Site of Denis Maksudov (Ufa, Russia)

PDF-version of this page

Here I collected my following works:

1) extended arrows notation (December 2016)

2) the extended Wilfried Buchholz's functions (April 2017)

3) fundamental sequences for the functions collapsing {\displaystyle \alpha }\alpha -weakly inaccessible cardinals (August 2017).


Extended arrow notation

We define for non-zero natural numbers {\displaystyle n}n{\displaystyle b}b and for ordinal number {\displaystyle \alpha \geq 0:}{\displaystyle \alpha \geq 0:}

1) {\displaystyle n\uparrow ^{\alpha }b=nb}{\displaystyle n\uparrow ^{\alpha }b=nb} if {\displaystyle \alpha =0}\alpha =0

2) {\displaystyle n\uparrow ^{\alpha +1}b=n}{\displaystyle n\uparrow ^{\alpha +1}b=n} if {\displaystyle b=1}b=1

3) {\displaystyle n\uparrow ^{\alpha +1}b=n\uparrow ^{\alpha }(n\uparrow ^{\alpha +1}(b-1))}{\displaystyle n\uparrow ^{\alpha +1}b=n\uparrow ^{\alpha }(n\uparrow ^{\alpha +1}(b-1))} if {\displaystyle b>1}b>1

4) {\displaystyle n\uparrow ^{\alpha }b=n\uparrow ^{\alpha [b]}n}{\displaystyle n\uparrow ^{\alpha }b=n\uparrow ^{\alpha [b]}n} if {\displaystyle \alpha }\alpha  is a countable limit ordinal

where {\displaystyle \alpha [b]}{\displaystyle \alpha [b]} denotes the b-th element of the fundamental sequence assigned to the limit ordinal {\displaystyle \alpha }\alpha .

The fast-growing hierarchy

The fast-growing hierarchy is defined as follows:

  1. {\displaystyle f_{0}(n)=n+1}{\displaystyle f_{0}(n)=n+1}
  2. {\displaystyle f_{\alpha }^{m+1}(n)=f_{\alpha }(f_{\alpha }^{m}(n))}{\displaystyle f_{\alpha }^{m+1}(n)=f_{\alpha }(f_{\alpha }^{m}(n))}
  3. {\displaystyle f_{\alpha }^{0}(n)=n}{\displaystyle f_{\alpha }^{0}(n)=n}
  4. {\displaystyle f_{\alpha +1}(n)=f_{\alpha }^{n}(n)}{\displaystyle f_{\alpha +1}(n)=f_{\alpha }^{n}(n)}
  5. {\displaystyle f_{\alpha }(n)=f_{\alpha [n]}(n)}{\displaystyle f_{\alpha }(n)=f_{\alpha [n]}(n)} if {\displaystyle \alpha }\alpha  is a countable limit ordinal

where {\displaystyle n}n{\displaystyle m}m are non-negative integers and {\displaystyle \alpha [n]}{\displaystyle \alpha [n]} is the {\displaystyle n}n-th element of the fundamental sequence assigned to the limit ordinal {\displaystyle \alpha }\alpha .

The extended Wilfried Buchholz's functions

We rewrite Buchholz's definition as follows:

  • {\displaystyle C_{\nu }^{0}(\alpha )=\{\beta |\beta <\Omega _{\nu }\}}{\displaystyle C_{\nu }^{0}(\alpha )=\{\beta |\beta <\Omega _{\nu }\}}
  • {\displaystyle C_{\nu }^{n+1}(\alpha )=\{\beta +\gamma ,\psi _{\mu }(\eta )|\mu ,\beta ,\gamma ,\eta \in C_{\nu }^{n}(\alpha )\wedge \eta <\alpha \}}{\displaystyle C_{\nu }^{n+1}(\alpha )=\{\beta +\gamma ,\psi _{\mu }(\eta )|\mu ,\beta ,\gamma ,\eta \in C_{\nu }^{n}(\alpha )\wedge \eta <\alpha \}}
  • {\displaystyle C_{\nu }(\alpha )=\bigcup _{n<\omega }C_{\nu }^{n}(\alpha )}{\displaystyle C_{\nu }(\alpha )=\bigcup _{n<\omega }C_{\nu }^{n}(\alpha )}
  • {\displaystyle \psi _{\nu }(\alpha )=\min\{\gamma |\gamma \not \in C_{\nu }(\alpha )\}}{\displaystyle \psi _{\nu }(\alpha )=\min\{\gamma |\gamma \not \in C_{\nu }(\alpha )\}}

