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This kind of things have been discussed here.

Current classification

Currently, we have these size classes (in ascending order):

  1. Class 0 (< 6)
  2. Class 1 (6 ~ 106)
  3. Numbers with 7 to 21 digits
  4. Numbers with 22 to 100 digits
  5. Numbers with 101 to 309 digits
  6. Numbers with 309 to 4933 digits
  7. Numbers with 4933 to 1000000 digits (#3 ~ #7 are also called "Class 2")
  8. Class 3 (\(10^{10^6}\) ~ \(10^{10^{10^6}}\))
  9. Class 4 (\(10^{10^{10^6}}\) ~ \(10^{10^{10^{10^6}}}\))
  10. Class 5 (\(10^{10^{10^{10^6}}}\) ~ \(10^{10^{10^{10^{10^6}}}}\))
  11. Exponentiation level (\(10^{10^{10^{10^{10^6}}}}\) ~ \(10\uparrow\uparrow10\))
  12. Tetration level
  13. Up-arrow notation level
  14. Chained arrow notation level
  15. 5-6 entry linear array notation level
  16. 7+ entry linear array notation level
  17. Two row array notation level
  18. Planar array notation level
  19. Higher dimensional array notation level
  20. Superdimensional array level
  21. Trimensional array level
  22. Quadramensional array level
  23. Higher tetrational array level (#20 ~ #23 are also called "Tetrational array notation level")
  24. Higher array notation level
  25. Legiattic array notation level
  26. Beyond legiattic array notation level
  27. Uncomputable

Problem of BEAF

BEAF is ill-defined beyond tetrational arrays, so #24 ~ #26 are bad classified.

My suggestion is using names of set theories and number theories as the names of classes. e.g. "ATR0 level", "KP level", "\(\Pi_1^1-\text{CA}_0\) level", "\(\Pi_1^1-\text{TR}_0\) level" and "Higher second-order arithmetic level".

An alternative choice is using Bird's array notation below \(\theta(\Omega_\Omega,0)\), because it's consistant with BEAF below tetrational array notation. e.g. "Nested array notation level" (instead of "Tetrational array notation level"), "Hyper-nested array notation level", "Hierarchical hyper-nested array notation level" and "Nested hierarchical hyper-nested array notation level".

Problem of "Exponentiation level"

"Exponentiation level" might not be a good class.

I think exponentiation is \(a^b\) just as tetration is \(^ba\), and no "nested exponentiation" or "nested tetration" here. In that sense, the upper bound of class 5, \(10^{10^{10^{10^{10^6}}}}\), is already larger than the reach of "exponentiation". So beyond class 5 it should be higher tetration level instead of higher exponentiation level.

Boundaries

Here's the biggest case of this blog post.

Now look at Robert Munafo's reason about the boundaries of class 1, 2 and 3. (Class 1 number) objects can be seen by human eyes. A class 2 number can be represented exactly in decimal place-value notation, so in this case there are (class 1 number) digits. A class 3 number can be represented inexactly in scientific notation, so in this case there are (class 1 number) digits in the exponent of 10.

Further, if a notation need n objects as ascending toward the limit, the class limit will be at n = 106 case; if a notation have an index n (written in a number form) as ascending toward the limit, the class limit will be at n = \(10^{10^6}\) case. The former case applies on upper bounds of "Chained arrow notation level", "7+ entry linear array notation level", "Two row array notation level", "Planar array notation level", "Superdimensional array level", "Quadramensional array level" and "Higher tetrational array level"; the latter case applies on upper bounds of "Tetration level", "Up-arrow notation level", "5-6 entry linear array notation level", "Higher dimensional array notation level" and "Trimensional array level".

Boundaries of classes beyond tetrational array notation level

Before discussion about boundaries, we need to choose the notation we use. Notations beyond tetrational array notation level are shown below.

Lower bound of "Uncomputable"

Currently, the strongest but computable googologism is greedy clique sequence, and it certainly should not be classified into "uncomputable". However, it's a combinatorial googologism and hard to calculate, which make it unsuitable to be a boundary between classes.

New classification

Here is my suggestion about new classes.

