User blog:Hyp cos/Suggestions about Size Classes of Numbers

This kind of things have been discussed here.

Current classes
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.
 * Hyper-Extended Cascading-E Notation with limit \(\varphi(1,0,0,0)\).
 * Bird's array notation with limit \(\theta(\Omega_\Omega,0)\).
 * Hyperfactorial array notation with limit \(\psi(I_\omega)\).
 * Aarex's Array Notation
 * Strong array notation
 * Fast-growing hierarchy. Note that only the ordinal notations with fundamental sequences can be used.
 * Veblen's phi function (FS)
 * \(\vartheta\) function (FS)
 * Taranovsky's ordinal notation (FS)