where

{\displaystyle \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.}{\displaystyle \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.}

and {\displaystyle \omega }\omega  is the smallest infinite ordinal.

There is only one little detail difference with original Buchholz definition: ordinal {\displaystyle \mu }\mu  is not limited by {\displaystyle \omega }\omega , now ordinal {\displaystyle \mu }\mu  belongs to previous set {\displaystyle C_{n}}C_{n}. Limit of this notation must be omega fixed point {\displaystyle \psi _{0}(\Omega _{\Omega _{\Omega _{...}}})=\psi _{0}(\psi _{\psi _{...}(0)}(0))}{\displaystyle \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 {\displaystyle \alpha }\alpha  is a nonzero ordinal number {\displaystyle \alpha <\Lambda ={\text{min}}\{\beta |\psi _{\beta }(0)=\beta \}}{\displaystyle \alpha <\Lambda ={\text{min}}\{\beta |\psi _{\beta }(0)=\beta \}} then the normal form for {\displaystyle \alpha }\alpha  is {\displaystyle \alpha =\psi _{\nu _{1}}(\beta _{1})+\psi _{\nu _{2}}(\beta _{2})+\cdots +\psi _{\nu _{k}}(\beta _{k})}{\displaystyle \alpha =\psi _{\nu _{1}}(\beta _{1})+\psi _{\nu _{2}}(\beta _{2})+\cdots +\psi _{\nu _{k}}(\beta _{k})}where {\displaystyle k}k is a positive integer and {\displaystyle \psi _{\nu _{1}}(\beta _{1})\geq \psi _{\nu _{2}}(\beta _{2})\geq \cdots \geq \psi _{\nu _{k}}(\beta _{k})}{\displaystyle \psi _{\nu _{1}}(\beta _{1})\geq \psi _{\nu _{2}}(\beta _{2})\geq \cdots \geq \psi _{\nu _{k}}(\beta _{k})} and each {\displaystyle \nu _{i},\beta _{i}}{\displaystyle \nu _{i},\beta _{i}} are also written in normal form.

Fundamental sequences for the extended Wilfried Buchholz's functions

The fundamental sequence for an ordinal number {\displaystyle \alpha }\alpha  with cofinality {\displaystyle {\text{cof}}(\alpha )=\beta }{\displaystyle {\text{cof}}(\alpha )=\beta } is a strictly increasing sequence {\displaystyle (\alpha [\eta ])_{\eta <\beta }}{\displaystyle (\alpha [\eta ])_{\eta <\beta }} with length {\displaystyle \beta }\beta  and with limit {\displaystyle \alpha }\alpha , where {\displaystyle \alpha [\eta ]}{\displaystyle \alpha [\eta ]} is the {\displaystyle \eta }\eta -th element of this sequence. If {\displaystyle \alpha }\alpha  is a successor ordinal then {\displaystyle {\text{cof}}(\alpha )=1}{\displaystyle {\text{cof}}(\alpha )=1} and the fundamental sequence has only one element {\displaystyle \alpha [0]=\alpha -1}{\displaystyle \alpha [0]=\alpha -1}. If {\displaystyle \alpha }\alpha  is a limit ordinal then {\displaystyle {\text{cof}}(\alpha )\in \{\omega \}\cup \{\Omega _{\mu +1}|\mu \geq 0\}}{\displaystyle {\text{cof}}(\alpha )\in \{\omega \}\cup \{\Omega _{\mu +1}|\mu \geq 0\}}