  1. Class 0 (< 6)
  2. Class 1 (6 ~ 106)
  3. Class 2 (106 ~ \(10^{10^6}\))
  4. Class 3 (\(10^{10^6}\) ~ \(10^{10^{10^6}}\))
  5. Class 4 (\(10^{10^{10^6}}\) ~ \(10^{10^{10^{10^6}}}\))
  6. Class 5 (\(10^{10^{10^{10^6}}}\) ~ \(10^{10^{10^{10^{10^6}}}}\))
  7. Tetration level (\(10^{10^{10^{10^{10^6}}}}\) ~ \(10\uparrow\uparrow(10^{10^6})\))
  8. Up-arrow notation level (\(10\uparrow\uparrow(10^{10^6})\) ~ \(10\uparrow^{10^{10^6}}10\))
  9. Chained arrow notation level (\(10\uparrow^{10^{10^6}}10\) ~ \(\underbrace{10\rightarrow10\rightarrow\cdots10\rightarrow10}_{10^6\;10's}\))
  10. 5-6 entry linear array notation level (\(\underbrace{10\rightarrow10\rightarrow\cdots10\rightarrow10}_{10^6\;10's}\) ~ \(\{10,10,10,10,10,10^{10^6}\}\))
  11. 7+ entry linear array notation level (\(\{10,10,10,10,10,10^{10^6}\}\) ~ \(\{10,10^6(1)2\}\))
  12. Two row array notation level (\(\{10,10^6(1)2\}\) ~ \(\{10,10^6(1)(1)2\}\))
  13. Planar array notation level (\(\{10,10^6(1)(1)2\}\) ~ \(\{10,10^6(2)2\}\))
  14. Higher dimensional array notation level (\(\{10,10^6(2)2\}\) ~ \(\{10,10(10^{10^6})2\}\))
  15. Superdimensional array level (\(\{10,10(10^{10^6})2\}\) ~ \(\{10,10^6((1)1)2\}\))
  16. Trimensional array level (\(\{10,10^6((1)1)2\}\) ~ \(\{10,10((10^{10^6})1)2\}\))
  17. Quadramensional array level (\(\{10,10((10^{10^6})1)2\}\) ~ \(\{10,10^6(((1)1)1)2\}\))
  18. Higher tetrational array level (\(\{10,10^6(((1)1)1)2\}\) ~ \(\{10,10((\cdots(((\underbrace{1)1)1)\cdots1)1)}_{10^6\;1's}2\}\))
  19. \(\text{ACA}_0^+\) level (\(\{10,10((\cdots(((\underbrace{1)1)1)\cdots1)1)}_{10^6\;1's}2\}\) ~ \(f_{\varphi(2,0)}(10^6)\)) (The limit of \(\text{ACA}_0^+\) is \(\zeta_0\))
  20. ATR0 level (\(f_{\varphi(2,0)}(10^6)\) ~ \(f_{\varphi(1,0,0)}(10^6)\))
  21. KP level (\(f_{\varphi(1,0,0)}(10^6)\) ~ \(f_{\vartheta(\varphi(1,\Omega+1))}(10^6)\))
  22. \(\Pi_1^1-\text{CA}_0\) level (\(f_{\vartheta(\varphi(1,\Omega+1))}(10^6)\) ~ \(f_{\vartheta(\Omega_\omega)}(10^{10^6})\))
  23. \(\Pi_1^1-\text{TR}_0\) level (\(f_{\vartheta(\Omega_\omega)}(10^{10^6})\) ~ \(f_{\vartheta(\underbrace{\Omega_{\Omega_{\cdots_{\Omega_\Omega}}}}_{10^6\;\Omega's})}(10^6)\))
  24. Beyond \(\Pi_1^1-\text{TR}_0\) level (\(>f_{\vartheta(\underbrace{\Omega_{\Omega_{\cdots_{\Omega_\Omega}}}}_{10^6\;\Omega's})}(10^6)\) but computable)
  25. Uncomputable

Another new classification

In this version of classification, we only use FGH beyond "Class 5".

