User blog:Hyp cos/Advanced approximation project

Current approximations can be further improved, and this will take large amount of edits, so I start this discussion.

BEAF
BEAF is not well-defined beyond tetrational arrays. More accurately, \(X\uparrow\uparrow A\&n\) where \(A\) is not a single number (e.g. \(X+1\), \(2X\), \(X^2\), \(X^X\), etc.) and beyond are not well-defined. But \(X\uparrow\uparrow m\&n\) and \(X\uparrow\uparrow X\&n\) are still considered as well-defined.

My suggestion about that is: Do you agree the first operation? Yes No Do you agree the second operation? Yes No If you have another way, please comment.
 * 1) Approximations of numbers defined in ill-defined BEAF should be removed.
 * 2) Approximations in ill-defined BEAF of any numbers should be removed.

FGH, HH and SGH
FGH, HH and SGH strongly depend on the fundamental sequence of ordinal notations, and there are many fundamental sequence systems.

Some FS systems do not get their limit ordinals, such as this one and this one. The ordinal notations are \(\vartheta\) function (e.g. \(\vartheta(\Omega^\omega+1)=\sup\{\vartheta(\Omega^n+\vartheta(\Omega^\omega))|n<\omega\}\)), but the FS systems still behave in \(\psi\) ways (e.g. the FS for \(\vartheta(\Omega^\omega+1)\) is \(\vartheta(\Omega^\omega+1)[n]=\vartheta(\Omega^\omega)\times n\)). These FS systems should not be used.

Some FS systems are not complete (i.e. not defined for some ordinals), such as Fundamental sequence of \(\theta\) function. These FS systems should not be used.

Some well-defined FS systems are shown below: Any two of the last 5 systems are inconsistent. What's more, system 7 ~ 9 even use the same notation, but different meaning. So we need to determine the FS system (and notation system) we use in approximation tables. For ordinals between \(\varepsilon_0\) (including) and \(\Gamma_0\) Use system 4 Use system 7 Use system 8 Use system 9 Use system 10 Use system 4 if it's expressed in \(\varphi\) function, system 7 if it's expressed in \(\psi\) function, and system 10 if it's expressed in C function Use system 4 if it's expressed in \(\varphi\) function, system 8 if it's expressed in \(\psi\) function, and system 10 if it's expressed in C function Use system 4 if it's expressed in \(\varphi\) function, system 9 if it's expressed in \(\psi\) function, and system 10 if it's expressed in C function For ordinals between \(\Gamma_0\) (including) and SVO Use system 5 Use system 7 Use system 8 Use system 9 Use system 10 Use system 5 if it's expressed in \(\varphi\) function, system 7 if it's expressed in \(\psi\) function, and system 10 if it's expressed in C function Use system 5 if it's expressed in \(\varphi\) function, system 8 if it's expressed in \(\psi\) function, and system 10 if it's expressed in C function Use system 5 if it's expressed in \(\varphi\) function, system 9 if it's expressed in \(\psi\) function, and system 10 if it's expressed in C function For ordinals between \(\Gamma_0\) (including) and LVO Use system 6 Use system 7 Use system 8 Use system 9 Use system 10 Use system 6 if it's expressed in Veblen function, system 7 if it's expressed in \(\psi\) function, and system 10 if it's expressed in C function Use system 6 if it's expressed in Veblen function, system 8 if it's expressed in \(\psi\) function, and system 10 if it's expressed in C function Use system 6 if it's expressed in Veblen function, system 9 if it's expressed in \(\psi\) function, and system 10 if it's expressed in C function For ordinals between LVO (including) and \(\psi(\beta=\Omega_\beta)\) Use system 7 Use system 8 Use system 9 Use system 10 Use system 7 if it's expressed in \(\psi\) function, and system 10 if it's expressed in C function Use system 8 if it's expressed in \(\psi\) function, and system 10 if it's expressed in C function Use system 9 if it's expressed in \(\psi\) function, and system 10 if it's expressed in C function For ordinals between \(\psi(\beta=\Omega_\beta)\) (including) and \(\psi(\beta=I_\beta)\) Use system 8 Use system 9 Use system 10 Use system 8 if it's expressed in \(\psi\) function, and system 10 if it's expressed in C function Use system 9 if it's expressed in \(\psi\) function, and system 10 if it's expressed in C function For ordinals between \(\psi(\beta=I_\beta)\) (including) and \(\psi(\psi_{I(1,0,0)}(0))\) Use system 9 Use system 10 Use system 9 if it's expressed in \(\psi\) function, and system 10 if it's expressed in C function If you have another way, please comment.
 * 1) Wainer hierarchy for ordinals \(<\varepsilon_0\)
 * 2) Veblen hierarchy for ordinals \(<\Gamma_0\). It's consistent with system 1 if we write \(\omega^\alpha\) as \(\varphi_0(\alpha)\).
 * 3) Another fundamental sequence system for Veblen hierarchy, but \(\varphi_\alpha(\beta)\) is written in \(\varphi(\alpha,\beta)\).
 * 4) Fundamental sequence of \(\Gamma\) function for ordinals \(<\varphi(1,1,0)\).
 * 5) Finitary Veblen hierarchy for ordinals < SVO. It's consistent with system 3 and 4, but also 1 and 2 in values. If we consider values only, system 2 and 4 are identical.
 * 6) Transfinitary Veblen hierarchy for ordinals < LVO. It's conststent with system 1 ~ 5 in values.
 * 7) Extended Buchholz hierarchy for ordinals \(< \psi(\Omega_{\Omega_{\Omega_\cdots}})\). It's consistent with system 1, but inconsistent with system 2 ~ 6.
 * 8) Fundamental sequence system for \(\psi\) function up to weakly inaccessibles for ordinals \(<\psi(I_{I_{I_\cdots}})\). It's consistent with system 1, but inconsistent with system 2 ~ 7. The point starting to be inconsistent with system 7 is \(\psi(\Omega)=\varepsilon_0\).
 * 9) Fundamental sequence system for \(\psi\) function up to \(\alpha\)-weakly inaccessibles for ordinals \(<\psi(\psi_{I(1,0,0)}(0))\). It's consistent with system 1, but inconsistent with system 2 ~ 8. The point starting to be inconsistent with system 7 is \(\psi(\Omega)=\varepsilon_0\), and the point starting to be inconsistent with system 8 is \(\psi(\psi_{\Omega_2}(\omega))=\varepsilon_{\omega^\omega}\).
 * 10) Fundamental sequence system for Taranovsky's ordinal notation for ordinals in the range of this notation. It's inconsistent with system 1 ~ 9.