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The beginning of third attempt of second generation dropping notation. In second generation of my array notation, (A){B} is a separator, where (A) is a dropper, meaning how many times it finds an outer smaller separator until expanding. Here, I use hydra-based notation instead of array-based notation. S(B1,B2,...Bk-1,Bk) is the separator analogue, where B1,B2,...Bk-1,Bk are sections; B2,...Bk-1,Bk are droppers. That's where the name of the notation comes from.

Definition

Syntax: nRA

where n is a positive integer and A is a term. This "n" is called the base number.

Definition of term:

  1. 0 is a term
  2. A+B is a term, where A is a term and B is a principle term
  3. S(T,T,...,T,B1,B2,...Bk-1,Bk) is a term, where k is a positive integer, B1,B2,...Bk-1,Bk are terms. This kind of term is called a principle term.

Notes:

  • The whole thing after the "R" is called the expression.
  • The expression uses symbol "0", "+", "(", ")", ",", "S" and "T".
  • Each of the Bi in a principle term S(B1,B2...,Bk) is called section. Some section could be T.
  • For a principle term A = S(B1,B2...,Bk), define \(A|_i^j=S(B_i,B_{i+1}\cdots,B_{j-1},B_j)\). i could be negative; in this case, see it as \(k+1+i\); this also work for j. So for instance, \(A|_{-1}^{-1}\) takes the rightmost section of A, forming a new term with one section.

Rules and process

  1. (zero rule) nR0 = 10n
  2. (successor rule) nRA S(0) = (...((nRA 0)RA 0)...RA 0)RA 0 with n R's, where A is a string to make "A S(0)" the expression
  3. (tailing rule) "+0" and "0+" anywhere are deleted

If none of the 3 rules above applies, do the main process below. When some rule says "process ends", process exits, and the 3 rules are checked again before next process.

Consider the rightmost ")"s in the expression. There should be at least two of them, and there should be a "0" immediately left to those ")"s. Suppose there are m ")"s, define Ai to be the principle term ending with the i-th ")" counted from the right. Note that changes on Ai's mean modifying on the original expression.

  • Case A: Am = S(0) contains one section
    • Case A1: Am-1 = S(B S(0)) contains one section, where B is a string to make "B S(0)" a term
      1. Change the Am-1 into En, where n is the base number, E0 = 0 and Ei+1 = Ei+S(B 0)
      2. The process ends
    • Case A2: Am-1 = S(B1,B2...,Bk,B S(0)) contains more than one sections, where B is a string to make "B S(0)" a term
      1. Let \(t=\max\{i|A_i<_\text{lv}A_{m-1}\}\), i.e. At is the innermost smaller term than Am-1
      • Case A2a: if \(A_t|_{-1}^{-1}<_\text{lv}f_{A_{m-1}|_{-2}^{-2}}(0)\)
        1. Let \(u=\min\{i|i>t\land A_i<_\text{lv}p(A_{m-1}|_1^{-2})\}\)
        2. Change the Au into \(f_{A_{m-1}|_1^{-2}}(A_u)\)
        3. The process ends
      • Case A2b: if \(A_t|_{-1}^{-1}\ge_\text{lv}f_{A_{m-1}|_{-2}^{-2}}(0)\)
        1. Suppose At=S(C1,C2...,Cj), where those Ci's are sections of At
        2. Change the At into S(C1,C2...,Cj,B 0)
        3. The process ends
  • Case B: Am = S(B1,B2...,Bk,0) contains more than one sections
    1. Change the Am into \(f_{A_m|_1^{-2}}^n(0)\), where n is the base number
    2. The process ends

Level comparisons

Level comparisons are the \(<_\text{lv}\), \(>_\text{lv}\) and \(=_\text{lv}\) things. \(A\ge_\text{lv}B\) if \(A>_\text{lv}B\) or \(A=_\text{lv}B\). And similar to \(A\le_\text{lv}B\).

Firstly, if "A0" or "\(A_0|_{-1}^{-1}\)" are used in the main process, they are defined to be \(<_\text{lv}\) than any term or T. T is \(>_\text{lv}\) than any term. Terms are compared when tailing rule cannot apply. Among terms, 0 is the smallest.

Comparison of \(A=A_1+A_2\cdots+A_{l_A}\) and \(B=B_1+B_2\cdots+B_{l_B}\), where \(A_i\)'s and \(B_i\)'s are principle term and (\(l_A>1\) or \(l_B>1\)):

  1. Let \(i_C=\min\{i\in\{1,2\cdots,l_C\}|\forall j\in\{1,2\cdots,l_C\}(C_i\ge_\text{lv}C_j)\}\) for \(C\in\{A,B\}\)
  2. If \(A_{i_A}>_\text{lv}B_{i_B}\) then \(A>_\text{lv}B\); if \(A_{i_A}<_\text{lv}B_{i_B}\) then \(A<_\text{lv}B\); else
  3. If \(A_{i_A+1}+A_{i_A+2}\cdots+A_{l_A}>_\text{lv}B_{i_B+1}+B_{i_B+2}\cdots+B_{l_B}\) then \(A>_\text{lv}B\); if \(A_{i_A+1}+A_{i_A+2}\cdots+A_{l_A}<_\text{lv}B_{i_B+1}+B_{i_B+2}\cdots+B_{l_B}\) then \(A<_\text{lv}B\); if \(A_{i_A+1}+A_{i_A+2}\cdots+A_{l_A}=_\text{lv}B_{i_B+1}+B_{i_B+2}\cdots+B_{l_B}\) then \(A=_\text{lv}B\)

Comparison of principle term \(A=S(A_1,A_2\cdots,A_{l_A})\) and \(B=S(B_1,B_2\cdots,B_{l_B})\):

  1. If \(A_1>_\text{lv}B_1\) then \(A>_\text{lv}B\); if \(A_1<_\text{lv}B_1\) then \(A<_\text{lv}B\); else
  2. If \(l_A=1<l_B\) then \(A>_\text{lv}B\); if \(l_B=1<l_A\) then \(A<_\text{lv}B\); if \(l_A=l_B=1\) then \(A=_\text{lv}B\); else
  3. If \(A|_2^{-1}>_\text{lv}B|_2^{-1}\) then \(A>_\text{lv}B\); if \(A|_2^{-1}<_\text{lv}B|_2^{-1}\) then \(A<_\text{lv}B\); if \(A|_2^{-1}=_\text{lv}B|_2^{-1}\) then \(A=_\text{lv}B\)

Subfunctions

2 subfunctions appear in the main process: f and p function. fterm is a function from term to term. p(term) may be a term or T. To get fA and p(A), consider the rightmost ")"s in A. Suppose there are m ")"s, define Ai to be the principle term ending with the i-th ")" counted from the right. There should be a "0" or a "T" immediately left to those ")"s.

  • Case A: "0" is immidiately left to those ")"s
    • Case A1: Am = S(0) contains one section
      1. Suppose Am-1 = S(B1,B2...,Bk,B S(0)) contains one or more sections, where B is a string to make "B S(0)" a term
      2. Define p(A) = Am-1, fA(x) is A but with Am-1 changed into S(B1,B2...,Bk,B 0)+x
    • Case A2: Am = S(B1,B2...,Bk,0) contains more than one sections
      1. Define p(A) = Am, fA(x) is A but with Am changed into x
  • Case B: "T" is immidiately left to those ")"s
    1. Suppose Am = S(T,T...,T,T) with k T's
    2. Define p(A) = T, fA(x) is A but with Am changed into S(T,T...,T,x) with k-1 T's

Explanation

In the main process, case A1 is the case for "times by ω". Case B is the expanding, after all dropping done. Case A2b is the actual dropping, while case A2a is the "adding rule" when the outer term has a wrong (too small in the final section) level. So, S(A,B) means B times of dropping, where the "B" is mainly decreased by main case A2b.

The subfunctions are the linking between different "drop down process". Subfunctions used in a later dropping is usually the same thing happens in a former dropping.

Compared with NDAN (second attempt of second generation array notation), this notation uses two subfunctions instead of four. The previous X(A) and Y(A) are a pair, simplified into a single fA. The previous D(A) is now fA(0), derived from the fA. The previous U(A) is now p(A), unchanged.

The essential difference between nested dropper notation and NDAN is the definition of fA in case A1. Equivalently, second attempt uses

fA(x) is A but with Am-1 changed into x

which makes the fA(0) (or D(A)) of S(S(T,α+1)) (or {1(α+1)△2}) be S(0) (or {1}), so it allows many different high-leveled separators to target at the same layer. In third attempt, the fA(0) (or D(A)) of S(S(T,α+1)) (or {1(α+1)△2}) is S(S(T,α)) (or {1(α)△2}), generating different minimal requirements for different term. This canceled the infinite loop around {1(1,2)△3}. Thanks to the help of discord user Reflecting Ordinal, I got this form of fA(0).

But it still doesn't work

Here is an example of infinite loop.

First period: S(S(T,S(T,1)+1)) = S(S(T,S(T,1)+1),S(T,1)) = S(S(T,S(T,1)+1),S(T,1),0), reduced into nests of f(x) = S(S(T,S(T,1)+1),S(T,0)+x). Consider a smaller term S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1)+1). It adds a layer of f(x) = S(S(T,S(T,1))+x), resulting S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1)+1)).

Second period: S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1)+1)) = S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1)+1),S(T,0)+S(S(T,S(T,1)+1),1)) = S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1)+1),S(T,0)+S(S(T,S(T,1)+1),1),0), reduced into nests of f(x) = S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1)+1),S(T,0)+S(S(T,S(T,1)+1),0)+x). Consider a smaller term S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1)+1),S(T,0)+S(S(T,S(T,1)+1),0)+S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1)+1),1)+1). It adds a layer of f(x) = S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1))+x), resulting S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1))+S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1)+1),S(T,0)+S(S(T,S(T,1)+1),0)+S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1)+1),1)+1)).

Third period: S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1))+S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1)+1),S(T,0)+S(S(T,S(T,1)+1),0)+S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1)+1),1)+1)) = S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1))+S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1)+1),S(T,0)+S(S(T,S(T,1)+1),0)+S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1)+1),1)+1),S(T,0)+S(S(T,S(T,1)+1),0)+S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1)+1),1)) = S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1))+S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1)+1),S(T,0)+S(S(T,S(T,1)+1),0)+S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1)+1),1)+1),S(T,0)+S(S(T,S(T,1)+1),0)+S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1)+1),1),0), reduced into nests of f(x) = S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1))+S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1)+1),S(T,0)+S(S(T,S(T,1)+1),0)+S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1)+1),1)+1),S(T,0)+S(S(T,S(T,1)+1),0)+S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1)+1),0)+x). Consider a smaller term S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1))+S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1)+1),S(T,0)+S(S(T,S(T,1)+1),0)+S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1)+1),1)+1),S(T,0)+S(S(T,S(T,1)+1),0)+S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1)+1),0)+S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1))+S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1)+1),S(T,0)+S(S(T,S(T,1)+1),0)+S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1)+1),1)+1),1)+1). It adds a layer of f(x) = S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1))+S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1)+1),S(T,0)+S(S(T,S(T,1)+1),0)+S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1)+1),1))+x), resulting S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1))+S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1)+1),S(T,0)+S(S(T,S(T,1)+1),0)+S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1)+1),1))+S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1))+S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1)+1),S(T,0)+S(S(T,S(T,1)+1),0)+S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1)+1),1)+1),S(T,0)+S(S(T,S(T,1)+1),0)+S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1)+1),0)+S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1))+S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1)+1),S(T,0)+S(S(T,S(T,1)+1),0)+S(S(T,S(T,1))+S(S(T,S(T,1)+1),S(T,0)+S(S(T,S(T,1)+1),1)+1),1)+1),1)+1)).

Generally, the n-th period starts with S(Bn+S(An,Cn+S(An,1)+1)), where \(f_{S(A_n)}(0)\) = S(Bn), Cn starting with "S(T,0)" so it is larger than Bn. S(Bn+S(An,Cn+S(An,1)+1)) = S(Bn+S(An,Cn+S(An,1)+1),Cn+S(An,1)) = S(Bn+S(An,Cn+S(An,1)+1),Cn+S(An,1),0), reduced into nests of f(x) = S(Bn+S(An,Cn+S(An,1)+1),Cn+S(An,0)+x). Consider a smaller term S(Bn+S(An,Cn+S(An,1)+1),Cn+S(An,0)+S(Bn+S(An,Cn+S(An,1)+1),1)+1), it adds a layer of f(x) = S(Bn+S(An,Cn+S(An,1))+x), resulting S(Bn+S(An,Cn+S(An,1))+S(Bn+S(An,Cn+S(An,1)+1),Cn+S(An,0)+S(Bn+S(An,Cn+S(An,1)+1),1)+1)). The new An+1 = Bn+S(An,Cn+S(An,1)+1), Bn+1 = Bn+S(An,Cn+S(An,1)), Cn+1 = Cn+S(An,0). So S(Bn+S(An,Cn+S(An,1))+S(Bn+S(An,Cn+S(An,1)+1),Cn+S(An,0)+S(Bn+S(An,Cn+S(An,1)+1),1)+1)) is S(Bn+1+S(An+1,Cn+1+S(An+1,1)+1)); \(f_{S(A_{n+1})}(0)\) = S(Bn+1) and Cn+1 starting with "S(T,0)" still hold.

Analysis

The growth rate is counted starting from exponentiation, so that nR0 has growth rate 0; nRS(0) has growth rate 1; nRS(0)+S(0) has growth rate 2, and so on. Additional, there is ordinal correspondence, meaning how a term acts in terms of large ordinals. The subject of "has growth rate" would be the expression instead of the function f(x)=xRthis expression.

Due to a higher fA(0) value and thus more frequent application of adding rules, ordinal correspondence of this notation becomes much more complicated than (first generation) array notation.

Without occurence of T, the unary S() and 0 and addition can only reach \(\varepsilon_0\), because of this ordinal correspondence:

If B < S(A+B), S(A) means \(\alpha\) and B means \(\beta\), then S(A+B) means \(\alpha\cdot\omega^\beta\)

So, from now on, we will use some shorthands. S(0)+S(0)+...S(0)+S(0) (with n S(0)'s) is denoted by positive integer n; S(S(0)) is denoted by ω. "+" means ordinal addition; ordinal multiplication and exponentiation are derived from the S(A+B) property. Especially, for a principle term A, A+A+...A+A (with n A's) is denoted by An.