  1. Class 0 (< 6)
  2. Class 1 (6 ~ 106)
  3. Class 2 (106 ~ \(10^{10^6}\))
  4. Class 3 (\(10^{10^6}\) ~ \(10^{10^{10^6}}\))
  5. Class 4 (\(10^{10^{10^6}}\) ~ \(10^{10^{10^{10^6}}}\))
  6. Class 5 (\(10^{10^{10^{10^6}}}\) ~ \(10^{10^{10^{10^{10^6}}}}\))
  7. Tetration level (\(10^{10^{10^{10^{10^6}}}}\) ~ \(f_3(10^{10^6})\))
  8. Up-arrow notation level (\(f_3(10^{10^6})\) ~ \(f_\omega(10^{10^6})\))
  9. Chained arrow notation level (\(f_\omega(10^{10^6})\) ~ \(f_{\omega^2}(10^{10^6})\))
  10. 5-6 entry linear array notation level (\(f_{\omega^2}(10^{10^6})\) ~ \(f_{\omega^4}(10^{10^6})\))
  11. 7+ entry linear array notation level (\(f_{\omega^4}(10^{10^6})\) ~ \(f_{\omega^\omega}(10^{10^6})\))
  12. Two row array notation level (\(f_{\omega^\omega}(10^{10^6})\) ~ \(f_{\omega^{\omega2}}(10^{10^6})\))
  13. Planar array notation level (\(f_{\omega^{\omega2}}(10^{10^6})\) ~ \(f_{\omega^{\omega^2}}(10^{10^6})\))
  14. Higher dimensional array notation level (\(f_{\omega^{\omega^2}}(10^{10^6})\) ~ \(f_{\omega^{\omega^\omega}}(10^{10^6})\))
  15. Superdimensional array level (\(f_{\omega^{\omega^\omega}}(10^{10^6})\) ~ \(f_{\omega^{\omega^{\omega^\omega}}}(10^{10^6})\))
  16. Trimensional array level (\(f_{\omega^{\omega^{\omega^\omega}}}(10^{10^6})\) ~ \(f_{\omega^{\omega^{\omega^{\omega^\omega}}}}(10^{10^6})\))
  17. Quadramensional array level (\(f_{\omega^{\omega^{\omega^{\omega^\omega}}}}(10^{10^6})\) ~ \(f_{\omega^{\omega^{\omega^{\omega^{\omega^\omega}}}}}(10^{10^6})\))
  18. Higher tetrational array level (\(f_{\omega^{\omega^{\omega^{\omega^{\omega^\omega}}}}}(10^{10^6})\) ~ \(f_{\varphi(1,0)}(10^6)\))
  19. \(\text{ACA}_0^+\) level (\(f_{\varphi(1,0)}(10^6)\) ~ \(f_{\varphi(2,0)}(10^6)\))
  20. ATR0 level (\(f_{\varphi(2,0)}(10^6)\) ~ \(f_{\varphi(1,0,0)}(10^6)\))
  21. KP level (\(f_{\varphi(1,0,0)}(10^6)\) ~ \(f_{\vartheta(\varphi(1,\Omega+1))}(10^6)\))
  22. \(\Pi_1^1-\text{CA}_0\) level (\(f_{\vartheta(\varphi(1,\Omega+1))}(10^6)\) ~ \(f_{\vartheta(\Omega_\omega)}(10^{10^6})\))
  23. \(\Pi_1^1-\text{TR}_0\) level (\(f_{\vartheta(\Omega_\omega)}(10^{10^6})\) ~ \(f_{\vartheta(\underbrace{\Omega_{\Omega_{\cdots_{\Omega_\Omega}}}}_{10^6\;\Omega's})}(10^6)\))
  24. Beyond \(\Pi_1^1-\text{TR}_0\) level (\(>f_{\vartheta(\underbrace{\Omega_{\Omega_{\cdots_{\Omega_\Omega}}}}_{10^6\;\Omega's})}(10^6)\) but computable)
  25. Uncomputable

Third new classification

In this version of classification, the boundary indexes are always locked at 106.