For nonzero ordinals {\displaystyle \alpha <\Lambda }{\displaystyle \alpha <\Lambda }, written in normal form, fundamental sequences are defined as follows:

  1. If {\displaystyle \alpha =\psi _{\nu _{1}}(\beta _{1})+\psi _{\nu _{2}}(\beta _{2})+\cdots +\psi _{\nu _{k}}(\beta _{k})}{\displaystyle \alpha =\psi _{\nu _{1}}(\beta _{1})+\psi _{\nu _{2}}(\beta _{2})+\cdots +\psi _{\nu _{k}}(\beta _{k})} where {\displaystyle k\geq 2}{\displaystyle k\geq 2} then {\displaystyle {\text{cof}}(\alpha )={\text{cof}}(\psi _{\nu _{k}}(\beta _{k}))}{\displaystyle {\text{cof}}(\alpha )={\text{cof}}(\psi _{\nu _{k}}(\beta _{k}))} and {\displaystyle \alpha [\eta ]=\psi _{\nu _{1}}(\beta _{1})+\cdots +\psi _{\nu _{k-1}}(\beta _{k-1})+(\psi _{\nu _{k}}(\beta _{k})[\eta ])}{\displaystyle \alpha [\eta ]=\psi _{\nu _{1}}(\beta _{1})+\cdots +\psi _{\nu _{k-1}}(\beta _{k-1})+(\psi _{\nu _{k}}(\beta _{k})[\eta ])}
  2. If {\displaystyle \alpha =\psi _{0}(0)=1}{\displaystyle \alpha =\psi _{0}(0)=1}, then {\displaystyle {\text{cof}}(\alpha )=1}{\displaystyle {\text{cof}}(\alpha )=1} and {\displaystyle \alpha [0]=0}{\displaystyle \alpha [0]=0}
  3. If {\displaystyle \alpha =\psi _{\nu +1}(0)}{\displaystyle \alpha =\psi _{\nu +1}(0)}, then {\displaystyle {\text{cof}}(\alpha )=\Omega _{\nu +1}}{\displaystyle {\text{cof}}(\alpha )=\Omega _{\nu +1}} and {\displaystyle \alpha [\eta ]=\Omega _{\nu +1}[\eta ]=\eta }{\displaystyle \alpha [\eta ]=\Omega _{\nu +1}[\eta ]=\eta }
  4. If {\displaystyle \alpha =\psi _{\nu }(0)}{\displaystyle \alpha =\psi _{\nu }(0)} and {\displaystyle {\text{cof}}(\nu )\in \{\omega \}\cup \{\Omega _{\mu +1}|\mu \geq 0\}}{\displaystyle {\text{cof}}(\nu )\in \{\omega \}\cup \{\Omega _{\mu +1}|\mu \geq 0\}}, then {\displaystyle {\text{cof}}(\alpha )={\text{cof}}(\nu )}{\displaystyle {\text{cof}}(\alpha )={\text{cof}}(\nu )} and {\displaystyle \alpha [\eta ]=\psi _{\nu [\eta ]}(0)=\Omega _{\nu [\eta ]}}{\displaystyle \alpha [\eta ]=\psi _{\nu [\eta ]}(0)=\Omega _{\nu [\eta ]}}
  5. If {\displaystyle \alpha =\psi _{\nu }(\beta +1)}{\displaystyle \alpha =\psi _{\nu }(\beta +1)} then {\displaystyle {\text{cof}}(\alpha )=\omega }{\displaystyle {\text{cof}}(\alpha )=\omega } and {\displaystyle \alpha [\eta ]=\psi _{\nu }(\beta )\cdot \eta }{\displaystyle \alpha [\eta ]=\psi _{\nu }(\beta )\cdot \eta } (and note: {\displaystyle \psi _{\nu }(0)=\Omega _{\nu }}{\displaystyle \psi _{\nu }(0)=\Omega _{\nu }})
  6. If {\displaystyle \alpha =\psi _{\nu }(\beta )}{\displaystyle \alpha =\psi _{\nu }(\beta )} and {\displaystyle {\text{cof}}(\beta )\in \{\omega \}\cup \{\Omega _{\mu +1}|\mu <\nu \}}{\displaystyle {\text{cof}}(\beta )\in \{\omega \}\cup \{\Omega _{\mu +1}|\mu <\nu \}} then {\displaystyle {\text{cof}}(\alpha )={\text{cof}}(\beta )}{\displaystyle {\text{cof}}(\alpha )={\text{cof}}(\beta )} and {\displaystyle \alpha [\eta ]=\psi _{\nu }(\beta [\eta ])}{\displaystyle \alpha [\eta ]=\psi _{\nu }(\beta [\eta ])}
  7. If {\displaystyle \alpha =\psi _{\nu }(\beta )}{\displaystyle \alpha =\psi _{\nu }(\beta )} and {\displaystyle {\text{cof}}(\beta )\in \{\Omega _{\mu +1}|\mu \geq \nu \}}{\displaystyle {\text{cof}}(\beta )\in \{\Omega _{\mu +1}|\mu \geq \nu \}} then {\displaystyle {\text{cof}}(\alpha )=\omega }{\displaystyle {\text{cof}}(\alpha )=\omega } and {\displaystyle \alpha [\eta ]=\psi _{\nu }(\beta [\gamma [\eta ]])}{\displaystyle \alpha [\eta ]=\psi _{\nu }(\beta [\gamma [\eta ]])} where {\displaystyle \left\{{\begin{array}{lcr}\gamma [0]=\Omega _{\mu }\\\gamma [\eta +1]=\psi _{\mu }(\beta [\gamma [\eta ]])\\\end{array}}\right.}{\displaystyle \left\{{\begin{array}{lcr}\gamma [0]=\Omega _{\mu }\\\gamma [\eta +1]=\psi _{\mu }(\beta [\gamma [\eta ]])\\\end{array}}\right.}

