Snabbväxande hierarki

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Den snabbt växande hierarkin (även kallad den utökade Grzegorczyk-hierarkin ) är en familj av snabbväxande funktioner indexerade med ordningstal . Det mest kända specialfallet av en snabbt växande hierarki är Loeb -Weiner-hierarkin.

Definition

En snabbt växande hierarki definieras av följande regler

$f_{0}(n)=n+1$ (kan i allmänhet vara vilken växande funktion som helst), $f_{0}(n)$
$f_{\alpha +1}(n)=f_{\alpha }^{n}(n)=\underbrace {f_{\alpha }(f_{\alpha }(\cdots (f_{\alpha } (f_{\alpha }(} _{n{\mbox{ gånger))}n))\cdots ))$ ,
$f_{\alpha }(n)=f_{\alpha [n]}(n)$ om gränsen ordinal, $\alfa$
- var är det n :te elementet i grundsekvensen som fastställts för någon limitordinal . $\alpha[n]$ $\alfa$
- Det finns olika versioner av den snabbt växande hierarkin, men den mest kända är Loeb-Weiner-hierarkin, där de grundläggande sekvenserna för gränsordningstal skrivna i Cantors normala form definieras av följande regler:
$\omega [n]=n$ ,
$(\omega ^{\alpha _{1))+\omega ^{\alpha _{2))+\cdots +\omega ^{\alpha _{k-1))+\omega ^{\ alpha _{k)))[n]=\omega ^{\alpha _{1}}+\omega ^{\alpha _{2}}+\cdots +\omega ^{\alpha _{k-1} }+\omega ^{\alpha _{k}}[n]$
- för , ${\displaystyle \alpha _{1}\geq \alpha _{2}\geq \cdots \geq \alpha _{k))$
$\omega ^{\alpha +1}[n]=\omega ^{\alpha }n$ ,
$\omega ^{\alpha [n]=\omega ^{\alpha [n]}$ om gränsen ordinal, $\alfa$
$\varepsilon _{0}[0]=0$ och . $\varepsilon _{0}[n+1]=\omega ^{\varepsilon _{0}[n]}=\omega \uparrow \uparrow n$

Grundläggande sekvenser för gränsordningstal ovan ges i artiklarna om Veblen- funktioner och Buchholz-funktioner $\varepsilon_{0}$

Exempel

$f_{1}(n)=f_{0}^{n}(n)=\underbrace {f_{0}(f_{0}(\cdots (f_{0}(f_{0}(} _{n\quad f_{0}}n))\cdots ))=2\ gånger n$ ,

$f_{2}(n)=f_{1}^{n}(n)=\underbrace {f_{1}(f_{1}(\cdots (f_{1}(f_{1}(} _{n\quad f_{1}}n))\cdots ))=2^{n}\ gånger n$ .

För funktioner indexerade med ändliga ordningstal , $\alfa >1$

$f_{m}(n)>2\uparrow ^{m-1}n$ .

I synnerhet för n =10:

$f_{3}(10)=f_{2}^{10}(10)\approx \underbrace {10^{10^{10^{\cdots {^{10^{10^{3.489}} }}}}}} _{10\quad {\text{ten}}}\approx 10\uparrow \uparrow 11$ ,

$f_{4}(10)=f_{3}^{10}(10)\approx \left.{\begin{matrix}&&\underbrace {10^{10^{10^{\cdots {^ {10^{10^{3.489}}}}}}}} \\&&\underbrace {10^{10^{10^{\cdots {^{10^{10^{3.489}}}}}}} } \\&&\underbrace {\qquad \;\;\vdots \qquad \;\;} \\&&\underbrace {10^{10^{10^{\cdots {^{10^{10^{3.489} )))))))} \\&&10\quad {\text{ten}}\end{matris}}\right\}{\text{10 nedre hängslen}}\ca 10\uparrow \uparrow \uparrow 11$ ,

$f_{m}(10)\approx 10\uparrow ^{m-1}11$ .