  1. Class 0 (< 6)
  2. Class 1 (6 ~ 106)
  3. Class 2 (106 ~ \(10^{10^6}\))
  4. Class 3 (\(10^{10^6}\) ~ \(10^{10^{10^6}}\))
  5. Class 4 (\(10^{10^{10^6}}\) ~ \(10^{10^{10^{10^6}}}\))
  6. Class 5 (\(10^{10^{10^{10^6}}}\) ~ \(10^{10^{10^{10^{10^6}}}}\))
  7. Tetration level (\(10^{10^{10^{10^{10^6}}}}\) ~ \(10\uparrow\uparrow(10^6)\))
  8. Up-arrow notation level (\(10\uparrow\uparrow(10^6)\) ~ \(10\uparrow^{10^6}10\))
  9. Chained arrow notation level (\(10\uparrow^{10^6}10\) ~ \(\underbrace{10\rightarrow10\rightarrow\cdots10\rightarrow10}_{10^6\;10's}\))
  10. 5-6 entry linear array notation level (\(\underbrace{10\rightarrow10\rightarrow\cdots10\rightarrow10}_{10^6\;10's}\) ~ \(\{10,10,10,10,10,10^6\}\))
  11. 7+ entry linear array notation level (\(\{10,10,10,10,10,10^6\}\) ~ \(\{10,10^6(1)2\}\))
  12. Two row array notation level (\(\{10,10^6(1)2\}\) ~ \(\{10,10^6(1)(1)2\}\))
  13. Planar array notation level (\(\{10,10^6(1)(1)2\}\) ~ \(\{10,10^6(2)2\}\))
  14. Higher dimensional array notation level (\(\{10,10^6(2)2\}\) ~ \(\{10,10(10^6)2\}\))
  15. Superdimensional array level (\(\{10,10(10^6)2\}\) ~ \(\{10,10^6((1)1)2\}\))
  16. Trimensional array level (\(\{10,10^6((1)1)2\}\) ~ \(\{10,10((10^6)1)2\}\))
  17. Quadramensional array level (\(\{10,10((10^6)1)2\}\) ~ \(\{10,10^6(((1)1)1)2\}\))
  18. Higher tetrational array level (\(\{10,10^6(((1)1)1)2\}\) ~ \(\{10,10((\cdots(((\underbrace{1)1)1)\cdots1)1)}_{10^6\;1's}2\}\))
  19. \(\text{ACA}_0^+\) level (\(\{10,10((\cdots(((\underbrace{1)1)1)\cdots1)1)}_{10^6\;1's}2\}\) ~ \(f_{\varphi(2,0)}(10^6)\))
  20. ATR0 level (\(f_{\varphi(2,0)}(10^6)\) ~ \(f_{\varphi(1,0,0)}(10^6)\))
  21. KP level (\(f_{\varphi(1,0,0)}(10^6)\) ~ \(f_{\vartheta(\varphi(1,\Omega+1))}(10^6)\))
  22. \(\Pi_1^1-\text{CA}_0\) level (\(f_{\vartheta(\varphi(1,\Omega+1))}(10^6)\) ~ \(f_{\vartheta(\Omega_\omega)}(10^6)\))
  23. \(\Pi_1^1-\text{TR}_0\) level (\(f_{\vartheta(\Omega_\omega)}(10^6)\) ~ \(f_{\vartheta(\underbrace{\Omega_{\Omega_{\cdots_{\Omega_\Omega}}}}_{10^6\;\Omega's})}(10^6)\))
  24. Beyond \(\Pi_1^1-\text{TR}_0\) level (\(>f_{\vartheta(\underbrace{\Omega_{\Omega_{\cdots_{\Omega_\Omega}}}}_{10^6\;\Omega's})}(10^6)\) but computable)
  25. Uncomputable

Fourth new classification

A modification of "another new classification".

  1. Class 0 (< 6)
  2. Class 1 (6 ~ 106)
  3. Class 2 (106 ~ \(10^{10^6}\))
  4. Class 3 (\(10^{10^6}\) ~ \(10^{10^{10^6}}\))
  5. Class 4 (\(10^{10^{10^6}}\) ~ \(10^{10^{10^{10^6}}}\))
  6. Class 5 (\(10^{10^{10^{10^6}}}\) ~ \(10^{10^{10^{10^{10^6}}}}\))
  7. Tetration level (\(10^{10^{10^{10^{10^6}}}}\) ~ \(f_3(10^{10^6})\))
  8. Up-arrow notation level (\(f_3(10^{10^6})\) ~ \(f_\omega(10^{10^6})\))
  9. Chained arrow notation level (\(f_\omega(10^{10^6})\) ~ \(f_{\omega^2}(10^{10^6})\))
  10. Bowers' 5-6 entry linear array level (\(f_{\omega^2}(10^{10^6})\) ~ \(f_{\omega^4}(10^{10^6})\))
  11. Bowers' 7+ entry linear array level (\(f_{\omega^4}(10^{10^6})\) ~ \(f_{\omega^\omega}(10^{10^6})\))
  12. Bowers' two row array level (\(f_{\omega^\omega}(10^{10^6})\) ~ \(f_{\omega^{\omega2}}(10^{10^6})\))
  13. Bowers' planar array level (\(f_{\omega^{\omega2}}(10^{10^6})\) ~ \(f_{\omega^{\omega^2}}(10^{10^6})\))
  14. Bowers' dimensional array level (\(f_{\omega^{\omega^2}}(10^{10^6})\) ~ \(f_{\omega^{\omega^\omega}}(10^{10^6})\))
  15. Bowers' superdimensional array level (\(f_{\omega^{\omega^\omega}}(10^{10^6})\) ~ \(f_{\omega^{\omega^{\omega^\omega}}}(10^{10^6})\))
  16. Bowers' trimensional array level (\(f_{\omega^{\omega^{\omega^\omega}}}(10^{10^6})\) ~ \(f_{\omega^{\omega^{\omega^{\omega^\omega}}}}(10^{10^6})\))
  17. ACA0 level (\(f_{\omega^{\omega^{\omega^{\omega^\omega}}}}(10^{10^6})\) ~ \(f_{\varphi(1,0)}(10^6)\))
  18. \(\text{ACA}_0^+\) level (\(f_{\varphi(1,0)}(10^6)\) ~ \(f_{\varphi(2,0)}(10^6)\))
  19. ATR0 level (\(f_{\varphi(2,0)}(10^6)\) ~ \(f_{\varphi(1,0,0)}(10^6)\))
  20. KP level (\(f_{\varphi(1,0,0)}(10^6)\) ~ \(f_{\vartheta(\varphi(1,\Omega+1))}(10^6)\))
  21. Higher computable level (\(>f_{\vartheta(\varphi(1,\Omega+1))}(10^6)\) but computable)
  22. Uncomputable