Limit of this notation {\displaystyle \Lambda ={\text{min}}\{\beta |\psi _{\beta }(0)=\beta \}}{\displaystyle \Lambda ={\text{min}}\{\beta |\psi _{\beta }(0)=\beta \}}. If {\displaystyle \alpha =\Lambda }{\displaystyle \alpha =\Lambda } then {\displaystyle {\text{cof}}(\alpha )=\omega }{\displaystyle {\text{cof}}(\alpha )=\omega } and {\displaystyle \alpha [0]=0}{\displaystyle \alpha [0]=0} and {\displaystyle \alpha [\eta +1]=\psi _{\alpha [\eta ]}(0)=\Omega _{\alpha [\eta ]}}{\displaystyle \alpha [\eta +1]=\psi _{\alpha [\eta ]}(0)=\Omega _{\alpha [\eta ]}}

Fundamental sequences for the functions collapsing {\displaystyle \alpha }\alpha -weakly inaccessible cardinals

Definition of the functions collapsing {\displaystyle \alpha }\alpha -weakly inaccessible cardinals

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

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

On this page the variables {\displaystyle \rho }\rho {\displaystyle \pi }\pi  are reserved for uncountable regular cardinals of the form {\displaystyle I(\alpha ,0)}{\displaystyle I(\alpha ,0)} or {\displaystyle I(\alpha ,\beta +1)}{\displaystyle I(\alpha ,\beta +1)}

Then,

{\displaystyle C_{0}(\alpha ,\beta )=\beta \cup \{0\}}{\displaystyle C_{0}(\alpha ,\beta )=\beta \cup \{0\}}

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

{\displaystyle \cup \{I(\gamma ,\delta )|\gamma ,\delta \in C_{n}(\alpha ,\beta )\}}{\displaystyle \cup \{I(\gamma ,\delta )|\gamma ,\delta \in C_{n}(\alpha ,\beta )\}}

