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\(\omega^{CK}_1\) is the ordinal strength of Turing Machines, and the smallest nonrecursive ordinal. What about \(\omega^{CK}_2\) or \(\omega^{CK}_n\)? If \(\omega^{CK}_1\) is the limit of A(0), B(0), C(0), D(0), and so on, where A, B, C, D... are normal functions, is the limit of \(A(\omega^{CK}_1+1)\), \(B(\omega^{CK}_1+1)\), \(C(\omega^{CK}_1+1)\), \(D(\omega^{CK}_1+1)\)... \(\omega^{CK}_2\) or it is smaller. Also is the limit of oracle machines with access to the halting problem \(\omega^{CK}_2\)?
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When I know an article has a lot of sections, I get excited for the content it has, only to click on the section, and to find nothing there. I then get disappointed at the fact that there is not content in the section, and want to add more to it. I feel like I am getting ripped off by the person who made the empty sections, and get impatient for them to add the content in.
Also, I don't like "Coming soon" tags placed in articles, I don't know when the content is coming out. It come out tomorrow, in a few years, or never, and most of the time it is the latter. By making a coming soon tag or an empty section, you are committing yourself to adding the content in the near future.
Googolists often plan some content, and never create it, because tâ€¦
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Did you know you can make ~ behave like ,, in SAN if you change a few rules.
Everything up to [1[1~3]2] remains unchanged.
Comparisons between mine and Bird's expression.
[1[1[1~3]2~2]2[1~3]2] in my system is equal to [1[1~3]3] in Bird's system.
[1[1[1~3]2~2]3[1~3]2] in my system is equal to [1[1~3]4] in Bird's system.
[1[1[1~3]2~2]1[1[1~3]2~2]2[1~3]2] in my system is equal to [1[1~3]1[1~3]2] in Bird's system.
[1[2[2~3]2~2]2[1~3]2] in my system is equal to [1[2~3]2] in Bird's system.
[1[1\2[1~3]2~2]2[1~3]2] in my system is equal to [1[1/2~3]2] in Bird's system.
[1[1[1[1~3]2~2]2[1~3]2~2]2[1~3]2] in my system is equal to [1[1[1~3]2~3]2] in Bird's system.
[1[1[1[1[1~3]2~2]2[1~3]2~2]2[1~3]2~2]2[1~3]2] in my system is equal to [1[1[1[1~3]2~3]2~3]2] in Bâ€¦
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(0,0,0)(1,1,1) has level \(\psi(\Omega_\omega)\)
(0,0,0)(1,1,1)(1,1,0) has level \(\psi(\Omega_\omega+1)\)
(0,0,0)(1,1,1)(1,1,0)(2,1,0) has level \(\psi(\Omega_\omega+\Omega)\)
(0,0,0)(1,1,1)(1,1,0)(2,2,0) has level \(\psi(\Omega_\omega+\psi_1(0))\)
(0,0,0)(1,1,1)(1,1,0)(2,2,1) has level \(\psi(\Omega_\omega+\psi_1(\Omega_\omega))\)
(0,0,0)(1,1,1)(1,1,0)(2,2,1)(2,2,0) has level \(\psi(\Omega_\omega+\psi_1(\Omega_\omega+1))\)
(0,0,0)(1,1,1)(1,1,0)(2,2,1)(2,2,0)(3,2,0) has level \(\psi(\Omega_\omega+\Omega_2)\)
(0,0,0)(1,1,1)(1,1,0)(2,2,1)(2,2,0)(3,3,1)(3,3,0)(4,3,0) has level \(\psi(\Omega_\omega+\Omega_3)\)
(0,0,0)(1,1,1)(1,1,1) has level \(\psi(\Omega_\omega*2)\)
(0,0,0)(1,1,1)(2,0,0) has level \(\psi(\Omega_\omega*\omega)\)
(0,0,0)(1,1,1)(2,1,0) â€¦
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The function types diverge at \(\Omega_2\)
Here is the comparison.
Function A Function B
\(\psi(\psi_1(\Omega_2))\) \(\psi(\Omega_2)\)
\(\psi(\psi_1(\Omega_2)+1)\) \(\psi(\Omega_2+1)\)
\(\psi(\psi_1(\Omega_2)+\Omega)\) \(\psi(\Omega_2+\Omega)\)
\(\psi(\psi_1(\Omega_2)*2)\) \(\psi(\Omega_2+\psi_1(\Omega_2))\)
\(\psi(\psi_1(\Omega_2+1))\) \(\psi(\Omega_2+\psi_1(\Omega_2+1))\)
\(\psi(\psi_1(\Omega_2+\Omega))\) \(\psi(\Omega_2+\psi_1(\Omega_2+\Omega))\)
\( \psi(\psi_1(\Omega_2+\psi_1(\Omega_2)))\) \(\psi(\Omega_2+\psi_1(\Omega_2+\psi_1(\Omega_2)))\)
\(\psi(\psi_1(\Omega_2*2))\) \(\psi(\Omega_2*2)\)
So the function catches up at \(\Omega_2*2\) and all multiples of \(\Omega_2\).
I like Function B better because it is more extensible.
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