Redan den första transfinita ordinalen motsvarar alltså gränsen för Knuths pilnotation . $\omega$

Det berömda Graham-talet är mindre än . $f_{\omega +1}(64)$

På grund av definitionens enkelhet och tydlighet används den snabbt växande hierarkin för att analysera olika notationer för att skriva stora siffror .

	Knuth notation	Conway notation	Bowers notation
notationsgräns	$\omega$	$\omega ^{2}$	$\omega^\omega$
exempel	$f_{\omega }(10)\approx 10\uparrow ^{9}11=10\uparrow \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow \uparrow 11$	$f_{\omega ^{2}}(n)\approx \underbrace {n\rightarrow n\rightarrow n\cdots \rightarrow n} _{n+1\quad \rightarrow }$	$f_{\omega ^{\omega }}(n)\approx \underbrace {\{n,n,n,\cdots ,n,n\}} _{n+2\quad n's}$
exempel	$f_{\omega +1}(10)\approx \left.{\begin{matrix}&&\underbrace {10\uparrow \uparrow \cdots \uparrow \uparrow 11} \\&&\underbrace {10\uparrow \uparrow \cdots \uparrow \uparrow 11} \\&&\underbrace {\quad \quad \;\;\vdots \quad \quad \;\;} \\&&\underbrace {10\uparrow \cdots \uparrow \uparrow 11} \\&&{\text{9 pilar}}\end{matris}}\right\}{\text{10}}$	$f_{\omega ^{2}+1}(n)\approx \left.{\begin{matrix}&&\underbrace {n\högerpil n\högerpil \cdots n\högerpil n} \\&&\underbrace {n\högerpil n\högerpil \cdots n\högerpil n} \\&&\underbrace {\quad \quad \quad \;\;\vdots \quad \quad \quad \;\;} \\&&\underbrace {n \rightarrow n\rightarrow \cdots n\rightarrow n} \\&&{\text{n+1 pilar}}\end{matris}}\right\}{\text{n}}$	$f_{\omega ^{\omega }+1}(n)\approx \left.{\begin{matrix}&&\underbrace {\{n,n,n,\cdots ,n,n\)) \\&&\underbrace {\{n,n,n,\cdots ,n,n\)) \\&&\underbrace {\quad \quad \quad \;\;\vdots \quad \quad \quad \;\ ;} \\&&\underbrace {\{n,n,n,\cdots ,n,n\)) \\&&{\text{n+2 n's))\end{matris))\right\}{\ text{n}}$

Ovanstående definition definierar en snabbt växande hierarki upp till . För ytterligare tillväxt kan du använda Veblen-funktionen och andra ännu mer kraftfulla notationer för ordningstal [1] . $\varepsilon _{0}=\omega \uparrow \uparrow n$