{\displaystyle \cup \{\psi _{\pi }(\gamma )|\pi ,\gamma \in C_{n}(\alpha ,\beta )\wedge \gamma <\alpha \}}{\displaystyle \cup \{\psi _{\pi }(\gamma )|\pi ,\gamma \in C_{n}(\alpha ,\beta )\wedge \gamma <\alpha \}}

{\displaystyle C(\alpha ,\beta )=\bigcup _{n<\omega }C_{n}(\alpha ,\beta )}{\displaystyle C(\alpha ,\beta )=\bigcup _{n<\omega }C_{n}(\alpha ,\beta )}

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

Properties

  1. {\displaystyle I(0,\alpha )=\Omega _{1+\alpha }=\aleph _{1+\alpha }}{\displaystyle I(0,\alpha )=\Omega _{1+\alpha }=\aleph _{1+\alpha }}
  2. {\displaystyle I(1,\alpha )=I_{1+\alpha }}{\displaystyle I(1,\alpha )=I_{1+\alpha }}
  3. {\displaystyle \psi _{I(0,0)}(\alpha )=\omega ^{\alpha }}{\displaystyle \psi _{I(0,0)}(\alpha )=\omega ^{\alpha }} for {\displaystyle \alpha <\varepsilon _{0}}{\displaystyle \alpha <\varepsilon _{0}}
  4. {\displaystyle \psi _{I(0,\alpha +1)}(\beta )=\omega ^{I(0,\alpha )+1+\beta }}{\displaystyle \psi _{I(0,\alpha +1)}(\beta )=\omega ^{I(0,\alpha )+1+\beta }} for {\displaystyle \beta <\varepsilon _{I(0,\alpha )+1}}{\displaystyle \beta <\varepsilon _{I(0,\alpha )+1}}

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

  1. The standard form for 0 is 0
  2. If {\displaystyle \alpha }\alpha  is of the form {\displaystyle I(\beta ,\gamma )}{\displaystyle I(\beta ,\gamma )}, then the standard form for {\displaystyle \alpha }\alpha  is {\displaystyle \alpha =I(\beta ,\gamma )}{\displaystyle \alpha =I(\beta ,\gamma )} where {\displaystyle \beta ,\gamma <\alpha }{\displaystyle \beta ,\gamma <\alpha } and {\displaystyle \beta ,\gamma }{\displaystyle \beta ,\gamma } are expressed in standard form
  3. If {\displaystyle \alpha }\alpha  is not additively principal and {\displaystyle \alpha >0}\alpha >0, then the standard form for {\displaystyle \alpha }\alpha  is {\displaystyle \alpha =\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}}{\displaystyle \alpha =\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}}, where the {\displaystyle \alpha _{i}}\alpha _{i} are principal ordinals with {\displaystyle \alpha _{1}\geq \alpha _{2}\geq \cdots \geq \alpha _{n}}{\displaystyle \alpha _{1}\geq \alpha _{2}\geq \cdots \geq \alpha _{n}}, and the {\displaystyle \alpha _{i}}\alpha _{i} are expressed in standard form
  4. If {\displaystyle \alpha }\alpha  is an additively principal ordinal but not of the form {\displaystyle I(\beta ,\gamma )}{\displaystyle I(\beta ,\gamma )}, then {\displaystyle \alpha }\alpha  is expressible in the form {\displaystyle \psi _{\pi }(\delta )}{\displaystyle \psi _{\pi }(\delta )}. Then the standard form for {\displaystyle \alpha }\alpha  is {\displaystyle \alpha =\psi _{\pi }(\delta )}{\displaystyle \alpha =\psi _{\pi }(\delta )} where {\displaystyle \pi }\pi and {\displaystyle \delta }\delta  are expressed in standard form

Fundamental sequences

The fundamental sequence for an ordinal number {\displaystyle \alpha }\alpha  with cofinality {\displaystyle {\text{cof}}(\alpha )=\beta }{\displaystyle {\text{cof}}(\alpha )=\beta } is a strictly increasing sequence {\displaystyle (\alpha [\eta ])_{\eta <\beta }}{\displaystyle (\alpha [\eta ])_{\eta <\beta }} with length {\displaystyle \beta }\beta  and with limit {\displaystyle \alpha }\alpha , where {\displaystyle \alpha [\eta ]}{\displaystyle \alpha [\eta ]} is the {\displaystyle \eta }\eta -th element of this sequence.