  1. S(T,0) means and has growth rate \(\varepsilon_0\)
  2. S(S(T,0)) means and has growth rate \(\varepsilon_0\)
  3. S(S(T,1)) means and has growth rate \(\varepsilon_1\)
  4. S(T,1) has growth rate \(\varepsilon_1\)
  5. S(S(T,2),0) means and has growth rate \(\varepsilon_2\)
  6. S(S(T,1)+S(S(T,2),0)) means and has growth rate \(\varepsilon_2\)
  7. S(S(T,1)+S(S(T,2))) means and has growth rate \(\varepsilon_3\)
  8. S(T,2) has growth rate \(\varepsilon_3\)
  9. S(S(T,1)+S(S(T,2))+S(S(T,2),0)) means and has growth rate \(\varepsilon_3\cdot\varepsilon_2\)
  10. S(S(T,1)+S(S(T,2))2) means and has growth rate \(\varepsilon_4\)
  11. S(S(T,1)+S(S(T,2))3) means and has growth rate \(\varepsilon_5\)
  12. S(S(T,1)+S(S(T,2)+1)) means and has growth rate \(\varepsilon_\omega\)
  13. S(S(T,1)+S(S(T,2)+1)+S(S(T,2),0)) means and has growth rate \(\varepsilon_\omega\cdot\varepsilon_2\)
  14. S(S(T,1)+S(S(T,2)+1)+S(S(T,2),0)+S(S(T,1)+S(S(T,2)+1))) means and has growth rate \(\varepsilon_\omega^2\)
  15. S(S(T,1)+S(S(T,2)+1)+S(S(T,2))) means and has growth rate \(\varepsilon_{\omega+1}\)
  16. S(S(T,1)+S(S(T,2)+1)2) means and has growth rate \(\varepsilon_{\omega2}\)
  17. S(S(T,1)+S(S(T,2)+2)) means and has growth rate \(\varepsilon_{\omega^2}\)
  18. S(S(T,1)+S(S(T,2)+ω)) means and has growth rate \(\varepsilon_{\omega^\omega}\)
  19. S(S(T,1)+S(S(T,2)+S(S(T,0)))) means and has growth rate \(\varepsilon_{\varepsilon_0}\)
  20. S(S(T,1)+S(S(T,2)+S(S(T,0)))+S(S(T,2))) means and has growth rate \(\varepsilon_{\varepsilon_0+1}\)
  21. S(S(T,1)+S(S(T,2)+S(S(T,0))+1)) means and has growth rate \(\varepsilon_{\omega^{\varepsilon_0+1}}\)
  22. S(S(T,1)+S(S(T,2)+S(S(T,1)))) means and has growth rate \(\varepsilon_{\varepsilon_1}\)
  23. S(S(T,1)+S(S(T,2)+S(S(T,2),0))) means and has growth rate \(\varepsilon_{\varepsilon_2}\)
  24. S(S(T,1)+S(S(T,2)+S(S(T,2),0)+1)) means and has growth rate \(\varepsilon_{\omega^{\varepsilon_2+1}}\)
  25. S(S(T,1)+S(S(T,2)+S(S(T,2),0)+S(S(T,1)+S(S(T,2),0)))) means and has growth rate \(\varepsilon_{\omega^{\varepsilon_22}}\)
  26. S(S(T,1)+S(S(T,2)+S(S(T,2),0)+S(S(T,1)+S(S(T,2))))) means and has growth rate \(\varepsilon_{\varepsilon_3}\)
  27. S(S(T,1)+S(S(T,2)+S(S(T,2),0)+S(S(T,1)+S(S(T,2)+1)))) means and has growth rate \(\varepsilon_{\varepsilon_\omega}\)
  28. S(S(T,1)+S(S(T,2)+S(S(T,2),0)+S(S(T,1)+S(S(T,2)+S(S(T,2),0))))) means and has growth rate \(\varepsilon_{\varepsilon_{\varepsilon_2}}\)
  29. S(S(T,1)+S(S(T,2)+S(S(T,2),0)+S(S(T,1)+S(S(T,2)+S(S(T,2),0)+S(S(T,1)+S(S(T,2)+S(S(T,2),0))))))) means and has growth rate \(\varepsilon_{\varepsilon_{\varepsilon_{\varepsilon_2}}}\)
  30. S(S(T,1)+S(S(T,2)+S(S(T,2)))) means and has growth rate \(\zeta_0\)
  31. S(S(T,1)+S(S(T,2)+S(S(T,2)))+S(S(T,2),0)) means and has growth rate \(\zeta_0\cdot\varepsilon_2\)
  32. S(S(T,1)+S(S(T,2)+S(S(T,2)))+S(S(T,2))) means and has growth rate \(\varepsilon_{\zeta_0+1}\)
  33. S(S(T,1)+S(S(T,2)+S(S(T,2)))+S(S(T,2)+1)) means and has growth rate \(\varepsilon_{\zeta_0+\omega}\)
  34. S(S(T,1)+S(S(T,2)+S(S(T,2)))+S(S(T,2)+S(S(T,2),0))) means and has growth rate \(\varepsilon_{\zeta_0+\varepsilon_2}\)
  35. S(S(T,1)+S(S(T,2)+S(S(T,2)))+S(S(T,2)+S(S(T,2),0)+S(S(T,1)+S(S(T,2)+S(S(T,2),0))))) means and has growth rate \(\varepsilon_{\zeta_0+\varepsilon_{\varepsilon_2}}\)
  36. S(S(T,1)+S(S(T,2)+S(S(T,2)))+S(S(T,2)+S(S(T,2),0)+S(S(T,1)+S(S(T,2)+S(S(T,2)))))) means and has growth rate \(\varepsilon_{\zeta_02}\)
  37. S(S(T,1)+S(S(T,2)+S(S(T,2)))+S(S(T,2)+S(S(T,2),0)+S(S(T,1)+S(S(T,2)+S(S(T,2)))+S(S(T,2)+S(S(T,2),0))))) means and has growth rate \(\varepsilon_{\varepsilon_{\zeta_0+\varepsilon_2}}\)
  38. S(S(T,1)+S(S(T,2)+S(S(T,2)))2) means and has growth rate \(\zeta_1\)
  39. S(S(T,1)+S(S(T,2)+S(S(T,2)))2+S(S(T,2)+S(S(T,2),0))) means and has growth rate \(\varepsilon_{\zeta_1+\varepsilon_2}\)
  40. S(S(T,1)+S(S(T,2)+S(S(T,2)))3) means and has growth rate \(\zeta_2\)
  41. S(S(T,1)+S(S(T,2)+S(S(T,2))+1)) means and has growth rate \(\zeta_\omega\)
  42. S(S(T,1)+S(S(T,2)+S(S(T,2))+1)+S(S(T,2))) means and has growth rate \(\varepsilon_{\zeta_\omega+1}\)
  43. S(S(T,1)+S(S(T,2)+S(S(T,2))+1)+S(S(T,2)+S(S(T,2)))) means and has growth rate \(\zeta_{\omega+1}\)
  44. S(S(T,1)+S(S(T,2)+S(S(T,2))+2)) means and has growth rate \(\zeta_{\omega^2}\)
  45. S(S(T,1)+S(S(T,2)+S(S(T,2))+S(S(T,2),0))) means and has growth rate \(\zeta_{\varepsilon_2}\)
  46. S(S(T,1)+S(S(T,2)+S(S(T,2))+S(S(T,2),0)+1)) means and has growth rate \(\zeta_{\omega^{\varepsilon_2+1}}\)
  47. S(S(T,1)+S(S(T,2)+S(S(T,2))+S(S(T,2),0)+S(S(T,1)+S(S(T,2)+S(S(T,2)))))) means and has growth rate \(\zeta_{\zeta_0}\)
  48. S(S(T,1)+S(S(T,2)+S(S(T,2))+S(S(T,2),0)+S(S(T,1)+S(S(T,2)+S(S(T,2))+S(S(T,2),0))))) means and has growth rate \(\zeta_{\zeta_{\varepsilon_2}}\)
  49. S(S(T,1)+S(S(T,2)+S(S(T,2))2)) means and has growth rate \(\varphi(3,0)\)
  50. S(S(T,1)+S(S(T,2)+S(S(T,2))2)+S(S(T,2))) means and has growth rate \(\varepsilon_{\varphi(3,0)+1}\)
  51. S(S(T,1)+S(S(T,2)+S(S(T,2))2)+S(S(T,2)+1)) means and has growth rate \(\varepsilon_{\varphi(3,0)+\omega}\)
  52. S(S(T,1)+S(S(T,2)+S(S(T,2))2)+S(S(T,2)+S(S(T,2),0))) means and has growth rate \(\varepsilon_{\varphi(3,0)+\varepsilon_2}\)
  53. S(S(T,1)+S(S(T,2)+S(S(T,2))2)+S(S(T,2)+S(S(T,2),0)+S(S(T,1)+S(S(T,2)+S(S(T,2))2)+S(S(T,2)+S(S(T,2),0))))) means and has growth rate \(\varepsilon_{\varepsilon_{\varphi(3,0)+\varepsilon_2}}\)
  54. S(S(T,1)+S(S(T,2)+S(S(T,2))2)+S(S(T,2)+S(S(T,2)))) means and has growth rate \(\zeta_{\varphi(3,0)+1}\)
  55. S(S(T,1)+S(S(T,2)+S(S(T,2))2)+S(S(T,2)+S(S(T,2))+1)) means and has growth rate \(\zeta_{\varphi(3,0)+\omega}\)
  56. S(S(T,1)+S(S(T,2)+S(S(T,2))2)+S(S(T,2)+S(S(T,2))+S(S(T,2),0)+S(S(T,1)+S(S(T,2)+S(S(T,2))2)+S(S(T,2)+S(S(T,2))+1)))) means and has growth rate \(\zeta_{\zeta_{\varphi(3,0)+\omega}}\)
  57. S(S(T,1)+S(S(T,2)+S(S(T,2))2)2) means and has growth rate \(\varphi(3,1)\)
  58. S(S(T,1)+S(S(T,2)+S(S(T,2))2+1)) means and has growth rate \(\varphi(3,\omega)\)
  59. S(S(T,1)+S(S(T,2)+S(S(T,2))2+S(S(T,2),0))) means and has growth rate \(\varphi(3,\varepsilon_2)\)
  60. S(S(T,1)+S(S(T,2)+S(S(T,2))2+S(S(T,2),0)+S(S(T,1)+S(S(T,2)+S(S(T,2))2+S(S(T,2),0))))) means and has growth rate \(\varphi(3,\varphi(3,\varepsilon_2))\)
  61. S(S(T,1)+S(S(T,2)+S(S(T,2))3)) means and has growth rate \(\varphi(4,0)\)
  62. S(S(T,1)+S(S(T,2)+S(S(T,2))3)2) means and has growth rate \(\varphi(4,1)\)
  63. S(S(T,1)+S(S(T,2)+S(S(T,2))3+1)) means and has growth rate \(\varphi(4,\omega)\)
  64. S(S(T,1)+S(S(T,2)+S(S(T,2))4)) means and has growth rate \(\varphi(5,0)\)
  65. S(S(T,1)+S(S(T,2)+S(S(T,2)+1))) means and has growth rate \(\varphi(\omega,0)\)
  66. S(S(T,1)+S(S(T,2)+S(S(T,2)+1))+S(S(T,2))) means and has growth rate \(\varepsilon_{\varphi(\omega,0)+1}\)
  67. S(S(T,1)+S(S(T,2)+S(S(T,2)+1))+S(S(T,2)+1)) means and has growth rate \(\varepsilon_{\varphi(\omega,0)+\omega}\)
  68. S(S(T,1)+S(S(T,2)+S(S(T,2)+1))+S(S(T,2)+S(S(T,2)))) means and has growth rate \(\zeta_{\varphi(\omega,0)+1}\)
  69. S(S(T,1)+S(S(T,2)+S(S(T,2)+1))+S(S(T,2)+S(S(T,2))2)) means and has growth rate \(\varphi(3,\varphi(\omega,0)+1)\)
  70. S(S(T,1)+S(S(T,2)+S(S(T,2)+1))2) means and has growth rate \(\varphi(\omega,1)\)
  71. S(S(T,1)+S(S(T,2)+S(S(T,2)+1)+1)) means and has growth rate \(\varphi(\omega,\omega)\)
  72. S(S(T,1)+S(S(T,2)+S(S(T,2)+1)+S(S(T,2),0)+S(S(T,1)+S(S(T,2)+S(S(T,2)+1))))) means and has growth rate \(\varphi(\omega,\varphi(\omega,0))\)
  73. S(S(T,1)+S(S(T,2)+S(S(T,2)+1)+S(S(T,2)))) means and has growth rate \(\varphi(\omega+1,0)\)
  74. S(S(T,1)+S(S(T,2)+S(S(T,2)+1)+S(S(T,2))2)) means and has growth rate \(\varphi(\omega+2,0)\)
  75. S(S(T,1)+S(S(T,2)+S(S(T,2)+1)2)) means and has growth rate \(\varphi(\omega2,0)\)
  76. S(S(T,1)+S(S(T,2)+S(S(T,2)+2))) means and has growth rate \(\varphi(\omega^2,0)\)
  77. S(S(T,1)+S(S(T,2)+S(S(T,2)+S(S(T,2),0)))) means and has growth rate \(\varphi(\varepsilon_2,0)\)
  78. S(S(T,1)+S(S(T,2)+S(S(T,2)+S(S(T,2),0)+1))) means and has growth rate \(\varphi(\omega^{\varepsilon_2+1},0)\)
  79. S(S(T,1)+S(S(T,2)+S(S(T,2)+S(S(T,2),0)+S(S(T,1)+S(S(T,2)+S(S(T,2)+S(S(T,2),0))))))) means and has growth rate \(\varphi(\varphi(\varepsilon_2,0),0)\)
  80. S(S(T,1)+S(S(T,2)+S(S(T,2)+S(S(T,2))))) means and has growth rate \(\Gamma_0\)
  81. S(S(T,1)+S(S(T,2)+S(S(T,2)+S(S(T,2))))+S(S(T,2))) means and has growth rate \(\varepsilon_{\Gamma_0+1}\)
  82. S(S(T,1)+S(S(T,2)+S(S(T,2)+S(S(T,2))))+S(S(T,2)+S(S(T,2),0))) means and has growth rate \(\varepsilon_{\Gamma_0+\varepsilon_2}\)
  83. S(S(T,1)+S(S(T,2)+S(S(T,2)+S(S(T,2))))+S(S(T,2)+S(S(T,2),0)+S(S(T,1)+S(S(T,2)+S(S(T,2)+S(S(T,2))))))) means and has growth rate \(\varepsilon_{\Gamma_02}\)
  84. S(S(T,1)+S(S(T,2)+S(S(T,2)+S(S(T,2))))+S(S(T,2)+S(S(T,2)))) means and has growth rate \(\zeta_{\Gamma_0+1}\)
  85. S(S(T,1)+S(S(T,2)+S(S(T,2)+S(S(T,2))))+S(S(T,2)+S(S(T,2))2)) means and has growth rate \(\varphi(3,\Gamma_0+1)\)
  86. S(S(T,1)+S(S(T,2)+S(S(T,2)+S(S(T,2))))+S(S(T,2)+S(S(T,2)+1))) means and has growth rate \(\varphi(\omega,\Gamma_0+1)\)
  87. S(S(T,1)+S(S(T,2)+S(S(T,2)+S(S(T,2))))+S(S(T,2)+S(S(T,2)+S(S(T,2),0)))) means and has growth rate \(\varphi(\varepsilon_2,\Gamma_0+1)\)
  88. S(S(T,1)+S(S(T,2)+S(S(T,2)+S(S(T,2))))+S(S(T,2)+S(S(T,2)+S(S(T,2),0)+S(S(T,1)+S(S(T,2)+S(S(T,2)+S(S(T,2)))))))) means and has growth rate \(\varphi(\Gamma_0,1)\)
  89. S(S(T,1)+S(S(T,2)+S(S(T,2)+S(S(T,2))))+S(S(T,2)+S(S(T,2)+S(S(T,2),0)+S(S(T,1)+S(S(T,2)+S(S(T,2)+S(S(T,2))))+S(S(T,2)+S(S(T,2)+S(S(T,2),0))))))) means and has growth rate \(\varphi(\varphi(\varepsilon_2,\Gamma_0+1),0)\)
  90. S(S(T,1)+S(S(T,2)+S(S(T,2)+S(S(T,2))))2) means and has growth rate \(\Gamma_1\)
  91. S(S(T,1)+S(S(T,2)+S(S(T,2)+S(S(T,2)))+1)) means and has growth rate \(\Gamma_\omega\)
  92. S(S(T,1)+S(S(T,2)+S(S(T,2)+S(S(T,2)))+S(S(T,2),0)+S(S(T,1)+S(S(T,2)+S(S(T,2)+S(S(T,2)))+1)))) means and has growth rate \(\Gamma_{\Gamma_\omega}\)
  93. S(S(T,1)+S(S(T,2)+S(S(T,2)+S(S(T,2)))+S(S(T,2)))) means and has growth rate \(\varphi(1,1,0)\)
  94. S(S(T,1)+S(S(T,2)+S(S(T,2)+S(S(T,2)))+S(S(T,2))2)) means and has growth rate \(\varphi(1,2,0)\)
  95. S(S(T,1)+S(S(T,2)+S(S(T,2)+S(S(T,2)))+S(S(T,2)+1))) means and has growth rate \(\varphi(1,\omega,0)\)
  96. S(S(T,1)+S(S(T,2)+S(S(T,2)+S(S(T,2)))+S(S(T,2)+S(S(T,2),0)+S(S(T,1)+S(S(T,2)+S(S(T,2)+S(S(T,2)))+S(S(T,2))))))) means and has growth rate \(\varphi(1,\varphi(1,1,0),0)\)
  97. S(S(T,1)+S(S(T,2)+S(S(T,2)+S(S(T,2)))2)) means and has growth rate \(\varphi(2,0,0)\)
  98. S(S(T,1)+S(S(T,2)+S(S(T,2)+S(S(T,2))+1))) means and has growth rate \(\varphi(\omega,0,0)\)
  99. S(S(T,1)+S(S(T,2)+S(S(T,2)+S(S(T,2))+S(S(T,2),0)+S(S(T,1)+S(S(T,2)+S(S(T,2)+S(S(T,2))+1)))))) means and has growth rate \(\varphi(\varphi(\omega,0,0),0,0)\)
  100. S(S(T,1)+S(S(T,2)+S(S(T,2)+S(S(T,2))2))) means and has growth rate \(\varphi(1,0,0,0)\)
  101. S(S(T,1)+S(S(T,2)+S(S(T,2)+S(S(T,2))3))) means and has growth rate \(\varphi(1,0,0,0,0)\)
  102. S(S(T,1)+S(S(T,2)+S(S(T,2)+S(S(T,2)+1)))) means and has growth rate SVO