Exempel

$f_{0}(n)=n+1$
$f_{1}(n)=2n$
$f_{2}(n)=2^{n}n$
$f_{3}(n)>2\uparrow \uparrow n$ (se Knuth pilnotation )
$f_{4}(n)>2\uparrow \uparrow \uparrow n$
$f_{m}(n)>2\uparrow ^{m-1}n$
${\displaystyle f_{\omega }(n)>2\uparrow ^{n-1}n=\{n,n,n-1\))$ (se Bowers massiv notation )
$f_{\omega +1}(n)=\{n,n,1,2\}\approx G(n)$ (se Graham nummer )
${\displaystyle f_{\omega +2}(n)=\{n,n,2,2\))$
${\displaystyle f_{\omega +m}(n)=\{n,n,m,2\))$
${\displaystyle f_{\omega 2}(n)=\{n,n,n,2\))$
${\displaystyle f_{\omega 2+1}(n)=\{n,n,1,3\))$
${\displaystyle f_{\omega 3}(n)=\{n,n,n,3\))$
$f_{\omega {m}}(n)=\{n,n,n,m\}$
$f_{\omega {m+k}}(n)=\{n,n,k,m+1\}$
$f_{\omega ^{2}}(n)=\{n,n,n,n\}$
$f_{\omega ^{3}}(n)=\{n,n,n,n,n\}$
$f_{\omega ^{m}}(n)=\{n,n,n,...,n\}$ (m gånger)
$f_{\omega ^{\omega }}(n)=\{n,n,n,...,n\}$ (n gånger) $=\{n,n(1)2\}$
$f_{\omega ^{\omega +1}}(n)=\{n,n(1)1,2\}$
$f_{\omega ^{\omega +2}}(n)=\{n,n(1)1,1,2\}$
$f_{\omega ^{\omega 2}}(n)=\{n,n(1)1(1)2\}$
$f_{\omega ^{\omega 2+1}}(n)=\{n,n(1)1(1)1,2\}$
$f_{\omega ^{\omega 3}}(n)=\{n,n(1)1(1)1(1)2\}$
$f_{\omega ^{\omega ^{2}}}(n)=\{n,n(2)2\}$
$f_{\omega ^{\omega ^{3}}}(n)=\{n,n(3)2\}$
${\displaystyle f_{\omega ^{\omega ^{m))}(n)=\{n,n(m)2\))$
${\displaystyle f_{\omega ^{\omega ^{\omega ))}(n)=\{n,n[1,2]2\))$ (se Bird's Array Notation )
${\displaystyle f_{\omega ^{\omega ^{\omega ^{2))))(n)=\{n,n[1,1,2]2\))$
${\displaystyle f_{\omega ^{\omega ^{\omega ^{\omega ))))(n)=\{n,n[1[2]2]2\))$
$f_{\epsilon _{0}}(n)=\{n,n[1{\backslash }2]2\}$
$f_{\omega ^{\omega ^{\epsilon _{0}+1))}(n)=\{n,n[1[1{\backslash }2]2{\backslash }2] 2\}$
${\displaystyle f_{\omega ^{\omega ^{\omega ^{\epsilon _{0}+1))))(n)=\{n,n[1[1[1{\backslash }2] 2{\backslash }2]2{\backslash }2]2\))$
$f_{\epsilon _{1}}(n)=\{n,n[1{\backslash }3]2\}$
$f_{\epsilon _{2}}(n)=\{n,n[1{\backslash }4]2\}$
$f_{\epsilon _{\omega }}(n)=\{n,n[1{\backslash }1,2]2\}$
$f_{\epsilon _{\omega ^{\omega }}}(n)=\{n,n[1{\backslash }1[2]2]2\}$
${\displaystyle f_{\epsilon _{\epsilon _{0))}(n)=\{n,n[1{\backslash }1[1{\backslash }2]2]2\))$
$f_{\epsilon _{\epsilon _{\epsilon _{0))))(n)=\{n,n[1{\backslash }1[1{\backslash }1[1{\backslash }2]2]2]2\}$
$f_{\zeta _{0}}(n)=\{n,n[1{\backslash }1{\backslash }2]2\}$
$f_{\epsilon _{\zeta _{0}+1}}(n)=\{n,n[1{\backslash }2{\backslash }2]2\}$

$f_{\epsilon _{\zeta _{0}+\omega }}(n)=\{n,n[1{\backslash }1,2{\backslash }2]2\}$

