FANDOM


Hyperoperations can be extended to transfinite ordinals

Let's define

1) H(n,1,b) = n+b,

2) H(n,\alpha+1,b) = \underbrace{H(n,\alpha,H(n,\alpha,H(\cdots H(n,\alpha,n}_{b \quad n's})\cdots))),

3) H(n,\alpha,b) = H(n,\alpha [b],n) iff \alpha is a limit ordinal.

Or in such form

1) n \uparrow^{-1} b = n+b,

2) n \uparrow^{\alpha+1} b = \underbrace{n \uparrow^{\alpha} (n \uparrow^{\alpha} (\cdots (n \uparrow^{\alpha} n}_{b \quad n's})\cdots)),

3) n\uparrow^{\alpha} b = n\uparrow^{\alpha [b]} n iff \alpha is a limit ordinal,

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

  • Third rule was inspired by Aeton's work.

Examples

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


10\uparrow^{\omega+1} 3 = 10\uparrow^{\omega}(10\uparrow^{\omega}10)=\left. 
 \begin{matrix}
 &&10 \underbrace{\uparrow\uparrow\uparrow\uparrow\cdots\uparrow\uparrow\uparrow\uparrow}10\\
 & &10 \underbrace{\uparrow\uparrow\cdots\uparrow\uparrow}10 \\ 
    & &10  
\end{matrix} 
\right \} \text {3 layers}=

= \lbrace 10, 10, \lbrace 10,10 ,10\rbrace \rbrace = \lbrace 10, 3,1,2\rbrace

and for arbitrary power of omega

n \uparrow^{\omega^{k}.z+\cdots+\omega^2.d+\omega.c+a} b =\lbrace n,b,a,c+1,d+1,\cdots,z+1\rbrace,

a \uparrow^{\omega^k} b = \lbrace \underbrace {a,a,a,\cdots, a}_{k+1 \quad a's},b\rbrace .

Nomenclature

Abbreviations adopted for the building of names:

add=addition, ult=multiplication, ex=exponentiation, etr=tetration, om=omega, phi - binary Veblen function

un,b,tr,quadr,quint,sext,sept,oct,non,dek=1,2,3,4,5,6,7,8,9,10

Although names of those hyper operators are very similar to names of my numbers, do not confuse them - just for the creation of names of hyperoperations almost same abbreviations were used as in my nomenclature of numbers.

a\uparrow^{\omega+1} b unaddomation

a unaddomated to b

a\uparrow^{\omega+2} b baddomation

(a baddomated to b)

a\uparrow^{\omega.2} b bultomation

(a bultomated to b)

a\uparrow^{\omega^2} b bexomation

(a bexomated to b)

a\uparrow^{^2\omega} b betromation

(a betromated to b)

a\uparrow^{\omega+3} b traddomation

(a traddomated to b)

a\uparrow^{\omega.3} b trultomation

(a trultomated to b)

a\uparrow^{\omega^3} b trexomation

(a trexomated to b)

a\uparrow^{^3\omega} b tretromation

(a tretromated to b)

a\uparrow^{\omega+4} b quadraddomation

(a quadraddomated to b)

a\uparrow^{\omega.4} b quadrultomation

(a quadrultomated to b)

a\uparrow^{\omega^4} b quadrexomation

(a quadrexomated to b)

a\uparrow^{^4\omega} b quadretromation

(a quadretromated to b)

a\uparrow^{\omega+5} b quintaddomation

(a quintaddomated to b)

a\uparrow^{\omega.5} b quintultomation

(a quintultomated to b)

a\uparrow^{\omega^5} b quintexomation

(a quintexomated to b)

a\uparrow^{^5\omega} b quintetromation

(a quintetromated to b)

a\uparrow^{\omega+6} b sextaddomation

(a sextaddomated to b)

a\uparrow^{\omega.6} b sextultomation

(a sextultomated to b)

a\uparrow^{\omega^6} b sextexomation

(a sextexomated to b)

a\uparrow^{^6\omega} b sextetromation

(a sextetromated to b)

a\uparrow^{\omega+7} b septaddomation

(a septaddomated to b)

a\uparrow^{\omega.7} b septultomation

(a septultomated to b)

a\uparrow^{\omega^7} b septexomation

(a septexomated to b)

a\uparrow^{^7\omega} b septetromation

(a septetromated to b)

a\uparrow^{\omega+8} b octaddomation

(a octaddomated to b)

a\uparrow^{\omega.8} b octultomation

(a octultomated to b)

a\uparrow^{\omega^8} b octexomation

(a octexomated to b)

a\uparrow^{^8\omega} b octetromation

(a octetromated to b)

a\uparrow^{\omega+9} b nonaddomation

(a nonaddomated to b)

