpha_1+\cdots+\alpha_k\wedge\alpha_1,...,\alpha_k\in T\Rightarrow\alpha\in T</math>
 +
 
 +
3) <math>\alpha=_{NF}M^\beta\gamma\wedge\beta,\gamma\in T\Rightarrow\alpha\in T</math>
 +
 
 +
4) <math>\alpha=_{NF}\psi_\pi(\beta)\wedge\pi,\beta\in T\Rightarrow\alpha\in T</math>
 +
 
 +
5) <math>\alpha=_{NF}\chi(\beta)\wedge\beta\in T\Rightarrow\alpha\in T</math>
 +
 
 +
'''Definition''' of fundamental sequences for non-zero ordinals <math>\alpha\in T</math>:
 +
 
 +
1) <math>\alpha=\alpha_1+\cdots+\alpha_k\Rightarrow\text{cof}(\alpha)=\text{cof}(\alpha_k)\wedge\alpha[\eta]=\alpha_1+\cdots+(\alpha_k[\eta])</math>
 +
 
 +
2) <math>\alpha=\psi_{\chi(\beta+1)}(0)\Rightarrow\text{cof}(\alpha)=\omega\wedge\alpha[\eta]=\chi(\beta)\times \eta</math>
 +
 
 +
3) <math>\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])</math>
 +
 
 +
4) <math>\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]])</math>
 +
 
 +
5)  <math>\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</math>
 +
 
 +
6)  <math>\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]])</math>
 +
 
 +
7)  <math>\alpha=\psi_{\chi(0)}(0)=1\vee\alpha=\chi(\beta)\vee\alpha=M\Rightarrow\text{cof}(\alpha)=\alpha\wedge\alpha[\eta]=\eta</math>
 +
 
 +
8) <math>\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])</math>
 +
 
 +
9) <math>\alpha=M^{\beta+1}\times(\gamma+1) \Rightarrow \text{cof} (\alpha)=M \wedge\alpha[\eta]=M^{\beta+1}\times\gamma+M^\beta\times\eta</math>
 +
 
 +
10) <math>\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]}</math>
 +
 
 +
11) <math>\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])</math>
  
 +
12) <math>\alpha=\psi_\pi(\beta) \wedge \text{cof}(\beta)=\rho\geq\pi \Rightarrow \text{cof} (\alpha)=\omega \wedge \alpha[\eta]=\psi _\pi(\beta[\gamma[\eta]])</math> where  <math>\gamma[0]=1</math> and  <math>\gamma[z+1]=\psi_\rho(\beta[\gamma[z]])</math> for all integers <math>z \geq 0</math>
  
Limit of this notation is <math>\Lambda</math>
+
Below in the text the small Greek letter <math>\lambda</math> denotes the largest countable limit ordinal such that the ruleset in this section allows to assign fundamental sequences for all  limit ordinals less than <math>\lambda</math>.  Then
 +
<math> \text{cof} (\lambda)=\omega</math> and  <math>\lambda[\eta]=\psi_{\chi(0)} (\chi(\alpha[\eta]))</math> where <math>\alpha[0]=0</math> and <math>\alpha[z+1]=M^{\alpha[z]}</math> for all integers <math>z \geq 0</math>
  
<math>\alpha=\Lambda \Rightarrow \text{cof}(\alpha)=\omega \wedge \alpha[n]=\chi(\beta[n],0)\wedge\beta[0]=0 \wedge \beta[k+1]=M^{\beta[k]}</math>
+
Note  <math>M^0=1</math>
  
The system of fundamental sequences can be applied for defining the slow-growing hierarchy
+
Examples of numbers
  
<math>g_0(n)=0</math><math>g_{\alpha+1}(n)=g_\alpha(n)+1</math>;  <math>g_\alpha(n)=g_{\alpha[n]}(n)</math> iff <math>\text{cof}(\alpha)=\omega</math>,
+
<math>\lambda </math>-billion <math>=10_\lambda ^9=10\uparrow^\lambda 9</math>
  
Hardy hierarchy
+
<math>\lambda </math>-trillion <math>=10_\lambda ^{12}=10\uparrow^\lambda 12</math>
 +
 +
and so on up to <math>\lambda </math>-centillion <math>=10_\lambda ^{303}=10\uparrow^\lambda 303</math>
  
<math>H_0(n)=n</math>; <math>H_{\alpha+1}(n)=H_\alpha(n+1)</math>; <math>H_\alpha(n)=H_{\alpha[n]}(n)</math> iff <math>\text{cof}(\alpha)=\omega</math>,
+
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 <math>\lambda </math>-centillion? Just wishing to somehow apply huge numbers mentioned above we can define for example (under the assumption that our universe is infinite)
  
and fast-growing hierarchy
+
the name of a number <math>j</math>  + "er" is the set of all points of the physical space, which are located not further than <math>j</math> parsecs from the point of the Earth's center.
  
<math>f_0(n)=n+1</math>
+
For example, <math>\lambda</math>-billioner is the set of all points of the physical space which are located not further than <math>\lambda</math>-billion parsecs from the point of the Earth's center.
  
<math>f_{\alpha+1}(n)=f_\alpha^n(n)</math> where <math>f_\alpha^0(n)=n</math> and <math>f_\alpha^{m+1}(n)=f_\alpha(f_\alpha^m(n))</math>
+
Other examples. <math>\omega</math>-billioner, <math>\varepsilon_0</math>-trillioner, <math>\upsilon </math>-billioner, <math>\sigma</math>-trillioner, <math>\tau</math>-centillioner, <math>\Lambda </math>-billioner
  
<math>f_\alpha(n)=f_{\alpha[n]}(n)</math> iff <math>\text{cof}(\alpha)=\omega</math>.
 
  
For example, <math>f_{\psi(\Lambda)}^2(9)</math>
+
Author: Denis Maksudov (Ufa, Russia)
  
 +
E-mail: md77@list.ru
  
 
==References==
 
==References==
  
1. W.Buchholz. A New System of Proof-Theoretic Ordinal Functions. Annals of Pure and Applied Logic (1986),32
+
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. M.Jäger. <math>\rho</math>-inaccessible ordinals, collapsing functions and a recursive notation system. Arch. Math. Logik Grundlagenforsch (1984),24
+
2. W.Buchholz (1986). A New System of Proof-Theoretic Ordinal Functions. Annals of Pure and Applied Logic, Vol. 32, 195-207
  
3. M. Rathjen. Ordinal Notations Based on a Weakly Mahlo Cardinal. Arch. Math. Logic (1990).29 
+
3. M.Jäger (1984). <math>\rho</math>-inaccessible ordinals, collapsing functions and a recursive notation system. Arch. Math. Logik Grundlagenforsch, Vol. 24, 49-62
  
4. [http://googology.wikia.com/wiki/User:Denis_Maksudov My account in googology.wikia]
+
4. M. Rathjen (1990). Ordinal Notations Based on a Weakly Mahlo Cardinal. Arch. Math. Logic, Vol. 29, 249-263

Latest revision as of 05:01, 25 April 2019

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