Let {\displaystyle S=\{\alpha |{\text{cof}}(\alpha )=1\}}{\displaystyle S=\{\alpha |{\text{cof}}(\alpha )=1\}} and {\displaystyle L=\{\alpha |{\text{cof}}(\alpha )\geq \omega \}}{\displaystyle L=\{\alpha |{\text{cof}}(\alpha )\geq \omega \}} where {\displaystyle S}S denotes the set of successor ordinals and {\displaystyle L}L denotes the set of limit ordinals.

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

  1. If {\displaystyle \alpha =\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}}{\displaystyle \alpha =\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}} with {\displaystyle n\geq 2}n\geq 2 then {\displaystyle {\text{cof}}(\alpha )={\text{cof}}(\alpha _{n})}{\displaystyle {\text{cof}}(\alpha )={\text{cof}}(\alpha _{n})} and {\displaystyle \alpha [\eta ]=\alpha _{1}+\alpha _{2}+\cdots +(\alpha _{n}[\eta ])}{\displaystyle \alpha [\eta ]=\alpha _{1}+\alpha _{2}+\cdots +(\alpha _{n}[\eta ])}
  2. If {\displaystyle \alpha =\psi _{I(0,0)}(0)}{\displaystyle \alpha =\psi _{I(0,0)}(0)} then {\displaystyle \alpha ={\text{cof}}(\alpha )=1}{\displaystyle \alpha ={\text{cof}}(\alpha )=1} and {\displaystyle \alpha [0]=0}{\displaystyle \alpha [0]=0}
  3. If {\displaystyle \alpha =\psi _{I(0,\beta +1)}(0)}{\displaystyle \alpha =\psi _{I(0,\beta +1)}(0)} then {\displaystyle {\text{cof}}(\alpha )=\omega }{\displaystyle {\text{cof}}(\alpha )=\omega } and {\displaystyle \alpha [\eta ]=I(0,\beta )\cdot \eta }{\displaystyle \alpha [\eta ]=I(0,\beta )\cdot \eta }
  4. If {\displaystyle \alpha =\psi _{I(0,\beta )}(\gamma +1)}{\displaystyle \alpha =\psi _{I(0,\beta )}(\gamma +1)} and {\displaystyle \beta \in \{0\}\cup S}{\displaystyle \beta \in \{0\}\cup S} then {\displaystyle {\text{cof}}(\alpha )=\omega }{\displaystyle {\text{cof}}(\alpha )=\omega } and {\displaystyle \alpha [\eta ]=\psi _{I(0,\beta )}(\gamma )\cdot \eta }{\displaystyle \alpha [\eta ]=\psi _{I(0,\beta )}(\gamma )\cdot \eta }
  5. If {\displaystyle \alpha =\psi _{I(\beta +1,0)}(0)}{\displaystyle \alpha =\psi _{I(\beta +1,0)}(0)} then {\displaystyle {\text{cof}}(\alpha )=\omega }{\displaystyle {\text{cof}}(\alpha )=\omega } and {\displaystyle \alpha [0]=0}{\displaystyle \alpha [0]=0} and {\displaystyle \alpha [\eta +1]=I(\beta ,\alpha [\eta ])}{\displaystyle \alpha [\eta +1]=I(\beta ,\alpha [\eta ])}
  6. If {\displaystyle \alpha =\psi _{I(\beta +1,\gamma +1)}(0)}{\displaystyle \alpha =\psi _{I(\beta +1,\gamma +1)}(0)} then {\displaystyle {\text{cof}}(\alpha )=\omega }{\displaystyle {\text{cof}}(\alpha )=\omega } and {\displaystyle \alpha [0]=I(\beta +1,\gamma )+1}{\displaystyle \alpha [0]=I(\beta +1,\gamma )+1} and {\displaystyle \alpha [\eta +1]=I(\beta ,\alpha [\eta ])}{\displaystyle \alpha [\eta +1]=I(\beta ,\alpha [\eta ])}
  7. If {\displaystyle \alpha =\psi _{I(\beta +1,\gamma )}(\delta +1)}{\displaystyle \alpha =\psi _{I(\beta +1,\gamma )}(\delta +1)} and {\displaystyle \gamma \in \{0\}\cup S}{\displaystyle \gamma \in \{0\}\cup S} then {\displaystyle {\text{cof}}(\alpha )=\omega }{\displaystyle {\text{cof}}(\alpha )=\omega } and {\displaystyle \alpha [0]=\psi _{I(\beta +1,\gamma )}(\delta )+1}{\displaystyle \alpha [0]=\psi _{I(\beta +1,\gamma )}(\delta )+1} and {\displaystyle \alpha [\eta +1]=I(\beta ,\alpha [\eta ])}{\displaystyle \alpha [\eta +1]=I(\beta ,\alpha [\eta ])}