$f_{\epsilon _{\zeta _{0}+\epsilon _{0}}}(n)=\{n,n[1{\backslash }1{[1{\backslash }2]} 2{\backslash }2]2\}$
${\displaystyle f_{\epsilon _{\zeta _{0}2}}(n)=\{n,n[1{\backslash }1{[1{\backslash }1{\backslash }2]}2 {\backslash }2]2\))$
${\displaystyle f_{\epsilon _{\epsilon _{\zeta _{0}+1}}}(n)=\{n,n[1{\backslash }1{[1{\backslash }2{\ backslash }2]}2{\backslash }2]2\))$
$f_{\zeta _{1}}(n)=\{n,n[1{\backslash }1{\backslash }3]2\}$
$f_{\zeta _{\omega }}(n)=\{n,n[1{\backslash }1{\backslash }1,2]2\}$
$f_{\zeta _{\epsilon _{0}}}(n)=\{n,n[1{\backslash }1{\backslash }1[1{\backslash }2]2]2\ }$
$f_{\zeta _{\zeta _{0}}}(n)=\{n,n[1{\backslash }1{\backslash }1[1{\backslash}1{\backslash }2 ]2]2\}$
$f_{\eta _{0}}(n)=\{n,n[1{\backslash }1{\backslash }1{\backslash }2]2\}$
${\displaystyle f_{\varphi (4,0)}(n)=\{n,n[1{\backslash }1{\backslash }1{\backslash }1{\backslash }2]2\))$
${\displaystyle f_{\varphi (m,0)}(n)=\{n,n[1{\backslash }1{\backslash }1{\cdots }1{\backslash }2]2\))$ (m bakåtstreck)
${\displaystyle f_{\varphi (\omega ,0)}(n)=\{n,n[1[2{\neg }2]2]2\))$
$f_{\epsilon _{\varphi (\omega ,0)+1))(n)=\{n,n[1{\backslash }2[2{\neg }2]2]2\}$
$f_{\zeta _{\varphi (\omega ,0)+1))(n)=\{n,n[1{\backslash }1{\backslash }2[2{\neg }2] 2]2\}$
$f_{\eta _{\varphi (\omega ,0)+1))(n)=\{n,n[1{\backslash }1{\backslash }1{\backslash }2[2{ \neg }2]2]2\}$
${\displaystyle f_{\varphi (\omega ,1)}(n)=\{n,n[1[2{\neg }2]3]2\))$
${\displaystyle f_{\varphi (\omega ,2)}(n)=\{n,n[1[2{\neg }2]4]2\))$
${\displaystyle f_{\varphi (\omega ,\omega )}(n)=\{n,n[1[2{\neg }2]1,2]2\))$
$f_{\varphi (\omega ,\epsilon _{0})}(n)=\{n,n[1[2{\neg }2]1[1{\backslash }2]2]2 \}$
$f_{\varphi (\omega ,\zeta _{0})}(n)=\{n,n[1[2{\neg }2]1[1{\backslash }1{\backslash } 2]2]2\}$
$f_{\varphi (\omega ,\varphi (\omega ,0))}(n)=\{n,n[1[2{\neg }2]1[1[2{\neg }2 ]2]2]2\}$
${\displaystyle f_{\varphi (\omega ,\varphi (\omega ,\varphi (\omega ,0)))}(n)=\{n,n[1[2{\neg }2]1[1 [2{\neg }2]1[1[2{\neg }2]2]2]2]2\))$
${\displaystyle f_{\varphi (\omega +1,0)}(n)=\{n,n[1[2{\neg }2]1{\backslash }2]2\))$
$f_{\varphi (\omega +2,0)}(n)=\{n,n[1[2{\neg }2]1{\backslash }1{\backslash }2]2\}$
${\displaystyle f_{\varphi (\omega 2,0)}(n)=\{n,n[1[2{\neg }2]1[2{\neg }2]2]2\))$
${\displaystyle f_{\varphi (\omega ^{2},0)}(n)=\{n,n[1[3{\neg }2]2]2\))$
${\displaystyle f_{\varphi (\omega ^{3},0)}(n)=\{n,n[1[4{\neg }2]2]2\))$
${\displaystyle f_{\varphi (\omega ^{\omega },0)}(n)=\{n,n[1[1,2{\neg }2]2]2\))$
$f_{\varphi (\omega ^{\omega ^{\omega )),0)}(n)=\{n,n[1[1[2]2{\neg }2]2]2 \}$
$f_{\varphi ({\epsilon _{0)),0)}(n)=\{n,n[1[1[1{\backslash }2]2{\neg }2]2] 2\}$
$f_{\Gamma _{0}}(n)=\{n,n[1[1{\backslash }2{\neg }2]2]2\}$
$f_{\Gamma _{1}}(n)=\{n,n[1[1{\backslash }2{\neg }2]3]2\}$
${\displaystyle f_{\Gamma _{\Gamma _{0))}(n)=\{n,n[1[1{\backslash }2{\neg }2]1[1[1{\backslash } 2{\neg }2]2]2]2\))$