Analysis using inaccessibility OCF

Here I use the inaccessibility OCF to continue the analysis. Second generation still keeps some analysis of first generation array notation, except some initial S(S(A,2),0), S(S(A,3),0) and S(S(A,3),1) things. I'll focus on the part different from first generation.

Up to a S(S(T,2)+S(S(T,3)+1))

  1. S(S(T,1)+S(S(T,2),0)) means and has growth rate \(\psi(\Omega3)\)
  2. S(S(T,1)+S(S(T,2))) means and has growth rate \(\psi(\Omega4)\)
  3. S(S(T,1)+S(S(T,2)+1)) means and has growth rate \(\psi(\Omega\omega)\)
  4. S(S(T,1)+S(S(T,2)+S(S(T,2)))) means and has growth rate \(\psi(\Omega^2)\)
  5. S(S(T,1)+S(S(T,2)+S(S(T,2))+1)) means and has growth rate \(\psi(\Omega^2\omega)\)
  6. S(S(T,1)+S(S(T,2)+S(S(T,2))2)) means and has growth rate \(\psi(\Omega^3)\)
  7. S(S(T,1)+S(S(T,2)+S(S(T,2)+1))) means and has growth rate \(\psi(\Omega^\omega)\)
  8. S(S(T,1)+S(S(T,2)+S(S(T,2)+S(S(T,2))))) means and has growth rate \(\psi(\Omega^\Omega)\)
  9. S(S(T,1)+S(S(T,2)+S(S(T,2)+S(S(T,2)+1)))) means and has growth rate \(\psi(\Omega^{\Omega^\omega})\)
  10. S(S(T,1)+S(S(T,2)+S(S(T,2)+S(S(T,2)+S(S(T,2)))))) means and has growth rate \(\psi(\Omega^{\Omega^\Omega})\)
  11. S(S(T,1)+S(S(T,2)+S(S(T,3),0))) means and has growth rate \(\psi(\Omega_2)\), i.e. BHO
  12. S(S(T,2)+S(S(T,3),0))) has growth rate \(\psi(\Omega_2)\)
  13. S(S(T,2)) means 1st admissible ordinal, or the countable analogue of \(\Omega\)
  14. S(S(T,3),0) means the countable analogue of \(\psi_{\Omega_2}(\Omega_2)=\varepsilon_{\Omega+1}\)
  15. S(S(T,2)+S(S(T,3),0)) means the countable analogue of \(\psi_{\Omega_2}(\Omega_2)\)
  16. S(S(T,1)+S(S(T,2)+S(S(T,3),0))+S(S(T,2))) means and has growth rate \(\psi(\Omega_2+\Omega)\)
  17. S(S(T,1)+S(S(T,2)+S(S(T,3),0))+S(S(T,2)+S(S(T,2)))) means and has growth rate \(\psi(\Omega_2+\Omega^\Omega)\)
  18. S(S(T,1)+S(S(T,2)+S(S(T,3),0))2) means and has growth rate \(\psi(\Omega_2+\psi_{\Omega_2}(\Omega_2))\)
  19. S(S(T,1)+S(S(T,2)+S(S(T,3),0))3) means and has growth rate \(\psi(\Omega_2+\psi_{\Omega_2}(\Omega_2)2)\)
  20. S(S(T,1)+S(S(T,2)+S(S(T,3),0)+1)) means and has growth rate \(\psi(\Omega_2+\psi_{\Omega_2}(\Omega_2+1))\)
  21. S(S(T,2)+S(S(T,3),0)+S(S(T,2))) has growth rate \(\psi(\Omega_2+\psi_{\Omega_2}(\Omega_2+\Omega))\)
  22. S(S(T,2)+S(S(T,3),0)+S(S(T,2)+S(S(T,3),0))) has growth rate \(\psi(\Omega_2+\psi_{\Omega_2}(\Omega_2+\psi_{\Omega_2}(\Omega_2)))\)
  23. S(S(T,2)+S(S(T,3),1)) has growth rate \(\psi(\Omega_22)\)
  24. S(S(T,2)+S(S(T,3),1)) means the countable analogue of \(\psi_{\Omega_2}(\Omega_22)=\varepsilon_{\Omega+2}\)
  25. S(S(T,1)+S(S(T,2)+S(S(T,3),1))+S(S(T,2))) means and has growth rate \(\psi(\Omega_22+\Omega)\)
  26. S(S(T,1)+S(S(T,2)+S(S(T,3),1))2) means and has growth rate \(\psi(\Omega_22+\psi_{\Omega_2}(\Omega_22))\)
  27. S(S(T,1)+S(S(T,2)+S(S(T,3),1)+1)) means and has growth rate \(\psi(\Omega_22+\psi_{\Omega_2}(\Omega_22+1))\)
  28. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3),1))) has growth rate \(\psi(\Omega_22+\psi_{\Omega_2}(\Omega_22+\psi_{\Omega_2}(\Omega_22)))\)
  29. S(S(T,2)+S(S(T,3),2),0) has growth rate \(\psi(\Omega_23)\)
  30. S(S(T,2)+S(S(T,3),2),0) means the countable analogue of \(\psi_{\Omega_2}(\Omega_23)=\varepsilon_{\Omega+3}\)
  31. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3),2),0)) means the countable analogue of \(\psi_{\Omega_2}(\Omega_23)\)
  32. S(T,3) has growth rate \(\psi(\Omega_24)\)
  33. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)))) means the countable analogue of \(\psi_{\Omega_2}(\Omega_24)\)
  34. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)))2) has growth rate \(\psi(\Omega_25)\)
  35. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+1)) has growth rate \(\psi(\Omega_2\omega)\)
  36. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+S(S(T,2),0))) has growth rate \(\psi(\Omega_2\psi(\Omega3))\)
  37. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+S(S(T,2)))) has growth rate \(\psi(\Omega_2\Omega)\)
  38. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+S(S(T,3),0))) has growth rate \(\psi(\Omega_2\psi_{\Omega_2}(\Omega_2))\)
  39. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+S(S(T,3),0)+S(S(T,2)+S(S(T,3),1)))) has growth rate \(\psi(\Omega_2\psi_{\Omega_2}(\Omega_22))\)
  40. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+S(S(T,3),0)+S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3),2),0)))) has growth rate \(\psi(\Omega_2\psi_{\Omega_2}(\Omega_23))\)
  41. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+S(S(T,3),0)+S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)))))) has growth rate \(\psi(\Omega_2\psi_{\Omega_2}(\Omega_24))\)
  42. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+S(S(T,3),0)+S(S(T,2)+S(S(T,3))))) has growth rate \(\psi(\Omega_2^2)\)
  43. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+S(S(T,3),0)+S(S(T,2)+S(S(T,3))))) means the countable analogue of \(\psi_{\Omega_2}(\Omega_2^2)\)
  44. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+S(S(T,3),0)+S(S(T,2)+S(S(T,3))))+S(S(T,2)+S(S(T,3),2),0)) means the countable analogue of \(\psi_{\Omega_2}(\Omega_2^2+\psi_{\Omega_2}(\Omega_23))\)
  45. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+S(S(T,3),0)+S(S(T,2)+S(S(T,3))))+S(S(T,2)+S(S(T,3)))) means the countable analogue of \(\psi_{\Omega_2}(\Omega_2^2+\Omega_2)\)
  46. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+S(S(T,3),0)+S(S(T,2)+S(S(T,3))))+S(S(T,2)+S(S(T,3))+S(S(T,2)))) means the countable analogue of \(\psi_{\Omega_2}(\Omega_2^2+\Omega_2\Omega)\)
  47. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+S(S(T,3),0)+S(S(T,2)+S(S(T,3))))+S(S(T,2)+S(S(T,3))+S(S(T,3),0)+S(S(T,2)+S(S(T,3),2),0))) means the countable analogue of \(\psi_{\Omega_2}(\Omega_2^2+\Omega_2\psi_{\Omega_2}(\Omega_23))\)
  48. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+S(S(T,3),0)+S(S(T,2)+S(S(T,3))))2) means the countable analogue of \(\psi_{\Omega_2}(\Omega_2^22)\)
  49. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+S(S(T,3),0)+S(S(T,2)+S(S(T,3)))+1)) means the countable analogue of \(\psi_{\Omega_2}(\Omega_2^2\omega)\)
  50. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+S(S(T,3),0)+S(S(T,2)+S(S(T,3)))+S(S(T,2)))) means the countable analogue of \(\psi_{\Omega_2}(\Omega_2^2\Omega)\)
  51. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+S(S(T,3),0)+S(S(T,2)+S(S(T,3)))+S(S(T,2)+S(S(T,3),2),0))) means the countable analogue of \(\psi_{\Omega_2}(\Omega_2^2\psi_{\Omega_2}(\Omega_23))\)
  52. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+S(S(T,3),0)+S(S(T,2)+S(S(T,3)))2)) means the countable analogue of \(\psi_{\Omega_2}(\Omega_2^3)\)
  53. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+S(S(T,3),0)+S(S(T,2)+S(S(T,3))+1))) means the countable analogue of \(\psi_{\Omega_2}(\Omega_2^\omega)\)
  54. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+S(S(T,3),0)+S(S(T,2)+S(S(T,3))+S(S(T,2))))) means the countable analogue of \(\psi_{\Omega_2}(\Omega_2^\Omega)\)
  55. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+S(S(T,3),0)+S(S(T,2)+S(S(T,3))+S(S(T,3),0)+S(S(T,2)+S(S(T,3),2),0)))) means the countable analogue of \(\psi_{\Omega_2}(\Omega_2^{\psi_{\Omega_2}(\Omega_23)})\)
  56. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+S(S(T,3),0)+S(S(T,2)+S(S(T,3))+S(S(T,3),0)+S(S(T,2)+S(S(T,3)))))) means the countable analogue of \(\psi_{\Omega_2}(\Omega_2^{\Omega_2})\)
  57. S(S(T,2)+S(S(T,3))) means 2nd admissible ordinal, or the countable analogue of \(\Omega_2\)
  58. S(S(T,2)+S(S(T,3))+S(S(T,3),1)) has growth rate \(\psi(\Omega_3)\)
  59. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+S(S(T,3),1))) means the countable analogue of \(\psi_{\Omega_2}(\Omega_3)\)
  60. S(S(T,2)+S(S(T,3))+S(S(T,3),1)) means the countable analogue of \(\psi_{\Omega_3}(\Omega_3)\)
  61. S(S(T,2)+S(S(T,3))+S(S(T,3),2),0) means the countable analogue of \(\psi_{\Omega_3}(\Omega_32)\)
  62. S(S(T,2)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+S(S(T,3),2),0)) means the countable analogue of \(\psi_{\Omega_3}(\Omega_32)\)
  63. S(S(T,2)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3))2)) means the countable analogue of \(\psi_{\Omega_3}(\Omega_33)\)
  64. S(S(T,2)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3))2)2) means the countable analogue of \(\psi_{\Omega_3}(\Omega_34)\)
  65. S(S(T,2)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3))2+1)) means the countable analogue of \(\psi_{\Omega_3}(\Omega_3\omega)\)
  66. S(S(T,2)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3))2+S(S(T,2)))) means the countable analogue of \(\psi_{\Omega_3}(\Omega_3\Omega)\)
  67. S(S(T,2)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3))2+S(S(T,3),0))) means the countable analogue of \(\psi_{\Omega_3}(\Omega_3\psi_{\Omega_2}(\Omega_2))\)
  68. S(S(T,2)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3))2+S(S(T,3),0)+S(S(T,2)+S(S(T,3))))) means the countable analogue of \(\psi_{\Omega_3}(\Omega_3\Omega_2)\)
  69. S(S(T,2)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3))2+S(S(T,3),0)+S(S(T,2)+S(S(T,3))+S(S(T,3),1)))) means the countable analogue of \(\psi_{\Omega_3}(\Omega_3\psi_{\Omega_3}(\Omega_3))\)
  70. S(S(T,2)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3))2+S(S(T,3),0)+S(S(T,2)+S(S(T,3))+S(S(T,3),2),0))) means the countable analogue of \(\psi_{\Omega_3}(\Omega_3\psi_{\Omega_3}(\Omega_32))\)
  71. S(S(T,2)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3))2+S(S(T,3),0)+S(S(T,2)+S(S(T,3))+S(S(T,3),2),0)+S(S(T,2)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3))2+S(S(T,3),0)+S(S(T,2)+S(S(T,3))+S(S(T,3),2),0))))) means the countable analogue of \(\psi_{\Omega_3}(\Omega_3\psi_{\Omega_3}(\Omega_3\psi_{\Omega_3}(\Omega_32)))\)
  72. S(S(T,2)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3))2+S(S(T,3),0)+S(S(T,2)+S(S(T,3))2))) means the countable analogue of \(\psi_{\Omega_3}(\Omega_3^2)\)
  73. S(S(T,2)+S(S(T,3))2) means 3rd admissible ordinal, or the countable analogue of \(\Omega_3\)
  74. S(S(T,2)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3))2+S(S(T,3),1))) means the countable analogue of \(\psi_{\Omega_3}(\Omega_4)\)
  75. S(S(T,2)+S(S(T,3))2+S(S(T,3),1)) means the countable analogue of \(\psi_{\Omega_4}(\Omega_4)\)
  76. S(S(T,2)+S(S(T,3))2+S(S(T,3),2),0) means the countable analogue of \(\psi_{\Omega_4}(\Omega_42)\)
  77. S(S(T,2)+S(S(T,3))2+S(S(T,3),1)+S(S(T,2)+S(S(T,3))2+S(S(T,3),2),0)) means the countable analogue of \(\psi_{\Omega_4}(\Omega_42)\)
  78. S(S(T,2)+S(S(T,3))2+S(S(T,3),1)+S(S(T,2)+S(S(T,3))3)) means the countable analogue of \(\psi_{\Omega_4}(\Omega_43)\)
  79. S(S(T,2)+S(S(T,3))2+S(S(T,3),1)+S(S(T,2)+S(S(T,3))3+1)) means the countable analogue of \(\psi_{\Omega_4}(\Omega_4\omega)\)
  80. S(S(T,2)+S(S(T,3))3) means 4th admissible ordinal, or the countable analogue of \(\Omega_4\)
  81. S(S(T,2)+S(S(T,3))3+S(S(T,3),1)) means the countable analogue of \(\psi_{\Omega_4}(\Omega_5)\)
  82. S(S(T,2)+S(S(T,3))3+S(S(T,3),2),0) means the countable analogue of \(\psi_{\Omega_4}(\Omega_52)\)
  83. S(S(T,2)+S(S(T,3))3+S(S(T,3),1)+S(S(T,2)+S(S(T,3))4)) means the countable analogue of \(\psi_{\Omega_5}(\Omega_53)\)
  84. S(S(T,2)+S(S(T,3))4) means 5th admissible ordinal, or the countable analogue of \(\Omega_5\)
  85. S(S(T,2)+S(S(T,3)+1)) has growth rate \(\psi(\Omega_\omega)\)