a\uparrow^{\omega.9} b nonultomation

(a nonultomated to b)

a\uparrow^{\omega^9} b nonexomation

(a nonexomated to b)

a\uparrow^{^9\omega} b nonetromation

(a nonetromated to b)

a\uparrow^{\omega+10} b dekaddomation

(a dekaddomated to b)

a\uparrow^{\omega.10} b dekultomation

(a dekultomated to b)

a\uparrow^{\omega^{10}} b dekexomation

(a dekexomated to b)

a\uparrow^{^{10}\omega} b deketromation

(a deketromated to b)


a\uparrow^{\varphi(1,0)} b uniphiation a\uparrow^{\theta(\Omega_1)} b unomtation
a\uparrow^{\varphi(2,0)} b biphiation a\uparrow^{\theta(\Omega_2)} b bomtation
a\uparrow^{\varphi(3,0)} b triphiation a\uparrow^{\theta(\Omega_3)} b tromtation
a\uparrow^{\varphi(4,0)} b quadriphiation a\uparrow^{\theta(\Omega_4)} b quadromtation
a\uparrow^{\varphi(5,0)} b quintiphiation a\uparrow^{\theta(\Omega_5)} b quintomtation
a\uparrow^{\varphi(6,0)} b sextiphiation a\uparrow^{\theta(\Omega_6)} b sextomtation
a\uparrow^{\varphi(7,0)} b septiphiation a\uparrow^{\theta(\Omega_7)} b septomtation
a\uparrow^{\varphi(8,0)} b octiphiation a\uparrow^{\theta(\Omega_8)} b octomtation
a\uparrow^{\varphi(9,0)} b noniphiation a\uparrow^{\theta(\Omega_9)} b nonomtation
a\uparrow^{\varphi(10,0)} b dekophiation a\uparrow^{\theta(\Omega_{10})} b dekomtation

Note: \varphi(\alpha, \beta) is the Veblen function and \theta(\alpha) is abbreviation for Feferman theta function \theta(\alpha,0).

Example for using inaccessible cardinal: let's calculate n \uparrow^{\theta(\psi(\Iota))}n for n=1,2 (probably this operation should be named ipsitation). Remarkable that for this values of n we will obtain modest output.

Here \theta(\psi(\Iota)) is abriviation for \theta(\psi_\Iota(0)) and

\theta_1(\alpha)=\theta_1(\alpha,0), where\theta_1 is function such that \theta_1(0,\alpha)=\Omega^\alpha and \theta(\theta_1(\alpha,\beta))=\theta(\alpha,\beta) for \alpha\geq \Omega_2,

also note that \psi_\Iota(0)[0]=\theta(\Omega)[0]=\theta_1(\Omega_2)[0]=\varepsilon_0[0]=\varepsilon_{\Omega+1}[0]=0,

n=1 n=2
1 \uparrow^{\theta(\psi(\Iota))}1=

1 \uparrow^{\theta(\psi_\Iota(0))}1=

1\uparrow^{\theta(\psi_\Iota(0))[1]}1=

1\uparrow^{\theta(\Omega_0)[1]}1=

1\uparrow^{\theta(\omega)[1]}1=

1\uparrow^{\theta(\omega[1])}1=

1\uparrow^{\theta(1)[1]}1=

1\uparrow^{\omega[1]}1=

1\uparrow^{1}1=

1^1=1

2 \uparrow^{\theta(\psi_\Iota(0))}2=

2\uparrow^{\theta(\psi_\Iota(0))[2]}2=

2\uparrow^{\theta(\Omega_{\Omega_0})[2]}2=

2\uparrow^{\theta(\Omega_{\omega})[2]}2=

2\uparrow^{\theta(\Omega_{2})[2]}2=

2\uparrow^{\theta(\theta_1(\Omega_{2})[2]}2=

2\uparrow^{\theta(\theta_1(\Omega_{2})[2]}2=

2\uparrow^{\theta(\theta_1(\theta_1(0)))[2]}2=

2\uparrow^{\theta(\theta_1(\Omega^0))[2]}2=

2\uparrow^{\theta(\theta_1(1))[2]}2=

2\uparrow^{\theta(\varepsilon_{\Omega+1})[2]}2=

2\uparrow^{\theta(\Omega^{\Omega^0})[2]}2=

2\uparrow^{\theta(\Omega)[2]}2=

2\uparrow^{\theta(\theta(0))[2]}2=

2\uparrow^{\theta(1)[2]}2=

2\uparrow^{\varepsilon_0[2]}2=

2\uparrow^{\omega^{\omega^0}[2]}2=

2\uparrow^{\omega[2]}2=

2\uparrow^{2}2=

2^2=4


Post Scriptum: Also let's introduce function f(\alpha)=10 \uparrow^{\alpha} 10.

original on my site

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