${\displaystyle f_{\varphi (1,1,0)}(n)=\{n,n[1[1{\backslash }2{\neg }2]1{\backslash }2]2\))$
$f_{\varphi (1,2,0)}(n)=\{n,n[1[1{\backslash }2{\neg }2]1{\backslash }1{\backslash }2 ]2\}$
$f_{\varphi (2,0,0)}(n)=\{n,n[1[1{\backslash }2{\neg }2]1[1{\backslash }2{\neg }2]2]2\}$
${\displaystyle f_{\varphi (\omega ,0,0)}(n)=\{n,n[1[2{\backslash }2{\neg }2]2]2\))$
${\displaystyle f_{\varphi (\Gamma _{0},0,0)}(n)=\{n,n[1[1[1[1{\omvänt snedstreck }2{\neg }2]2] 2{\omvänt snedstreck }2{\neg }2]2]2\))$
${\displaystyle f_{\varphi (1,0,0,0)}(n)=\{n,n[1[1{\backslash }3{\neg }2]2]2\))$
${\displaystyle f_{\varphi (1,0,0,0,0)}(n)=\{n,n[1[1{\backslash }4{\neg }2]2]2\))$
$f_{\psi (\Omega ^{\Omega ^{\omega )))}(n)=\{n,n[1[1{\backslash }1,2{\neg }2]2] 2\}\approx TREE(n)$ (se TRÄD(3) )
${\displaystyle f_{\psi (\Omega ^{\Omega ^{\psi (\Omega ^{\Omega )))))}(n)=\{n,n[1[1{\backslash }1[ 1[1{\omvänt snedstreck }2{\neg }2]2]2{\neg }2]2]2\))$
$f_{\psi (\Omega ^{\Omega ^{\Omega )))}(n)=\{n,n[1[1{\backslash }1{\backslash }2{\neg }2 ]2]2\}$
$f_{\psi (\Omega ^{\Omega ^{\Omega ^{2))})}(n)=\{n,n[1[1{\backslash }1{\backslash }1{ \backslash }2{\neg }2]2]2\}$
$f_{\psi (\Omega ^{\Omega ^{\Omega ^{\omega ))})}(n)=\{n,n[1[1[2{\neg }2]2{ \neg }2]2]2\}$
${\displaystyle f_{\psi (\Omega ^{\Omega ^{\Omega ^{\psi (\Omega ^{\Omega ^{\Omega )))))})(n)=\{n,n [ 1[1[1[1[1{\backslash }1{\neg }2{\neg }2]2]2{\neg }2]2{\neg }2]2]2\))$
$f_{\psi (\Omega ^{\Omega ^{\Omega ^{\Omega ))})}(n)=\{n,n[1[1[1{\omvänt snedstreck }2{\neg }2]2{\neg }2]2]2\}$
${\displaystyle f_{\psi (\Omega _{2})}(n)=\{n,n[1[1{\neg }3]2]2\))$
${\displaystyle f_{\psi (\Omega _{2}+1)}(n)=\{n,n[1{\backslash }2[1{\neg }3]2]2\))$
$f_{\psi (\Omega _{2}+\omega )}(n)=\{n,n[1{\backslash }1,2[1{\neg }3]2]2\}$
${\displaystyle f_{\psi (\Omega _{2}+\psi (\Omega ^{\Omega }))}(n)=\{n,n[1{\backslash }1[1[1{\ omvänt snedstreck }2{\neg }2]2]2[1{\neg }3]2]2\))$
${\displaystyle f_{\psi (\Omega _{2}+\psi (\Omega ^{\Omega ^{\omega ))))}(n)=\{n,n[1{\backslash }1[ 1[1{\omvänt snedstreck }1,2{\neg }2]2]2[1{\neg }3]2]2\))$
${\displaystyle f_{\psi (\Omega _{2}+\psi (\Omega ^{\Omega ^{\Omega ))))}(n)=\{n,n[1{\backslash }1[ 1[1{\backslash }1{\backslash }2{\neg }2]2]2[1{\neg }3]2]2\))$
${\displaystyle f_{\psi (\Omega _{2}+\psi (\Omega ^{\Omega ^{\Omega ^{\Omega ))}))}(n)=\{n,n[1{ \backslash }1[1[1[1{\neg }2{\neg }2]2{\neg }2]2]2[1{\neg }3]2]2\))$
${\displaystyle f_{\psi (\Omega _{2}+\psi (\Omega _{2}))}(n)=\{n,n[1{\backslash }1[1[1{\neg }3]2]2[1{\neg }3]2]2\))$
$f_{\psi (\Omega _{2}+\Omega )}(n)=\{n,n[1{\backslash }1{\backslash }2[1{\neg }3]2] 2\}$
$f_{\psi (\Omega _{2}+\Omega ^{2})}(n)=\{n,n[1{\backslash }1{\backslash }1{\backslash }2[ 1{\neg }3]2]2\}$
$f_{\psi (\Omega _{2}+\Omega ^{\omega })}(n)=\{n,n[1[2{\neg }2]2[1{\neg } 3]2]2\}$