Beyond S(S(T,2)+S(S(T,3)+1))

Things toward S(S(T,2)+S(S(T,3)+1)+S(S(T,3),1)) is complicated, mainly in the part below S(S(T,2)+S(S(T,3)+1)+S(S(T,3),0)+S(S(T,2)+S(S(T,3)+1))).

  1. S(S(T,1)+S(S(T,2)+S(S(T,3)+1))) means and has growth rate \(\psi(\Omega_\omega)\)
  2. S(S(T,1)+S(S(T,2)+S(S(T,3)+1))+1) means and has growth rate \(\psi(\Omega_\omega+1)\)
  3. S(S(T,1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2),0)) means and has growth rate \(\psi(\Omega_\omega+\psi(\Omega3))\)
  4. S(S(T,1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2))) means and has growth rate \(\psi(\Omega_\omega+\Omega)\)
  5. S(S(T,1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3),0))) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_2))\)
  6. S(S(T,1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3),1))) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_22))\)
  7. S(S(T,1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))))) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_24))\)
  8. S(S(T,1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3)))) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_24))\)
  9. S(S(T,1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+1))) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_2\omega))\)
  10. S(S(T,1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+S(S(T,3),0)))) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_2\psi_{\Omega_2}(\Omega_2)))\)
  11. S(S(T,1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+S(S(T,3),0)+S(S(T,2)+S(S(T,3)))))) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_2^2))\)
  12. S(S(T,1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3))+S(S(T,3),0)+S(S(T,2)+S(S(T,3))))) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_2^2))\)
  13. S(S(T,1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3))+S(S(T,3),1))) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_3))\)
  14. S(S(T,1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+S(S(T,3),2),0))) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_32))\)
  15. S(S(T,1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3))2))) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_33))\)
  16. S(S(T,1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3))2)) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_33))\)
  17. S(S(T,1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3))2+S(S(T,3),0)+S(S(T,2)+S(S(T,3))2))) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_3^2))\)
  18. S(S(T,1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3))2+S(S(T,3),1))) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_4))\)
  19. S(S(T,1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3))3)) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_43))\)
  20. S(S(T,1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3))4)) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_53))\)
  21. S(S(T,1)+S(S(T,2)+S(S(T,3)+1))2) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega))\)
  22. S(S(T,1)+S(S(T,2)+S(S(T,3)+1))2+S(S(T,2))) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega)+\Omega)\)
  23. S(S(T,1)+S(S(T,2)+S(S(T,3)+1))2+S(S(T,2)+S(S(T,3),0))) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega)+\psi_{\Omega_2}(\Omega_2))\)
  24. S(S(T,1)+S(S(T,2)+S(S(T,3)+1))2+S(S(T,2)+S(S(T,3)))) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega)+\psi_{\Omega_2}(\Omega_24))\)
  25. S(S(T,1)+S(S(T,2)+S(S(T,3)+1))2+S(S(T,2)+S(S(T,3))2)) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega)+\psi_{\Omega_2}(\Omega_33))\)
  26. S(S(T,1)+S(S(T,2)+S(S(T,3)+1))2+S(S(T,2)+S(S(T,3))3)) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega)+\psi_{\Omega_2}(\Omega_43))\)
  27. S(S(T,1)+S(S(T,2)+S(S(T,3)+1))3) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega)2)\)
  28. S(S(T,1)+S(S(T,2)+S(S(T,3)+1))4) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega)3)\)
  29. S(S(T,1)+S(S(T,2)+S(S(T,3)+1)+1)) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega+1))\)
  30. S(S(T,1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,2),0))) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega+\psi(\Omega3)))\)
  31. S(S(T,1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,2),0)+S(S(T,1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,2),0))))) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega+\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega+\psi(\Omega3)))))\)
  32. S(S(T,1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,2)))) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega+\Omega))\)

In summary, S(S(T,2)+S(S(T,3)+1)+S(S(T,2))) has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega+\Omega))\). This is the n=0 step towards S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3)+1))). Next comes the n=1 step.

  1. S(S(T,1)+S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)))) means and has growth rate \(\psi(\Omega_\omega)\)
  2. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))) has growth rate \(\psi(\Omega_\omega)\)
  3. S(S(T,1)+S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)))+S(S(T,2))) means and has growth rate \(\psi(\Omega_\omega+\Omega)\)
  4. S(S(T,1)+S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)))+S(S(T,2)+S(S(T,3),1))) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_22))\)
  5. S(S(T,1)+S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)))+S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))))) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_24))\)
  6. S(S(T,1)+S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)))+S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+S(S(T,3),1)))) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_3))\)
  7. S(S(T,1)+S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)))+S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))2))) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_33))\)
  8. S(S(T,1)+S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)))+S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))3))) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_43))\)
  9. S(S(T,1)+S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)))2) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega))\)
  10. S(S(T,1)+S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)))3) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega)2)\)
  11. S(S(T,1)+S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))+1)) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega+1))\)
  12. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))) means \(\psi_{\Omega_2}(\Omega_\omega)\)
  13. S(S(T,1)+S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)))) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega+\Omega))\)
  14. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2))) has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega+\Omega))\)
  15. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+1)) has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega+\Omega\omega))\)
  16. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3),2),0)) has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega+\psi_{\Omega_2}(\Omega_23)))\)
  17. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3),2),0)+S(S(T,2))) has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega+\psi_{\Omega_2}(\Omega_23)+\Omega))\)
  18. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3),2),0)+S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)))) has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega)))\)
  19. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3)))) has growth rate \(\psi(\Omega_\omega+\Omega_2)\)
  20. S(S(T,1)+S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3))))) means and has growth rate \(\psi(\Omega_\omega+\Omega_2)\)
  21. S(S(T,1)+S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3))))+S(S(T,2))) means and has growth rate \(\psi(\Omega_\omega+\Omega_2+\Omega)\)
  22. S(S(T,1)+S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3))))+S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)))) means and has growth rate \(\psi(\Omega_\omega+\Omega_2+\psi_{\Omega_2}(\Omega_\omega))\)
  23. S(S(T,1)+S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3))))2) means and has growth rate \(\psi(\Omega_\omega+\Omega_2+\psi_{\Omega_2}(\Omega_\omega+\Omega_2))\)
  24. S(S(T,1)+S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3)))+1)) means and has growth rate \(\psi(\Omega_\omega+\Omega_2+\psi_{\Omega_2}(\Omega_\omega+\Omega_2+1))\)
  25. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3)))+S(S(T,2))) has growth rate \(\psi(\Omega_\omega+\Omega_2+\psi_{\Omega_2}(\Omega_\omega+\Omega_2+\Omega))\)
  26. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3)))+S(S(T,2)+S(S(T,3),2),0)) has growth rate \(\psi(\Omega_\omega+\Omega_2+\psi_{\Omega_2}(\Omega_\omega+\Omega_2+\psi_{\Omega_2}(\Omega_23)))\)
  27. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3)))2) has growth rate \(\psi(\Omega_\omega+\Omega_22)\)
  28. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3))+1)) has growth rate \(\psi(\Omega_\omega+\Omega_2\omega)\)
  29. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3))+S(S(T,3),1))) has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_3}(\Omega_3))\)
  30. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3))2))) has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_3}(\Omega_33))\)
  31. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3))2)) has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_3}(\Omega_33))\)
  32. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3))2+1))) has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_3}(\Omega_3\omega))\)
  33. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3))2+S(S(T,3),0)+S(S(T,2)+S(S(T,3))2)))) has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_3}(\Omega_3^2))\)
  34. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3))2+S(S(T,3),1)))) has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_3}(\Omega_4))\)
  35. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3))2+S(S(T,3),1))) has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_3}(\Omega_4))\)
  36. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3))3)) has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_3}(\Omega_43))\)
  37. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3))4)) has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_3}(\Omega_53))\)
  38. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))2) has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_3}(\Omega_\omega))\)
  39. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)+1)) has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_3}(\Omega_\omega+1))\)
  40. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,2)))) has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_3}(\Omega_\omega+\Omega))\)
  41. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3),2),0))) has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_3}(\Omega_\omega+\psi_{\Omega_2}(\Omega_23)))\)
  42. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3),2),0)+S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))2))) has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_3}(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega+\psi_{\Omega_3}(\Omega_\omega))))\)
  43. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3))))) has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_3}(\Omega_\omega+\Omega_2))\)

In summary, S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3)))) has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_3}(\Omega_\omega+\Omega_2))\). This is the n=1 step towards S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3)+1))). Next comes the n=2 step.

  1. S(S(T,1)+S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))))) means and has growth rate \(\psi(\Omega_\omega)\)
  2. S(S(T,1)+S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))))+S(S(T,2))) means and has growth rate \(\psi(\Omega_\omega+\Omega)\)
  3. S(S(T,1)+S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))))2) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega))\)
  4. S(S(T,1)+S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)))+1)) means and has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_2}(\Omega_\omega+1))\)
  5. S(S(T,1)+S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)))+S(S(T,2)+S(S(T,3))))) means and has growth rate \(\psi(\Omega_\omega+\Omega_2)\)
  6. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)))+S(S(T,2)+S(S(T,3)))) has growth rate \(\psi(\Omega_\omega+\Omega_2)\)
  7. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)))2) has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_3}(\Omega_\omega))\)
  8. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))+1)) has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_3}(\Omega_\omega+1))\)
  9. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3))))) has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_3}(\Omega_\omega+\Omega_2))\)
  10. S(S(T,2)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3)))) has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_3}(\Omega_\omega+\Omega_2))\)
  11. S(S(T,2)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3))2)) has growth rate \(\psi(\Omega_\omega+\Omega_3)\)
  12. S(S(T,2)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3))3)) has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_4}(\Omega_43))\)
  13. S(S(T,2)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3))4)) has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_4}(\Omega_53))\)
  14. S(S(T,2)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1))2) has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_4}(\Omega_\omega))\)
  15. S(S(T,2)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)+1)) has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_4}(\Omega_\omega+1))\)
  16. S(S(T,2)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3))))) has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_4}(\Omega_\omega+\Omega_2))\)
  17. S(S(T,2)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3))2))) has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_4}(\Omega_\omega+\Omega_3))\)

In summary, S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3))2)) has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_4}(\Omega_\omega+\Omega_3))\). This is the n=2 step towards S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3)+1))). And further,