  8. If {\displaystyle \alpha =\psi _{I(\beta ,0)}(0)}{\displaystyle \alpha =\psi _{I(\beta ,0)}(0)} and {\displaystyle \beta \in L}{\displaystyle \beta \in L} then {\displaystyle {\text{cof}}(\alpha )={\text{cof}}(\beta )}{\displaystyle {\text{cof}}(\alpha )={\text{cof}}(\beta )} and {\displaystyle \alpha [\eta ]=I(\beta [\eta ],0)}{\displaystyle \alpha [\eta ]=I(\beta [\eta ],0)}
  9. If {\displaystyle \alpha =\psi _{I(\beta ,\gamma +1)}(0)}{\displaystyle \alpha =\psi _{I(\beta ,\gamma +1)}(0)} and {\displaystyle \beta \in L}{\displaystyle \beta \in L} then {\displaystyle {\text{cof}}(\alpha )={\text{cof}}(\beta )}{\displaystyle {\text{cof}}(\alpha )={\text{cof}}(\beta )} and {\displaystyle \alpha [\eta ]=I(\beta [\eta ],I(\beta ,\gamma )+1)}{\displaystyle \alpha [\eta ]=I(\beta [\eta ],I(\beta ,\gamma )+1)}
  10. If {\displaystyle \alpha =\psi _{I(\beta ,\gamma )}(\delta +1)}{\displaystyle \alpha =\psi _{I(\beta ,\gamma )}(\delta +1)} and {\displaystyle \beta \in L}{\displaystyle \beta \in L} and {\displaystyle \gamma \in \{0\}\cup S}{\displaystyle \gamma \in \{0\}\cup S} then {\displaystyle {\text{cof}}(\alpha )={\text{cof}}(\beta )}{\displaystyle {\text{cof}}(\alpha )={\text{cof}}(\beta )} and {\displaystyle \alpha [\eta ]=I(\beta [\eta ],\psi _{I(\beta ,\gamma )}(\delta )+1)}{\displaystyle \alpha [\eta ]=I(\beta [\eta ],\psi _{I(\beta ,\gamma )}(\delta )+1)}
  11. If {\displaystyle \alpha =\pi }{\displaystyle \alpha =\pi } then {\displaystyle {\text{cof}}(\alpha )=\pi }{\displaystyle {\text{cof}}(\alpha )=\pi } and {\displaystyle \alpha [\eta ]=\eta }{\displaystyle \alpha [\eta ]=\eta }
  12. If {\displaystyle \alpha =I(\beta ,\gamma )}{\displaystyle \alpha =I(\beta ,\gamma )} and {\displaystyle \gamma \in L}{\displaystyle \gamma \in L} then {\displaystyle {\text{cof}}(\alpha )={\text{cof}}(\gamma )}{\displaystyle {\text{cof}}(\alpha )={\text{cof}}(\gamma )} and {\displaystyle \alpha [\eta ]=I(\beta ,\gamma [\eta ])}{\displaystyle \alpha [\eta ]=I(\beta ,\gamma [\eta ])}
  13. If {\displaystyle \alpha =\psi _{\pi }(\beta )}{\displaystyle \alpha =\psi _{\pi }(\beta )} and {\displaystyle \omega \leq {\text{cof}}(\beta )<\pi }{\displaystyle \omega \leq {\text{cof}}(\beta )<\pi } then {\displaystyle {\text{cof}}(\alpha )={\text{cof}}(\beta )}{\displaystyle {\text{cof}}(\alpha )={\text{cof}}(\beta )} and {\displaystyle \alpha [\eta ]=\psi _{\pi }(\beta [\eta ])}{\displaystyle \alpha [\eta ]=\psi _{\pi }(\beta [\eta ])}
  14. If {\displaystyle \alpha =\psi _{\pi }(\beta )}{\displaystyle \alpha =\psi _{\pi }(\beta )} and {\displaystyle {\text{cof}}(\beta )=\rho \geq \pi }{\displaystyle {\text{cof}}(\beta )=\rho \geq \pi } then {\displaystyle {\text{cof}}(\alpha )=\omega }{\displaystyle {\text{cof}}(\alpha )=\omega } and {\displaystyle \alpha [\eta ]=\psi _{\pi }(\beta [\gamma [\eta ]])}{\displaystyle \alpha [\eta ]=\psi _{\pi }(\beta [\gamma [\eta ]])} with {\displaystyle \gamma [0]=1}{\displaystyle \gamma [0]=1} and {\displaystyle \gamma [\eta +1]=\psi _{\rho }(\beta [\gamma [\eta ]])}{\displaystyle \gamma [\eta +1]=\psi _{\rho }(\beta [\gamma [\eta ]])}