$f_{\psi (\Omega _{2}+\Omega ^{\Omega })}(n)=\{n,n[1[1{\backslash }2{\neg }2]2[ 1{\neg }3]2]2\}$
$f_{\psi (\Omega _{2}+\psi _{1}(\Omega _{2}))}(n)=\{n,n[1[1{\neg }3] 3]2\}$
${\displaystyle f_{\psi (\Omega _{2}+\psi _{1}(\Omega _{2}+\psi _{1}(\Omega _{2})))}(n)= \{n,n[1[1{\neg }3]1[1{\neg }3]2]2\))$
${\displaystyle f_{\psi (\Omega _{2}+\psi _{1}(\Omega _{2}+\psi _{1}(\Omega _{2}+1)))}(n )=\{n,n[1[2{\neg }3]2]2\))$
${\displaystyle f_{\psi (\Omega _{2}+\psi _{1}(\Omega _{2}+\psi _{1}(\Omega _{2}+\psi _{1}( \Omega _{2}))))}(n)=\{n,n[1[1[1{\neg }3]2{\neg }3]2]2\))$
${\displaystyle f_{\psi (\Omega _{2}2)}(n)=\{n,n[1[1{\neg }4]2]2\))$
${\displaystyle f_{\psi (\Omega _{2}3)}(n)=\{n,n[1[1{\neg }5]2]2\))$
${\displaystyle f_{\psi (\Omega _{2}\omega )}(n)=\{n,n[1[1{\neg }1,2]2]2\))$
$f_{\psi (\Omega _{2}\psi (0))}(n)=\{n,n[1[1{\neg }1[1{\omvänt snedstreck }2]2]2 ]2\}$
${\displaystyle f_{\psi (\Omega _{2}\psi (\Omega ^{\Omega }))}(n)=\{n,n[1[1{\neg }1[1[1{ \backslash }2{\neg }2]2]2]2]2\))$
$f_{\psi (\Omega _{2}\psi (\Omega _{2}))}(n)=\{n,n[1[1{\neg }1[1[1{\ neg }3]2]2]2]2\}$
${\displaystyle f_{\psi (\Omega _{2}\Omega )}(n)=\{n,n[1[1{\neg }1{\omvänt snedstreck }2]2]2\))$
$f_{\psi (\Omega _{2}\Omega ^{\Omega })}(n)=\{n,n[1[1{\neg }1[1{\omvänt snedstreck }2{\ neg }2]2]2]2\}$
${\displaystyle f_{\psi (\Omega _{2}\psi _{1}(\Omega _{2}))}(n)=\{n,n[1[1{\neg }1[1 {\neg }3]2]2]2\))$
${\displaystyle f_{\psi (\Omega _{2}^{2})}(n)=\{n,n[1[1{\neg }1{\neg }2]2]2\))$
$f_{\psi (\Omega _{2}^{\omega })}(n)=\{n,n[1[1[2{\backslash }_{3}2]2]2] 2\}$
$f_{\psi (\Omega _{2}^{\Omega _{2))}(n)=\{n,n[1[1[1{\omvänt snedstreck }_{2}2{ \ omvänt snedstreck }_{3}2]2]2]2\}$
${\displaystyle f_{\psi (\Omega _{3})}(n)=\{n,n[1[1[1{\backslash }_{3}3]2]2]2\))$
${\displaystyle f_{\psi (\Omega _{3}^{\Omega _{3)))}(n)=\{n,n[1[1[1[1{\omvänt snedstreck }_{3} 2{\omvänt snedstreck }_{4}2]2]2]2]2\))$
$f_{\psi (\Omega _{4})}(n)=\{n,n[1[1[1[1{\backslash }_{4}3]2]2]2]2 \}$
${\displaystyle f_{\psi (\Omega _{\omega })}(n)=\{n,n[1[2{\backslash }_{1,2}2]2]2\))$
$f_{\psi (\Omega _{\psi (\Omega )})}(n)=\{n,n[1[2{\backslash }_{1[1{\backslash }2]2 }2]2]2\}$
$f_{\psi (\Omega _{\psi (\Omega _{\psi (\Omega )})})}(n)=\{n,n[1[2{\backslash }_{1 [2{\backslash _{1[1{\backslash }2]2}}2]2}2]2]2\}$
$f_{\psi (\Omega _{\Omega })}(n)=\{n,n[1[2{\backslash }_{1{\backslash }2}2]2]2\} =(0,0,0)(1,1,1)(2,1,1)(3,1,0)[n]$ (se Bashicu Matrix System )
$f_{\psi ({\Omega _{\Omega _{2))})}(n)=(0,0,0)(1,1,1)(2,1,1)(3 ,1,0)(1,1,0)(2,2,1)(3,2,1)(4,2,0)[n]$
$f_{\psi ({\Omega _{\Omega _{\omega ))})}(n)=(0,0,0)(1,1,1)(2,1,1)( 3,1,0)(1,1,1)[n]$
$f_{\psi ({\Omega _{\Omega _{\Omega ))})}(n)=(0,0,0)(1,1,1)(2,1,1)( 3,1,0)(1,1,1)(2,1,1)(3,1,0)[n]$
$f_{\psi (\psi _{I}(0))}(n)=(0,0,0)(1,1,1)(2,1,1)(3,1,0 )(2,0,0)[n]$