  1. S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3))3)) has growth rate \(\psi(\Omega_\omega+\psi_{\Omega_5}(\Omega_\omega+\Omega_4))\)
  2. S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3)+1))) has growth rate \(\psi(\Omega_\omega2)\)
  3. S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2))) has growth rate \(\psi(\Omega_\omega2+\psi_{\Omega_2}(\Omega_\omega2+\Omega))\)
  4. S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3)))) has growth rate \(\psi(\Omega_\omega2+\psi_{\Omega_3}(\Omega_\omega2+\Omega_2))\)
  5. S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3)+1))+S(S(T,2)+S(S(T,3))2)) has growth rate \(\psi(\Omega_\omega2+\psi_{\Omega_4}(\Omega_\omega2+\Omega_3))\)
  6. S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3)+1))2) has growth rate \(\psi(\Omega_\omega3)\)
  7. S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3)+1))3) has growth rate \(\psi(\Omega_\omega4)\)
  8. S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3)+1)+1)) has growth rate \(\psi(\Omega_\omega\omega)\)
  9. S(S(T,1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,2),0)))) means and has growth rate \(\psi(\Omega_\omega\psi(\Omega3))\)
  10. S(S(T,1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,2),0)+S(S(T,1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3)+1))))))) means and has growth rate \(\psi(\Omega_\omega\psi(\Omega_\omega2))\)
  11. S(S(T,1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,2))))) means and has growth rate \(\psi(\Omega_\omega\Omega)\)
  12. S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,2)))) has growth rate \(\psi(\Omega_\omega\Omega)\)
  13. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,2))))) has growth rate \(\psi(\Omega_\omega\Omega)\)
  14. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3),2),0)+S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3)+1))))))) has growth rate \(\psi(\Omega_\omega\psi_{\Omega_2}(\Omega_\omega2))\)
  15. S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3)))))) has growth rate \(\psi(\Omega_\omega\Omega_2)\)
  16. S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3))))) has growth rate \(\psi(\Omega_\omega\Omega_2)\)
  17. S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3))2))) has growth rate \(\psi(\Omega_\omega\Omega_3)\)
  18. S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3)+1)))) has growth rate \(\psi(\Omega_\omega^2)\)
  19. S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3)+1))))) has growth rate \(\psi(\Omega_\omega^{\Omega_\omega})\)
  20. S(S(T,2)+S(S(T,3)+1)) means 1st limit of admissible ordinals, or the countable analogue of \(\Omega_\omega\)
  21. S(S(T,2)+S(S(T,3)+1)+S(S(T,3),1)) has growth rate \(\psi(\Omega_{\omega+1})\)
  22. S(S(T,2)+S(S(T,3)+1)+S(S(T,3),1)) means the countable analogue of \(\psi_{\Omega_{\omega+1}}(\Omega_{\omega+1})=\varepsilon_{\Omega_\omega+1}\)
  23. S(S(T,2)+S(S(T,3)+1)+S(S(T,3),2),0) means the countable analogue of \(\psi_{\Omega_{\omega+1}}(\Omega_{\omega+1}2)\)
  24. S(S(T,2)+S(S(T,3)+1)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,3),2),0)) means the countable analogue of \(\psi_{\Omega_{\omega+1}}(\Omega_{\omega+1}2)\)
  25. S(S(T,2)+S(S(T,3)+1)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,3)))) means the countable analogue of \(\psi_{\Omega_{\omega+1}}(\Omega_{\omega+1}3)\)
  26. S(S(T,2)+S(S(T,3)+1)+S(S(T,3))) has growth rate \(\psi(\Omega_{\omega+1}3)\)
  27. S(S(T,2)+S(S(T,3)+1)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,3))+1)) means the countable analogue of \(\psi_{\Omega_{\omega+1}}(\Omega_{\omega+1}\omega)\)
  28. S(S(T,2)+S(S(T,3)+1)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,3))+S(S(T,2)+S(S(T,3)+1)+S(S(T,3))))) means the countable analogue of \(\psi_{\Omega_{\omega+1}}(\Omega_{\omega+1}^2)\)
  29. S(S(T,2)+S(S(T,3)+1)+S(S(T,3))) means the countable analogue of \(\Omega_{\omega+1}\)
  30. S(S(T,2)+S(S(T,3)+1)+S(S(T,3))+S(S(T,3),1)) means the countable analogue of \(\psi_{\Omega_{\omega+2}}(\Omega_{\omega+2})\)
  31. S(S(T,2)+S(S(T,3)+1)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,3))2)) means the countable analogue of \(\psi_{\Omega_{\omega+2}}(\Omega_{\omega+2}3)\)
  32. S(S(T,2)+S(S(T,3)+1)+S(S(T,3))2) has growth rate \(\psi(\Omega_{\omega+2}3)\)
  33. S(S(T,2)+S(S(T,3)+1)+S(S(T,3))3) has growth rate \(\psi(\Omega_{\omega+3}3)\)
  34. S(S(T,2)+S(S(T,3)+1)2) has growth rate \(\psi(\Omega_{\omega2})\)
  35. S(S(T,2)+S(S(T,3)+1)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)2)+S(S(T,2)+S(S(T,3)+1))) has growth rate \(\psi(\Omega_{\omega2}+\Omega_\omega)\)
  36. S(S(T,2)+S(S(T,3)+1)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)2)+S(S(T,2)+S(S(T,3)+1)+S(S(T,3),1))) has growth rate \(\psi(\Omega_{\omega2}+\psi_{\Omega_{\omega+1}}(\Omega_{\omega+1}))\)
  37. S(S(T,2)+S(S(T,3)+1)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)2)+S(S(T,2)+S(S(T,3)+1)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,3))))) has growth rate \(\psi(\Omega_{\omega2}+\psi_{\Omega_{\omega+1}}(\Omega_{\omega+1}3))\)
  38. S(S(T,2)+S(S(T,3)+1)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)2)+S(S(T,2)+S(S(T,3)+1)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,3))2))) has growth rate \(\psi(\Omega_{\omega2}+\psi_{\Omega_{\omega+1}}(\Omega_{\omega+2}3))\)
  39. S(S(T,2)+S(S(T,3)+1)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)2)+S(S(T,2)+S(S(T,3)+1)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)2))) has growth rate \(\psi(\Omega_{\omega2}+\psi_{\Omega_{\omega+1}}(\Omega_{\omega2}))\)
  40. S(S(T,2)+S(S(T,3)+1)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)2)+S(S(T,2)+S(S(T,3)+1)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)2)+1)) has growth rate \(\psi(\Omega_{\omega2}+\psi_{\Omega_{\omega+1}}(\Omega_{\omega2}+1))\)
  41. S(S(T,2)+S(S(T,3)+1)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)2)+S(S(T,2)+S(S(T,3)+1)+S(S(T,3)))) has growth rate \(\psi(\Omega_{\omega2}+\Omega_{\omega+1})\)
  42. S(S(T,2)+S(S(T,3)+1)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)2)+S(S(T,2)+S(S(T,3)+1)+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)+S(S(T,3))2))) has growth rate \(\psi(\Omega_{\omega2}+\psi_{\Omega_{\omega+2}}(\Omega_{\omega+2}3))\)
  43. S(S(T,2)+S(S(T,3)+1)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)2)+S(S(T,2)+S(S(T,3)+1)+S(S(T,3))2)) has growth rate \(\psi(\Omega_{\omega2}+\psi_{\Omega_{\omega+2}}(\Omega_{\omega+2}3))\)
  44. S(S(T,2)+S(S(T,3)+1)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)2)+S(S(T,2)+S(S(T,3)+1)+S(S(T,3))3)) has growth rate \(\psi(\Omega_{\omega2}+\psi_{\Omega_{\omega+2}}(\Omega_{\omega+3}3))\)
  45. S(S(T,2)+S(S(T,3)+1)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)2)2) has growth rate \(\psi(\Omega_{\omega2}+\psi_{\Omega_{\omega+2}}(\Omega_{\omega2}))\)
  46. S(S(T,2)+S(S(T,3)+1)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)2+1)) has growth rate \(\psi(\Omega_{\omega2}+\psi_{\Omega_{\omega+2}}(\Omega_{\omega2}+1))\)
  47. S(S(T,2)+S(S(T,3)+1)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)2+S(S(T,2)+S(S(T,3)+1)+S(S(T,3))))) has growth rate \(\psi(\Omega_{\omega2}+\psi_{\Omega_{\omega+2}}(\Omega_{\omega2}+\Omega_{\omega+1}))\)
  48. S(S(T,2)+S(S(T,3)+1)2+S(S(T,2)+S(S(T,3)+1)+S(S(T,3)))) has growth rate \(\psi(\Omega_{\omega2}+\psi_{\Omega_{\omega+2}}(\Omega_{\omega2}+\Omega_{\omega+1}))\)
  49. S(S(T,2)+S(S(T,3)+1)2+S(S(T,2)+S(S(T,3)+1)+S(S(T,3))2)) has growth rate \(\psi(\Omega_{\omega2}+\psi_{\Omega_{\omega+3}}(\Omega_{\omega2}+\Omega_{\omega+2}))\)
  50. S(S(T,2)+S(S(T,3)+1)2+S(S(T,2)+S(S(T,3)+1)2)) has growth rate \(\psi(\Omega_{\omega2}2)\)
  51. S(S(T,2)+S(S(T,3)+1)2+S(S(T,2)+S(S(T,3)+1)2)+S(S(T,2)+S(S(T,3)+1)+S(S(T,3)))) has growth rate \(\psi(\Omega_{\omega2}2+\psi_{\Omega_{\omega+2}}(\Omega_{\omega2}2+\Omega_{\omega+1}))\)
  52. S(S(T,2)+S(S(T,3)+1)2+S(S(T,2)+S(S(T,3)+1)2)2) has growth rate \(\psi(\Omega_{\omega2}3)\)
  53. S(S(T,2)+S(S(T,3)+1)2+S(S(T,2)+S(S(T,3)+1)2+1)) has growth rate \(\psi(\Omega_{\omega2}\omega)\)
  54. S(S(T,2)+S(S(T,3)+1)2+S(S(T,2)+S(S(T,3)+1)2+S(S(T,2)+S(S(T,3))))) has growth rate \(\psi(\Omega_{\omega2}\Omega_2)\)
  55. S(S(T,2)+S(S(T,3)+1)2+S(S(T,2)+S(S(T,3)+1)2+S(S(T,2)+S(S(T,3)+1)))) has growth rate \(\psi(\Omega_{\omega2}\Omega_\omega)\)
  56. S(S(T,2)+S(S(T,3)+1)2+S(S(T,2)+S(S(T,3)+1)2+S(S(T,2)+S(S(T,3)+1)+S(S(T,3))))) has growth rate \(\psi(\Omega_{\omega2}\Omega_{\omega+1})\)
  57. S(S(T,2)+S(S(T,3)+1)2+S(S(T,2)+S(S(T,3)+1)2+S(S(T,2)+S(S(T,3)+1)2))) has growth rate \(\psi(\Omega_{\omega2}^2)\)
  58. S(S(T,2)+S(S(T,3)+1)2+S(S(T,2)+S(S(T,3)+1)2+S(S(T,2)+S(S(T,3)+1)2+S(S(T,2)+S(S(T,3)+1)2)))) has growth rate \(\psi(\Omega_{\omega2}^{\Omega_{\omega2}})\)
  59. S(S(T,2)+S(S(T,3)+1)2) means 2nd limit of admissible ordinals, or the countable analogue of \(\Omega_{\omega2}\)
  60. S(S(T,2)+S(S(T,3)+1)2+S(S(T,3),1)) has growth rate \(\psi(\Omega_{\omega2+1})\)
  61. S(S(T,2)+S(S(T,3)+1)2+S(S(T,3))) has growth rate \(\psi(\Omega_{\omega2+1}3)\)
  62. S(S(T,2)+S(S(T,3)+1)2+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)2+S(S(T,3))+1)) has growth rate \(\psi(\Omega_{\omega2+1}\omega)\)
  63. S(S(T,2)+S(S(T,3)+1)2+S(S(T,3))+S(S(T,3),1)) has growth rate \(\psi(\Omega_{\omega2+2})\)
  64. S(S(T,2)+S(S(T,3)+1)2+S(S(T,3))2) has growth rate \(\psi(\Omega_{\omega2+2}3)\)
  65. S(S(T,2)+S(S(T,3)+1)2+S(S(T,3))3) has growth rate \(\psi(\Omega_{\omega2+3}3)\)
  66. S(S(T,2)+S(S(T,3)+1)3) has growth rate \(\psi(\Omega_{\omega3})\)
  67. S(S(T,2)+S(S(T,3)+1)3+S(S(T,3))) has growth rate \(\psi(\Omega_{\omega3+1}3)\)
  68. S(S(T,2)+S(S(T,3)+1)3+S(S(T,3))2) has growth rate \(\psi(\Omega_{\omega3+2}3)\)
  69. S(S(T,2)+S(S(T,3)+1)4) has growth rate \(\psi(\Omega_{\omega4})\)
  70. S(S(T,2)+S(S(T,3)+2)) has growth rate \(\psi(\Omega_{\omega^2})\)
  71. S(S(T,2)+S(S(T,3)+2)) means 1st limit of limits of admissibles, or the countable analogue of \(\Omega_{\omega^2}\)
  72. S(S(T,1)+S(S(T,2)+S(S(T,3)+2))) means and has growth rate \(\psi(\Omega_{\omega^2})\)
  73. S(S(T,1)+S(S(T,2)+S(S(T,3)+S(S(T,2),0)))) means and has growth rate \(\psi(\Omega_{\psi(\Omega3)})\)
  74. S(S(T,1)+S(S(T,2)+S(S(T,3)+S(S(T,2),0)+1))) means and has growth rate \(\psi(\Omega_{\psi(\Omega3+1)})\)
  75. S(S(T,1)+S(S(T,2)+S(S(T,3)+S(S(T,2),0)+S(S(T,1)+S(S(T,2)))))) means and has growth rate \(\psi(\Omega_{\psi(\Omega4)})\)
  76. S(S(T,1)+S(S(T,2)+S(S(T,3)+S(S(T,2),0)+S(S(T,1)+S(S(T,2)+S(S(T,3))))))) means and has growth rate \(\psi(\Omega_{\psi(\Omega_24)})\)
  77. S(S(T,1)+S(S(T,2)+S(S(T,3)+S(S(T,2),0)+S(S(T,1)+S(S(T,2)+S(S(T,3)+1)))))) means and has growth rate \(\psi(\Omega_{\psi(\Omega_\omega)})\)
  78. S(S(T,1)+S(S(T,2)+S(S(T,3)+S(S(T,2),0)+S(S(T,1)+S(S(T,2)+S(S(T,3)+S(S(T,2),0))))))) means and has growth rate \(\psi(\Omega_{\psi(\Omega_{\psi(\Omega3)})})\)
  79. S(S(T,1)+S(S(T,2)+S(S(T,3)+S(S(T,2))))) means and has growth rate \(\psi(\Omega_\Omega)\)
  80. S(S(T,2)+S(S(T,3)+S(S(T,2)))) has growth rate \(\psi(\Omega_\Omega)\)
  81. S(S(T,2)+S(S(T,3)+S(S(T,2)))) means the countable analogue of \(\Omega_\Omega\)
  82. S(S(T,2)+S(S(T,3)+S(S(T,2))+1)) means the countable analogue of \(\Omega_{\omega^{\Omega+1}}\)
  83. S(S(T,2)+S(S(T,3)+S(S(T,2)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)))))) means the countable analogue of \(\Omega_{\psi_{\Omega_2}(\Omega_\omega)}\)
  84. S(S(T,2)+S(S(T,3)+S(S(T,2)+S(S(T,3))))) means the countable analogue of \(\Omega_{\Omega_2}\)
  85. S(S(T,2)+S(S(T,3)+S(S(T,2)+S(S(T,3))2))) means the countable analogue of \(\Omega_{\Omega_3}\)
  86. S(S(T,2)+S(S(T,3)+S(S(T,2)+S(S(T,3)+1)))) means the countable analogue of \(\Omega_{\Omega_\omega}\)
  87. S(S(T,2)+S(S(T,3)+S(S(T,2)+S(S(T,3)+S(S(T,2)+S(S(T,3))))))) means the countable analogue of \(\Omega_{\Omega_{\Omega_2}}\)
  88. S(S(T,2)+S(S(T,3)+S(S(T,3),1))) has growth rate \(\psi(\psi_I(0))\)
  89. S(S(T,1)+S(S(T,2)+S(S(T,3)+S(S(T,3),1)))+S(S(T,2))) means and has growth rate \(\psi(\psi_I(0)+\Omega)\)
  90. S(S(T,1)+S(S(T,2)+S(S(T,3)+S(S(T,3),1)))+S(S(T,2)+S(S(T,3)))) means and has growth rate \(\psi(\psi_I(0)+\psi_{\Omega_2}(\Omega_24))\)