Limit of this notation {\displaystyle \psi _{I(1,0,0)}(0)}{\displaystyle \psi _{I(1,0,0)}(0)}. If {\displaystyle \alpha =\psi _{I(1,0,0)}(0)}{\displaystyle \alpha =\psi _{I(1,0,0)}(0)} then {\displaystyle {\text{cof}}(\alpha )=\omega }{\displaystyle {\text{cof}}(\alpha )=\omega } and {\displaystyle \alpha [0]=0}{\displaystyle \alpha [0]=0} and {\displaystyle \alpha [\eta +1]=I(\alpha [\eta ],0)}{\displaystyle \alpha [\eta +1]=I(\alpha [\eta ],0)}


Let X be equal to {\displaystyle f_{\psi _{I(0,0)}(\psi _{I(1,0,0)}(0))}(10)}{\displaystyle f_{\psi _{I(0,0)}(\psi _{I(1,0,0)}(0))}(10)} using the fast-growing hierarchy with fundamental sequences for the functions collapsing {\displaystyle \alpha }\alpha -weakly inaccessible cardinals. Let Rax be spherical region of the universe bounded by imaginary sphere with center at the Earth's center and with radius of X meters. Boundaries of this region are not  defined  by really existing  structures of the universe  or inhomogeneities in the distribution of matter because we nothing know about structures or inhomogeneities on so huge scales, but nevertheless this region can be defined at simple assumption that our universe is infinite.  The main motivation of this definition is wishing to apply somehow huge numbers, which can be obtained using the fast-growing hierarchy, to the real physical world.

Note 1: a sphere with center {\displaystyle c}c and with radius {\displaystyle r}r is the set of all points that are at distance {\displaystyle r}r from {\displaystyle c}c.