Se även

Anteckningar

↑ Kerr, Josh Mind blown: den snabbt växande hierarkin för lekmän - aka enorma antal . Hämtad 7 oktober 2016. Arkiverad från originalet 13 juli 2019. (obestämd)

Länkar

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Cichon, EA & Wainer, SS (1983), The slow-growing and the Grzegorczyk hierarchies , The Journal of Symbolic Logic vol. 48 (2): 399–408, ISSN 0022-4812 , DOI 10.2307/2273557
Gallier, Jean H. (1991), Vad är så speciellt med Kruskals sats och ordningen Γ 0 ? En översikt över några resultat i bevisteori , Ann. Rent äpple. Logic T. 53(3): 199–260, doi : 10.1016/0168-0072 ( >http://stinet.dtic.mil/oai/oai?verb=getRecord&metadataPrefix=html&identifier=ADA2900387, <91)90022-E PDF:er: del 1 2 3 . (Särskilt del 3, avsnitt 12, s. 59–64, "A Glimt at Hierarchies of Fast and Slow Growing Functions".)
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Wainer, S.S. (1989), " Långsamt växande kontra snabbväxande ". Journal of Symbolic Logic 54 (2): 608-614.

Stora siffror
Tal	googol Shannon nummer Googolplex Skeves nummer Moser nummer Graham nummer TRÄD(3) Rayo nummer
Funktioner	Ackermann funktion Veblen funktion Psi-funktioner för Buchholz tetration Pentation Hyperoperator Snabbväxande hierarki Långsamt växande hierarki Härlig hierarki
Noteringar	Steinhaus-Moser notation Knuth pilnotation Conway pil notation Massiv Bowers Notation