Fifth new classification

In this version of classification, ranges of ordinals of growth rates are used on the names of classes beyond "Up-arrow notation level".

  1. Class 0 (< 6)
  2. Class 1 (6 ~ 106)
  3. Class 2 (106 ~ \(10^{10^6}\))
  4. Class 3 (\(10^{10^6}\) ~ \(10^{10^{10^6}}\))
  5. Class 4 (\(10^{10^{10^6}}\) ~ \(10^{10^{10^{10^6}}}\))
  6. Class 5 (\(10^{10^{10^{10^6}}}\) ~ \(10^{10^{10^{10^{10^6}}}}\))
  7. Tetration level (\(10^{10^{10^{10^{10^6}}}}\) ~ \(f_3(10^{10^6})\))
  8. Up-arrow notation level (\(f_3(10^{10^6})\) ~ \(f_\omega(10^{10^6})\))
  9. Linear omega level (\(f_\omega(10^{10^6})\) ~ \(f_{\omega^2}(10^{10^6})\))
  10. Quadratic omega level (\(f_{\omega^2}(10^{10^6})\) ~ \(f_{\omega^3}(10^{10^6})\))
  11. Polynomial omega level (\(f_{\omega^3}(10^{10^6})\) ~ \(f_{\omega^\omega}(10^{10^6})\))
  12. Exponentiated linear omega level (\(f_{\omega^\omega}(10^{10^6})\) ~ \(f_{\omega^{\omega^2}}(10^{10^6})\))
  13. Exponentiated polynomial omega level (\(f_{\omega^{\omega^2}}(10^{10^6})\) ~ \(f_{\omega^{\omega^\omega}}(10^{10^6})\))
  14. Double exponentiated polynomial omega level (\(f_{\omega^{\omega^\omega}}(10^{10^6})\) ~ \(f_{\omega^{\omega^{\omega^\omega}}}(10^{10^6})\))
  15. Triple exponentiated polynomial omega level (\(f_{\omega^{\omega^{\omega^\omega}}}(10^{10^6})\) ~ \(f_{\omega^{\omega^{\omega^{\omega^\omega}}}}(10^{10^6})\))
  16. Iterated Cantor normal form level (\(f_{\omega^{\omega^{\omega^{\omega^\omega}}}}(10^{10^6})\) ~ \(f_{\varphi(1,0)}(10^6)\))
  17. Epsilon level (\(f_{\varphi(1,0)}(10^6)\) ~ \(f_{\varphi(2,0)}(10^6)\))
  18. Binary phi level (\(f_{\varphi(2,0)}(10^6)\) ~ \(f_{\varphi(1,0,0)}(10^6)\))
  19. Bachmann's collapsing level (\(f_{\varphi(1,0,0)}(10^6)\) ~ \(f_{\vartheta(\varphi(1,\Omega+1))}(10^6)\))
  20. Higher computable level (\(>f_{\vartheta(\varphi(1,\Omega+1))}(10^6)\) but computable)
  21. Uncomputable
Which classification is better?
 
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0
 
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The poll was created at 15:06 on July 22, 2017, and so far 5 people voted.

Final plan

I will use the "fifth new classification". All the size-category will be change.

But there are some technical difficulty - Categories cannot be renamed, then I have to edit every number page to the new classification.

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