  91. S(S(T,1)+S(S(T,2)+S(S(T,3)+S(S(T,3),1)))+S(S(T,2)+S(S(T,3))2)) means and has growth rate \(\psi(\psi_I(0)+\psi_{\Omega_2}(\Omega_33))\)
  92. S(S(T,1)+S(S(T,2)+S(S(T,3)+S(S(T,3),1)))+S(S(T,2)+S(S(T,3)+1))) means and has growth rate \(\psi(\psi_I(0)+\psi_{\Omega_2}(\Omega_\omega))\)
  93. S(S(T,1)+S(S(T,2)+S(S(T,3)+S(S(T,3),1)))+S(S(T,2)+S(S(T,3)+S(S(T,2)+S(S(T,3)))))) means and has growth rate \(\psi(\psi_I(0)+\psi_{\Omega_2}(\Omega_{\Omega_2}))\)
  94. S(S(T,1)+S(S(T,2)+S(S(T,3)+S(S(T,3),1)))2) means and has growth rate \(\psi(\psi_I(0)+\psi_{\Omega_2}(\psi_I(0)))\)
  95. S(S(T,1)+S(S(T,2)+S(S(T,3)+S(S(T,3),1))+1)) means and has growth rate \(\psi(\psi_I(0)+\psi_{\Omega_2}(\psi_I(0)+1))\)
  96. S(S(T,1)+S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)))) means and has growth rate \(\psi(\psi_I(0)+\psi_{\Omega_2}(\psi_I(0)+\Omega))\)
  97. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2))) has growth rate \(\psi(\psi_I(0)+\psi_{\Omega_2}(\psi_I(0)+\Omega))\)
  98. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)))) has growth rate \(\psi(\psi_I(0)+\psi_{\Omega_3}(\psi_I(0)+\Omega_2))\)
  99. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3))2)) has growth rate \(\psi(\psi_I(0)+\psi_{\Omega_4}(\psi_I(0)+\Omega_3))\)
  100. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+1))) has growth rate \(\psi(\psi_I(0)+\Omega_\omega)\)
  101. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+1)+S(S(T,3)))) has growth rate \(\psi(\psi_I(0)+\psi_{\Omega_{\omega+2}}(\psi_I(0)+\Omega_{\omega+1}))\)
  102. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+2))) has growth rate \(\psi(\psi_I(0)+\Omega_{\omega^2})\)
  103. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,2)+S(S(T,3)))))) has growth rate \(\psi(\psi_I(0)+\Omega_{\Omega_2})\)
  104. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3),1)))) has growth rate \(\psi(\psi_I(0)2)\)
  105. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3),1)))+S(S(T,2)+S(S(T,3)))) has growth rate \(\psi(\psi_I(0)2+\psi_{\Omega_3}(\psi_I(0)2+\Omega_2))\)
  106. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3),1)))+S(S(T,2)+S(S(T,3)+1))) has growth rate \(\psi(\psi_I(0)2+\Omega_\omega)\)
  107. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3),1)))+S(S(T,2)+S(S(T,3)+S(S(T,2)+S(S(T,3)))))) has growth rate \(\psi(\psi_I(0)2+\Omega_{\Omega_2})\)
  108. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3),1)))2) has growth rate \(\psi(\psi_I(0)3)\)
  109. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3),1))+1)) has growth rate \(\psi(\psi_I(0)\omega)\)
  110. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)))) has growth rate \(\psi(\psi_I(0)\Omega)\)
  111. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3))))) has growth rate \(\psi(\psi_I(0)\Omega_2)\)
  112. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3),1))))) has growth rate \(\psi(\psi_I(0)^2)\)
  113. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3),1))+1))) has growth rate \(\psi(\psi_I(0)^\omega)\)
  114. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3),1)))))) has growth rate \(\psi(\psi_I(0)^{\psi_I(0)})\)
  115. S(S(T,2)+S(S(T,3)+S(S(T,3),1))) means 1st (limit of)(1,0) admissibles, or the countable analogue of \(\psi_I(0)\)
  116. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3),1)) has growth rate \(\psi(\Omega_{\psi_I(0)+1})\)
  117. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),1)))) has growth rate \(\psi(\Omega_{\psi_I(0)+1}+\psi_I(0))\)
  118. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3),1))) has growth rate \(\psi(\Omega_{\psi_I(0)+1}+\psi_{\Omega_{\psi_I(0)+1}}(\Omega_{\psi_I(0)+1}))\)
  119. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3),1)) means the countable analogue of \(\psi_{\Omega_{\psi_I(0)+1}}(\Omega_{\psi_I(0)+1})\)
  120. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3),2),0) has growth rate \(\psi(\Omega_{\psi_I(0)+1}2)\)
  121. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3),2),0) means the countable analogue of \(\psi_{\Omega_{\psi_I(0)+1}}(\Omega_{\psi_I(0)+1}2)\)
  122. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)))) means the countable analogue of \(\psi_{\Omega_{\psi_I(0)+1}}(\Omega_{\psi_I(0)+1}3)\)
  123. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3))+1)) means the countable analogue of \(\psi_{\Omega_{\psi_I(0)+1}}(\Omega_{\psi_I(0)+1}\omega)\)
  124. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3))) means the countable analogue of \(\Omega_{\psi_I(0)+1}\)
  125. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3))+S(S(T,3),1)) means the countable analogue of \(\psi_{\Omega_{\psi_I(0)+2}}(\Omega_{\psi_I(0)+2})\)
  126. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3))+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3))2)) means the countable analogue of \(\psi_{\Omega_{\psi_I(0)+2}}(\Omega_{\psi_I(0)+2}3)\)
  127. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3))2) means the countable analogue of \(\Omega_{\psi_I(0)+2}\)
  128. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+1)) has growth rate \(\psi(\Omega_{\psi_I(0)+\omega})\)
  129. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+1)) means the countable analogue of \(\Omega_{\psi_I(0)+\omega}\)
  130. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+S(S(T,2)))) means the countable analogue of \(\Omega_{\psi_I(0)+\Omega}\)
  131. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+S(S(T,2)+S(S(T,3))))) means the countable analogue of \(\Omega_{\psi_I(0)+\Omega_2}\)
  132. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+S(S(T,2)+S(S(T,3)+S(S(T,2)+S(S(T,3))))))) means the countable analogue of \(\Omega_{\psi_I(0)+\Omega_{\Omega_2}}\)
  133. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+S(S(T,2)+S(S(T,3)+S(S(T,3),1))))) means the countable analogue of \(\Omega_{\psi_I(0)2}\)
  134. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+S(S(T,2)+S(S(T,3)+S(S(T,3),1))))+S(S(T,3),1)) means the countable analogue of \(\psi_{\Omega_{\psi_I(0)2+1}}(\Omega_{\psi_I(0)2+1})\)
  135. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+S(S(T,2)+S(S(T,3)+S(S(T,3),1))))+S(S(T,3))) means the countable analogue of \(\Omega_{\psi_I(0)2+1}\)
  136. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+S(S(T,2)+S(S(T,3)+S(S(T,3),1))))+S(S(T,3)+1)) means the countable analogue of \(\Omega_{\psi_I(0)2+\omega}\)
  137. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+S(S(T,2)+S(S(T,3)+S(S(T,3),1))))+S(S(T,3)+S(S(T,2)+S(S(T,3))))) means the countable analogue of \(\Omega_{\psi_I(0)2+\Omega_2}\)
  138. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+S(S(T,2)+S(S(T,3)+S(S(T,3),1))))2) means the countable analogue of \(\Omega_{\psi_I(0)3}\)
  139. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+S(S(T,2)+S(S(T,3)+S(S(T,3),1)))+1)) means the countable analogue of \(\Omega_{\psi_I(0)\omega}\)
  140. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+S(S(T,2)+S(S(T,3)+S(S(T,3),1)))+S(S(T,2)+S(S(T,3))))) means the countable analogue of \(\Omega_{\psi_I(0)\Omega_2}\)
  141. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+S(S(T,2)+S(S(T,3)+S(S(T,3),1)))2)) means the countable analogue of \(\Omega_{\psi_I(0)^2}\)
  142. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+S(S(T,2)+S(S(T,3)+S(S(T,3),1))+1))) means the countable analogue of \(\Omega_{\psi_I(0)^\omega}\)
  143. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3),1)))))) means the countable analogue of \(\Omega_{\psi_I(0)^{\psi_I(0)}}\)
  144. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3),1)))) means the countable analogue of \(\Omega_{\psi_{\Omega_{\psi_I(0)+1}}(\Omega_{\psi_I(0)+1})}\)
  145. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3),1)) ))) means the countable analogue of \(\psi_{\Omega_{\psi_I(0)+1}}(\Omega_{\psi_{\Omega_{\psi_I(0)+1}}(\Omega_{\psi_I(0)+1})})\)
  146. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)))) ))) means the countable analogue of \(\psi_{\Omega_{\psi_I(0)+1}}(\Omega_{\psi_{\Omega_{\psi_I(0)+1}}(\Omega_{\psi_I(0)+1}3)})\)
  147. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3))2)) ))) means the countable analogue of \(\psi_{\Omega_{\psi_I(0)+1}}(\Omega_{\psi_{\Omega_{\psi_I(0)+1}}(\Omega_{\psi_I(0)+2}3)})\)
  148. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+1))) ))) means the countable analogue of \(\psi_{\Omega_{\psi_I(0)+1}}(\Omega_{\psi_{\Omega_{\psi_I(0)+1}}(\Omega_{\psi_I(0)+\omega})})\)
  149. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1))) ))) ))) means the countable analogue of \(\psi_{\Omega_{\psi_I(0)+1}}(\Omega_{\psi_{\Omega_{\psi_I(0)+1}}(\Omega_{\psi_I(0)2})})\)
  150. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3))) ))) means the countable analogue of \(\psi_{\Omega_{\psi_I(0)+1}}(\Omega_{\Omega_{\psi_I(0)+1}})\)
  151. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3))) )) means the countable analogue of \(\Omega_{\Omega_{\psi_I(0)+1}}\)
  152. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3))+S(S(T,3),1)) )) means the countable analogue of \(\Omega_{\psi_{\Omega_{\psi_I(0)+2}}(\Omega_{\psi_I(0)+2})}\)
  153. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3))2) )) means the countable analogue of \(\Omega_{\Omega_{\psi_I(0)+2}}\)
  154. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+1)) )) means the countable analogue of \(\Omega_{\Omega_{\psi_I(0)+\omega}}\)
  155. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+ S(S(T,2)+S(S(T,3))) )) )) means the countable analogue of \(\Omega_{\Omega_{\psi_I(0)+\Omega_2}}\)
  156. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1))) )) )) means the countable analogue of \(\Omega_{\Omega_{\psi_I(0)2}}\)
  157. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3))) )) )) means the countable analogue of \(\Omega_{\Omega_{\Omega_{\psi_I(0)+1}}}\)
  158. S(S(T,2)+S(S(T,3)+S(S(T,3),1))2) has growth rate \(\psi(\psi_I(1))\)
  159. S(S(T,2)+S(S(T,3)+S(S(T,3),1))2) means 2nd (limit of)(1,0) admissibles, or the countable analogue of \(\psi_I(1)\)
  160. S(S(T,2)+S(S(T,3)+S(S(T,3),1))2+S(S(T,3),1)) means the countable analogue of \(\psi_{\Omega_{\psi_I(1)+1}}(\Omega_{\psi_I(1)+1})\)
  161. S(S(T,2)+S(S(T,3)+S(S(T,3),1))2+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),1))2+S(S(T,3)))) means the countable analogue of \(\psi_{\Omega_{\psi_I(1)+1}}(\Omega_{\psi_I(1)+1}3)\)
  162. S(S(T,2)+S(S(T,3)+S(S(T,3),1))2+S(S(T,3))) means the countable analogue of \(\Omega_{\psi_I(1)+1}\)
  163. S(S(T,2)+S(S(T,3)+S(S(T,3),1))2+S(S(T,3)+S(S(T,2)+S(S(T,3))))) means the countable analogue of \(\Omega_{\psi_I(1)+\Omega_2}\)
  164. S(S(T,2)+S(S(T,3)+S(S(T,3),1))2+S(S(T,3)+S(S(T,2)+S(S(T,3)+S(S(T,3),1))))) means the countable analogue of \(\Omega_{\psi_I(1)+\psi_I(0)}\)
  165. S(S(T,2)+S(S(T,3)+S(S(T,3),1))2+S(S(T,3)+S(S(T,2)+S(S(T,3)+S(S(T,3),1))2))) means the countable analogue of \(\Omega_{\psi_I(1)2}\)
  166. S(S(T,2)+S(S(T,3)+S(S(T,3),1))2+S(S(T,3)+S(S(T,2)+S(S(T,3)+S(S(T,3),1))2+S(S(T,3))))) means the countable analogue of \(\Omega_{\Omega_{\psi_I(1)+1}}\)
  167. S(S(T,2)+S(S(T,3)+S(S(T,3),1))3) means 3rd (limit of)(1,0) admissibles, or the countable analogue of \(\psi_I(2)\)
  168. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+1)) means 1st (limit of)(1,1) admissibles, or the countable analogue of \(\psi_I(\omega)\)
  169. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)))) means the countable analogue of \(\psi_I(\Omega)\)
  170. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3))))) means the countable analogue of \(\psi_I(\Omega_2)\)
  171. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+1)))) means the countable analogue of \(\psi_I(\Omega_\omega)\)
  172. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),1))))) means the countable analogue of \(\psi_I(\psi_I(0))\)
  173. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),1))))))) means the countable analogue of \(\psi_I(\psi_I(\psi_I(0)))\)
  174. S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0) has growth rate \(\psi(I)\)
  175. S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0) means the countable analogue of \(\psi_I(I)\)

Beyond S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0)

There are different ways how S(S(T,2)+S(S(T,3)+S(S(T,3)))) could appear and work.

  1. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0)) has growth rate \(\psi(I)\)
  2. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0)) means the countable analogue of \(\psi_{\Omega_{\psi_I(0)+1}}(I)\)
  3. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0)+S(S(T,2)+S(S(T,3)+S(S(T,3),1)))) has growth rate \(\psi(I+\psi_I(0))\)
  4. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0)) ) has growth rate \(\psi(I+\psi_{\Omega_{\psi_I(0)+1}}(I))\)
  5. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0)+1) ) has growth rate \(\psi(I+\psi_{\Omega_{\psi_I(0)+1}}(I+1))\)
  6. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0)) ) ) has growth rate \(\psi(I+\psi_{\Omega_{\psi_I(0)+1}}(I+\psi_{\Omega_{\psi_I(0)+1}}(I)))\)
  7. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3))))) has growth rate \(\psi(I+\Omega_{\psi_I(0)+1})\)
  8. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3))))+S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)))) has growth rate \(\psi(I+\Omega_{\psi_I(0)+1}2)\)
  9. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3))))+S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3))+1)) has growth rate \(\psi(I+\Omega_{\psi_I(0)+1}\omega)\)
  10. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3))))+S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3))2)) has growth rate \(\psi(I+\psi_{\Omega_{\psi_I(0)+2}}(\Omega_{\psi_I(0)+2}3))\)
  11. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3))))+S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3))3)) has growth rate \(\psi(I+\psi_{\Omega_{\psi_I(0)+2}}(\Omega_{\psi_I(0)+3}3))\)
  12. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3))))+S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+1))) has growth rate \(\psi(I+\psi_{\Omega_{\psi_I(0)+2}}(\Omega_{\psi_I(0)+\omega}))\)
  13. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3))))+S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+S(S(T,2)+S(S(T,3)+S(S(T,3),1)))))) has growth rate \(\psi(I+\psi_{\Omega_{\psi_I(0)+2}}(\Omega_{\psi_I(0)2}))\)
  14. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3))))+S(S(T,2)+S(S(T,3)+S(S(T,3),1))2)) has growth rate \(\psi(I+\psi_{\Omega_{\psi_I(0)+2}}(\psi_I(1)))\)
  15. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3))))+S(S(T,2)+S(S(T,3)+S(S(T,3),1)+1))) has growth rate \(\psi(I+\psi_{\Omega_{\psi_I(0)+2}}(\psi_I(\omega)))\)
  16. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3))))+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0)) has growth rate \(\psi(I+\psi_{\Omega_{\psi_I(0)+2}}(I))\)
  17. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3))))2) has growth rate \(\psi(I+\psi_{\Omega_{\psi_I(0)+2}}(I)+\Omega_{\psi_I(0)+1})\)
  18. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3))))3) has growth rate \(\psi(I+\psi_{\Omega_{\psi_I(0)+2}}(I)2+\Omega_{\psi_I(0)+1})\)
  19. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3)))+1)) has growth rate \(\psi(I+\psi_{\Omega_{\psi_I(0)+2}}(I+1))\)
  20. S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3))))) has growth rate \(\psi(I+\psi_{\Omega_{\psi_I(0)+2}}(I+\Omega_{\psi_I(0)+1}))\)
  21. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)))) has growth rate \(\psi(I+\psi_{\Omega_{\psi_I(0)+2}}(I+\Omega_{\psi_I(0)+1}))\)

Up to this point, it appears that S(S(T,2)+S(S(T,3)+S(S(T,3)))) works as a \(\psi_{\Omega_{\psi_I(0)+2}}(I)\), but it could be stronger. Unlike S(S(T,2)+S(S(T,3)+1)), S(S(T,2)+S(S(T,3)+S(S(T,3)))) reduces into S(S(T,2)+S(S(T,3)+S(S(T,3),2)),1), which still search out for a layer, and it has a static layer to add, wherever a S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,2)+S(S(T,3)+S(S(T,3))))) appears.

  1. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))) has growth rate \(\psi(I)\)
  2. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))+S(S(T,2)+S(S(T,3)+1))) has growth rate \(\psi(I+\Omega_\omega)\)
  3. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))+S(S(T,2)+S(S(T,3)+S(S(T,3),1)))) has growth rate \(\psi(I+\psi_I(0))\)
  4. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))+S(S(T,2)+S(S(T,3)+S(S(T,3),1)+1))) has growth rate \(\psi(I+\psi_I(\omega))\)
  5. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))+S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),1)))))) has growth rate \(\psi(I+\psi_I(\psi_I(0)))\)
  6. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))+S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0)))) has growth rate \(\psi(I+\psi_I(I))\)
  7. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))+S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))+1)) has growth rate \(\psi(I+\psi_I(I)\omega)\)
  8. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))+S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))+S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))))) has growth rate \(\psi(I+\psi_I(I)^2)\)
  9. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))+S(S(T,3),1)) has growth rate \(\psi(I+\Omega_{\psi_I(I)+1})\)
  10. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))+S(S(T,3),2),0) has growth rate \(\psi(I+\Omega_{\psi_I(I)+1}2)\)
  11. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))+S(S(T,3))) has growth rate \(\psi(I+\Omega_{\psi_I(I)+1}3)\)
  12. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))+S(S(T,3))) means the countable analogue of \(\Omega_{\psi_I(I)+1}\)
  13. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))+S(S(T,3))2) means the countable analogue of \(\Omega_{\psi_I(I)+2}\)
  14. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))+S(S(T,3)+1)) means the countable analogue of \(\Omega_{\psi_I(I)+\omega}\)
  15. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))+S(S(T,3)+S(S(T,2)+S(S(T,3)+S(S(T,3),1))))) means the countable analogue of \(\Omega_{\psi_I(I)+\psi_I(0)}\)
  16. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))+S(S(T,3)+S(S(T,2)+S(S(T,3)+S(S(T,3),1)+1)))) means the countable analogue of \(\Omega_{\psi_I(I)+\psi_I(\omega)}\)
  17. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))) )) means the countable analogue of \(\Omega_{\psi_I(I)2}\)
  18. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0)))+1 )) means the countable analogue of \(\Omega_{\psi_I(I)\omega}\)
  19. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))+1) )) means the countable analogue of \(\Omega_{\psi_I(I)^\omega}\)
  20. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))+S(S(T,3),1)) )) means the countable analogue of \(\Omega_{\psi_{\Omega_{\psi_I(I)+1}}(\Omega_{\psi_I(I)+1})}\)
  21. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))+S(S(T,3))) )) means the countable analogue of \(\Omega_{\Omega_{\psi_I(I)+1}}\)
  22. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))+S(S(T,3)+S(S(T,3),1))) means the countable analogue of \(\psi_I(I+1)\)
  23. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))+S(S(T,3)+S(S(T,3),1)+1)) means the countable analogue of \(\psi_I(I+\omega)\)
  24. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))2) means the countable analogue of \(\psi_I(I+\psi_I(I))\)
  25. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))2+S(S(T,3))) means the countable analogue of \(\Omega_{\psi_I(I+\psi_I(I))+1}\)
  26. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))2+S(S(T,3)+S(S(T,3),1))) means the countable analogue of \(\psi_I(I+\psi_I(I)+1)\)
  27. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))3) means the countable analogue of \(\psi_I(I+\psi_I(I)2)\)
  28. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0)+1)) means the countable analogue of \(\psi_I(I+\psi_I(I)\omega)\)
  29. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))) )) means the countable analogue of \(\psi_I(I+\psi_I(I)^2)\)
  30. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))+S(S(T,3),1)) )) means the countable analogue of \(\psi_I(I+\psi_{\Omega_{\psi_I(I)+1}}(\Omega_{\psi_I(I)+1}))\)
  31. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))+S(S(T,3))) )) means the countable analogue of \(\psi_I(I+\Omega_{\psi_I(I)+1})\)
  32. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))+S(S(T,3)+S(S(T,3),1))) )) means the countable analogue of \(\psi_I(I+\psi_I(I+1))\)
  33. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))2) )) means the countable analogue of \(\psi_I(I+\psi_I(I+\psi_I(I)))\)
  34. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0)+1)) )) means the countable analogue of \(\psi_I(I+\psi_I(I+\psi_I(I)\omega))\)
  35. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))))) has growth rate \(\psi(I2)\)
  36. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))))) means the countable analogue of \(\psi_I(I2)\)
  37. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))))+S(S(T,3))) means the countable analogue of \(\Omega_{\psi_I(I2)+1}\)
  38. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1))) )) means the countable analogue of \(\Omega_{\psi_I(I2)+\psi_I(0)}\)
  39. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))))) )) means the countable analogue of \(\Omega_{\psi_I(I2)2}\)
  40. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))))+S(S(T,3)+S(S(T,3),1))) means the countable analogue of \(\psi_I(I2+1)\)
  41. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))))+S(S(T,3)+S(S(T,3),1)+1)) means the countable analogue of \(\psi_I(I2+\omega)\)
  42. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0) )) means the countable analogue of \(\psi_I(I2+\psi_I(I))\)
  43. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))))) )) means the countable analogue of \(\psi_I(I2+\psi_I(I2))\)
  44. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))))+S(S(T,3)+S(S(T,3),1))) )) means the countable analogue of \(\psi_I(I2+\psi_I(I2+1))\)
  45. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))))2) means the countable analogue of \(\psi_I(I3)\)
  46. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))))3) means the countable analogue of \(\psi_I(I4)\)
  47. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))))+1)) means the countable analogue of \(\psi_I(I\omega)\)
  48. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))))+ S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0) )) means the countable analogue of \(\psi_I(I\psi_I(I))\)
  49. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))))+ S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))))+1)) )) means the countable analogue of \(\psi_I(I\psi_I(I\omega))\)
  50. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))))2)) means the countable analogue of \(\psi_I(I^2)\)
  51. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))))2)2) means the countable analogue of \(\psi_I(I^22)\)
  52. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))))2+1)) means the countable analogue of \(\psi_I(I^2\omega)\)
  53. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))))3)) means the countable analogue of \(\psi_I(I^3)\)
  54. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+1) )) means the countable analogue of \(\psi_I(I^\omega)\)
  55. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0) ) )) means the countable analogue of \(\psi_I(I^{\psi_I(I)})\)
  56. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))+1))) ) )) means the countable analogue of \(\psi_I(I^{\psi_I(I^\omega)})\)
  57. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+ S(S(T,2)+S(S(T,3)+S(S(T,3)))) ) )) means the countable analogue of \(\psi_I(I^I)\)
  58. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+ S(S(T,2)+S(S(T,3)+S(S(T,3)))) )+1 )) means the countable analogue of \(\psi_I(I^I\omega)\)
  59. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+ S(S(T,2)+S(S(T,3)+S(S(T,3))))+1 ) )) means the countable analogue of \(\psi_I(I^{I\omega})\)
  60. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+ S(S(T,2)+S(S(T,3)+S(S(T,3))))2 ) )) means the countable analogue of \(\psi_I(I^{I^2})\)
  61. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+1) ) )) means the countable analogue of \(\psi_I(I^{I^\omega})\)
  62. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+ S(S(T,2)+S(S(T,3)+S(S(T,3)))) ) ) )) means the countable analogue of \(\psi_I(I^{I^I})\)
  63. S(S(T,2)+S(S(T,3)+S(S(T,3)))) means 1st recursively inaccessible ordinal, or the countable analogue of \(I\)
  64. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3),1)) has growth rate \(\psi(\Omega_{I+1})\)
  65. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3),1)) means the countable analogue of \(\psi_{\Omega_{I+1}}(\Omega_{I+1})\)
  66. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3),2),0) means the countable analogue of \(\psi_{\Omega_{I+1}}(\Omega_{I+1}2)\)
  67. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3))) ) means the countable analogue of \(\psi_{\Omega_{I+1}}(\Omega_{I+1}3)\)
  68. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3))) means 1st admissible ordinal after recursively inaccessible ordinal, or the countable analogue of \(\Omega_{I+1}\)
  69. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+1)) means the countable analogue of \(\Omega_{I+\omega}\)
  70. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1))) )) means the countable analogue of \(\Omega_{I+\psi_I(0)}\)
  71. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))) )) means the countable analogue of \(\Omega_{I+\psi_I(I)}\)
  72. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))))) )) means the countable analogue of \(\Omega_{I+\psi_I(I2)}\)
  73. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3))))) )) means the countable analogue of \(\Omega_{I+\psi_I(\Omega_{I+1})}\)
  74. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))) )) means the countable analogue of \(\Omega_{I2}\)
  75. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))) )2) means the countable analogue of \(\Omega_{I3}\)
  76. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3))))+1 )) means the countable analogue of \(\Omega_{I\omega}\)
  77. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3))))2 )) means the countable analogue of \(\Omega_{I^2}\)
  78. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+1) )) means the countable analogue of \(\Omega_{I^\omega}\)
  79. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3),1)) )) means the countable analogue of \(\Omega_{\psi_{\Omega_{I+1}}(\Omega_{I+1})}\)
  80. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)))) )) means the countable analogue of \(\Omega_{\psi_{\Omega_{I+1}}(\Omega_{I+1}3)}\)
  81. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))) )) ) )) means the countable analogue of \(\Omega_{\psi_{\Omega_{I+1}}(\Omega_{I2})}\)
  82. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3))) )) means the countable analogue of \(\Omega_{\Omega_{I+1}}\)
  83. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))) )) )) means the countable analogue of \(\Omega_{\Omega_{I2}}\)
  84. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3))) )) )) means the countable analogue of \(\Omega_{\Omega_{\Omega_{I+1}}}\)
  85. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1))) means the countable analogue of \(\psi_{I_2}(0)\)
  86. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1))+S(S(T,3))) means the countable analogue of \(\Omega_{\psi_{I_2}(0)+1}\)
  87. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))) )) means the countable analogue of \(\Omega_{\psi_{I_2}(0)+I}\)
  88. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3))) )) means the countable analogue of \(\Omega_{\psi_{I_2}(0)+\Omega_{I+1}}\)
  89. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3))))) )) means the countable analogue of \(\Omega_{\psi_{I_2}(0)+\Omega_{\Omega_{I+1}}}\)
  90. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1))) )) means the countable analogue of \(\Omega_{\psi_{I_2}(0)2}\)
  91. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1))+S(S(T,3))) )) means the countable analogue of \(\Omega_{\Omega_{\psi_{I_2}(0)+1}}\)
  92. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1))2) means the countable analogue of \(\psi_{I_2}(1)\)
  93. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+1)) means the countable analogue of \(\psi_{I_2}(\omega)\)
  94. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1))) )) means the countable analogue of \(\psi_{I_2}(\psi_I(0))\)
  95. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))) )) means the countable analogue of \(\psi_{I_2}(I)\)
  96. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3))) )) means the countable analogue of \(\psi_{I_2}(\Omega_{I+1})\)
  97. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1))) )) means the countable analogue of \(\psi_{I_2}(\psi_{I_2}(0))\)
  98. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1))+S(S(T,3))) )) means the countable analogue of \(\psi_{I_2}(\Omega_{\psi_{I_2}(0)+1})\)
  99. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1))2) )) means the countable analogue of \(\psi_{I_2}(\psi_{I_2}(1))\)
  100. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+1)) )) means the countable analogue of \(\psi_{I_2}(\psi_{I_2}(\omega))\)
  101. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1))) )) )) means the countable analogue of \(\psi_{I_2}(\psi_{I_2}(\psi_{I_2}(0)))\)
  102. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),2)),0) means the countable analogue of \(\psi_{I_2}(I_2)\)
  103. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),2)),0) )) means the countable analogue of \(\psi_{I_2}(I_2)\)
  104. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),2)),0) )+S(S(T,3))) means the countable analogue of \(\Omega_{\psi_{I_2}(I_2)+1}\)
  105. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),2)),0) )+S(S(T,3)+S(S(T,3),1))) means the countable analogue of \(\psi_{I_2}(I_2+1)\)
  106. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),2)),0) )2) means the countable analogue of \(\psi_{I_2}(I_2+\psi_{I_2}(I_2))\)
  107. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),2)),0)+1 )) means the countable analogue of \(\psi_{I_2}(I_2+\psi_{I_2}(I_2)\omega)\)
  108. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),2)),0)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),2)),0) )) )) means the countable analogue of \(\psi_{I_2}(I_2+\psi_{I_2}(I_2)^2)\)
  109. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),2)),0)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),2)),0) )+S(S(T,3))) )) means the countable analogue of \(\psi_{I_2}(I_2+\Omega_{\psi_{I_2}(I_2)+1})\)
  110. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),2)),0)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),2)),0)+1 )) )) means the countable analogue of \(\psi_{I_2}(I_2+\psi_{I_2}(I_2+\psi_{I_2}(I_2)\omega))\)
  111. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))2) )) means the countable analogue of \(\psi_{I_2}(I_22)\)
  112. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))2)+1 )) means the countable analogue of \(\psi_{I_2}(I_2\omega)\)
  113. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))2)2 )) means the countable analogue of \(\psi_{I_2}(I_2^2)\)
  114. S(S(T,2)+S(S(T,3)+S(S(T,3)))2) means 2nd recursively inaccessible ordinal, or the countable analogue of \(I_2\)
  115. S(S(T,2)+S(S(T,3)+S(S(T,3)))2+S(S(T,3))) means the countable analogue of \(\Omega_{I_2+1}\)
  116. S(S(T,2)+S(S(T,3)+S(S(T,3)))2+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))2+S(S(T,3))) )) means the countable analogue of \(\Omega_{\Omega_{I_2+1}}\)
  117. S(S(T,2)+S(S(T,3)+S(S(T,3)))2+S(S(T,3)+S(S(T,3),1))) means the countable analogue of \(\psi_{I_3}(0)\)
  118. S(S(T,2)+S(S(T,3)+S(S(T,3)))2+S(S(T,3)+S(S(T,3),1))2) means the countable analogue of \(\psi_{I_3}(1)\)
  119. S(S(T,2)+S(S(T,3)+S(S(T,3)))2+S(S(T,3)+S(S(T,3),1)+1)) means the countable analogue of \(\psi_{I_3}(\omega)\)
  120. S(S(T,2)+S(S(T,3)+S(S(T,3)))2+S(S(T,3)+S(S(T,3),2)),0) means the countable analogue of \(\psi_{I_3}(I_3)\)
  121. S(S(T,2)+S(S(T,3)+S(S(T,3)))2+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))3))) means the countable analogue of \(\psi_{I_3}(I_32)\)
  122. S(S(T,2)+S(S(T,3)+S(S(T,3)))3) means 3rd recursively inaccessible ordinal, or the countable analogue of \(I_3\)
  123. S(S(T,2)+S(S(T,3)+S(S(T,3))+1)) has growth rate \(\psi(I_\omega)\)

Beyond S(S(T,2)+S(S(T,3)+S(S(T,3))+1))

Similar to S(S(T,2)+S(S(T,3)+1)), S(S(T,2)+S(S(T,3)+S(S(T,3))+1)) doesn't have a static outer layer to add. We need to check S(S(T,2)+S(S(T,3)+S(S(T,3))+1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))n)) series to get the growth rate of S(S(T,2)+S(S(T,3)+S(S(T,3))+1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1))).

  1. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))) has growth rate \(\psi(I_\omega)\)
  2. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)) ) has growth rate \(\psi(I_\omega+\psi_{\Omega_2}(I_\omega+\Omega))\)
  3. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3))) ) has growth rate \(\psi(I_\omega+\psi_{\Omega_3}(I_\omega+\Omega_2))\)
  4. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+1)) ) has growth rate \(\psi(I_\omega+\Omega_\omega)\)
  5. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,2)))) ) has growth rate \(\psi(I_\omega+\Omega_\Omega)\)
  6. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,2)+S(S(T,3))))) ) has growth rate \(\psi(I_\omega+\Omega_{\Omega_2})\)
  7. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,2)+S(S(T,3)+S(S(T,2)+S(S(T,3))))))) ) has growth rate \(\psi(I_\omega+\Omega_{\Omega_{\Omega_2}})\)
  8. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1))) ) has growth rate \(\psi(I_\omega+\psi_I(0))\)
  9. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3))) ) has growth rate \(\psi(I_\omega+\Omega_{\psi_I(0)+1})\)
  10. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+S(S(T,2)+S(S(T,3)+S(S(T,3),1))))) ) has growth rate \(\psi(I_\omega+\Omega_{\psi_I(0)2})\)
  11. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1))2) ) has growth rate \(\psi(I_\omega+\psi_I(1))\)
  12. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+1)) ) has growth rate \(\psi(I_\omega+\psi_I(\omega))\)
  13. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),1))))) ) has growth rate \(\psi(I_\omega+\psi_I(\psi_I(0)))\)
  14. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))) ) has growth rate \(\psi(I_\omega+\psi_I(I))\)
  15. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))+S(S(T,3)+S(S(T,3),1))) ) has growth rate \(\psi(I_\omega+\psi_I(I+1))\)
  16. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))2) ) has growth rate \(\psi(I_\omega+\psi_I(I+\psi_I(I)))\)
  17. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0)+1)) ) has growth rate \(\psi(I_\omega+\psi_I(I+\psi_I(I)\omega))\)
  18. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))) )) ) has growth rate \(\psi(I_\omega+\psi_I(I+\psi_I(I)^2))\)
  19. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0))+S(S(T,3))) )) ) has growth rate \(\psi(I_\omega+\psi_I(I+\Omega_{\psi_I(I)+1}))\)
  20. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0)+1)) )) ) has growth rate \(\psi(I_\omega+\psi_I(I+\psi_I(I+\psi_I(I)\omega)))\)
  21. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))) )) ) has growth rate \(\psi(I_\omega+\psi_I(I2))\)
  22. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))) )2) ) has growth rate \(\psi(I_\omega+\psi_I(I3))\)
  23. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3))))+1 )) ) has growth rate \(\psi(I_\omega+\psi_I(I\omega))\)
  24. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3))))2 )) ) has growth rate \(\psi(I_\omega+\psi_I(I^2))\)
  25. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+1) )) ) has growth rate \(\psi(I_\omega+\psi_I(I^\omega))\)
  26. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+ S(S(T,2)+S(S(T,3)+S(S(T,3)))) ) )) ) has growth rate \(\psi(I_\omega+\psi_I(I^I))\)
  27. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3))) )) ) has growth rate \(\psi(I_\omega+\psi_I(\Omega_{I+1}3))\)
  28. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))) )) )) ) has growth rate \(\psi(I_\omega+\psi_I(\Omega_{I2}))\)
  29. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1))) )) ) has growth rate \(\psi(I_\omega+\psi_I(\psi_{I_2}(0)))\)
  30. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),2)),0) )) )) ) has growth rate \(\psi(I_\omega+\psi_I(I_2))\)
  31. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))2) )) ) has growth rate \(\psi(I_\omega+\psi_I(I_22))\)
  32. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))2+S(S(T,3))) )) ) has growth rate \(\psi(I_\omega+\psi_I(\Omega_{I_2+1}))\)
  33. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))2+S(S(T,3)+S(S(T,3),1))) )) ) has growth rate \(\psi(I_\omega+\psi_I(\psi_{I_3}(0)))\)
  34. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))3) )) ) has growth rate \(\psi(I_\omega+\psi_I(I_32))\)
  35. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))) ) has growth rate \(\psi(I_\omega+\psi_I(I_\omega))\)
  36. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+1) ) has growth rate \(\psi(I_\omega+\psi_I(I_\omega)\omega)\)
  37. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+1)) ) ) has growth rate \(\psi(I_\omega+\psi_I(I_\omega)\Omega_\omega)\)
  38. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1))) ) ) has growth rate \(\psi(I_\omega+\psi_I(I_\omega)\psi_I(0))\)
  39. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))))) ) ) has growth rate \(\psi(I_\omega+\psi_I(I_\omega)\psi_I(I2))\)
  40. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))) ) ) has growth rate \(\psi(I_\omega+\psi_I(I_\omega)^2)\)
  41. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))) ) ) ) has growth rate \(\psi(I_\omega+\psi_I(I_\omega)^{\psi_I(I_\omega)})\)
  42. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3),1)) has growth rate \(\psi(I_\omega+\Omega_{\psi_I(I_\omega)+1})\)
  43. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3),2),0) has growth rate \(\psi(I_\omega+\Omega_{\psi_I(I_\omega)+1}2)\)
  44. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3))) has growth rate \(\psi(I_\omega+\Omega_{\psi_I(I_\omega)+1}3)\)
  45. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3))2) has growth rate \(\psi(I_\omega+\Omega_{\psi_I(I_\omega)+2}3)\)
  46. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+1)) has growth rate \(\psi(I_\omega+\Omega_{\psi_I(I_\omega)+\omega})\)
  47. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1))) )) has growth rate \(\psi(I_\omega+\Omega_{\psi_I(I_\omega)+\psi_I(0)})\)
  48. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))))) )) has growth rate \(\psi(I_\omega+\Omega_{\psi_I(I_\omega)+\psi_I(I2)})\)
  49. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))) )) has growth rate \(\psi(I_\omega+\Omega_{\psi_I(I_\omega)2})\)
  50. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1))))+1 )) has growth rate \(\psi(I_\omega+\Omega_{\psi_I(I_\omega)\omega})\)
  51. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3),1)) )) has growth rate \(\psi(I_\omega+\Omega_{\psi_{\Omega_{\psi_I(I_\omega)+1}}(I_\omega+\Omega_{\psi_I(I_\omega)+1})})\)
  52. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3))) ) )) has growth rate \(\psi(I_\omega+\Omega_{\psi_{\Omega_{\psi_I(I_\omega)+1}}(I_\omega+\Omega_{\psi_I(I_\omega)+1}3)})\)
  53. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3))) )) has growth rate \(\psi(I_\omega+\Omega_{\Omega_{\psi_I(I_\omega)+1}})\)
  54. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3))) )) )) has growth rate \(\psi(I_\omega+\Omega_{\Omega_{\Omega_{\psi_I(I_\omega)+1}}})\)
  55. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+S(S(T,3),1))) has growth rate \(\psi(I_\omega+\psi_I(I_\omega+1))\)
  56. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+S(S(T,3),1)+1)) has growth rate \(\psi(I_\omega+\psi_I(I_\omega+\omega))\)
  57. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1))) )) has growth rate \(\psi(I_\omega+\psi_I(I_\omega+\psi_I(0)))\)
  58. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0) )) has growth rate \(\psi(I_\omega+\psi_I(I_\omega+\psi_I(I)))\)
  59. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))))) )) has growth rate \(\psi(I_\omega+\psi_I(I_\omega+\psi_I(I2)))\)
  60. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))) )) has growth rate \(\psi(I_\omega+\psi_I(I_\omega+\psi_I(I_\omega)))\)
  61. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+S(S(T,3),1))) )) has growth rate \(\psi(I_\omega+\psi_I(I_\omega+\psi_I(I_\omega+1)))\)
  62. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3),2)),0) )) )) has growth rate \(\psi(I_\omega+\psi_I(I_\omega+\psi_I(I_\omega+\psi_I(I))))\)
  63. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))))) has growth rate \(\psi(I_\omega+I)\)
  64. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))) ) has growth rate \(\psi(I_\omega+I+\psi_I(I_\omega))\)
  65. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))))+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))))) ) has growth rate \(\psi(I_\omega+I+\psi_I(I_\omega+I))\)
  66. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))))+ S(S(T,3)) ) has growth rate \(\psi(I_\omega+I+\Omega_{\psi_I(I_\omega+I)+1}3)\)
  67. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))))+ S(S(T,3)+S(S(T,3),1)) ) has growth rate \(\psi(I_\omega+I+\psi_I(I_\omega+I+1))\)
  68. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))))+ S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))))) ) ) has growth rate \(\psi(I_\omega+I+\psi_I(I_\omega+I+\psi_I(I_\omega+I)))\)
  69. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))))2 ) has growth rate \(\psi(I_\omega+I2)\)
  70. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3))))+1 )) has growth rate \(\psi(I_\omega+I\omega)\)
  71. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3))))2 )) has growth rate \(\psi(I_\omega+I^2)\)
  72. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3),1)) )) has growth rate \(\psi(I_\omega+\psi_{\Omega_{I+1}}(\Omega_{I+1}))\)
  73. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3))) ) )) has growth rate \(\psi(I_\omega+\psi_{\Omega_{I+1}}(\Omega_{I+1}3))\)
  74. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3))) )) has growth rate \(\psi(I_\omega+\psi_{\Omega_{I+1}}(\Omega_{I+1}3))\)
  75. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1))) )) has growth rate \(\psi(I_\omega+\psi_{\Omega_{I+1}}(\psi_{I_2}(0)))\)
  76. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))2) )) has growth rate \(\psi(I_\omega+\psi_{\Omega_{I+1}}(I_22))\)
  77. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))3) )) has growth rate \(\psi(I_\omega+\psi_{\Omega_{I+1}}(I_32))\)
  78. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))2) has growth rate \(\psi(I_\omega+\psi_{\Omega_{I+1}}(I_\omega))\)
  79. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))2+ S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))2) ) has growth rate \(\psi(I_\omega+\psi_{\Omega_{I+1}}(I_\omega)+\psi_I(I_\omega+\psi_{\Omega_{I+1}}(I_\omega)))\)
  80. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))2+ S(S(T,3))) has growth rate \(\psi(I_\omega+\psi_{\Omega_{I+1}}(I_\omega)+\Omega_{\psi_I(I_\omega+\psi_{\Omega_{I+1}}(I_\omega))+1}3)\)
  81. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))2+ S(S(T,3)+S(S(T,3),1))) has growth rate \(\psi(I_\omega+\psi_{\Omega_{I+1}}(I_\omega)+\psi_I(I_\omega+\psi_{\Omega_{I+1}}(I_\omega)+1))\)
  82. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))2+ S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))))) has growth rate \(\psi(I_\omega+\psi_{\Omega_{I+1}}(I_\omega)+I)\)
  83. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))3) has growth rate \(\psi(I_\omega+\psi_{\Omega_{I+1}}(I_\omega)2)\)
  84. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1))+1)) has growth rate \(\psi(I_\omega+\psi_{\Omega_{I+1}}(I_\omega+1))\)
  85. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3))+1))+S(S(T,2)+S(S(T,3)+S(S(T,3)))) )) has growth rate \(\psi(I_\omega+\psi_{\Omega_{I+1}}(I_\omega+I))\)
  86. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3))+1))+S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3))) )) has growth rate \(\psi(I_\omega+\psi_{\Omega_{I+1}}(I_\omega+\psi_{\Omega_{I+1}}(\Omega_{I+1}3)))\)
  87. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3))+1))+S(S(T,2)+S(S(T,3)+S(S(T,3)))2) )) has growth rate \(\psi(I_\omega+\psi_{\Omega_{I+1}}(I_\omega+\psi_{\Omega_{I+1}}(I_22)))\)
  88. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3))+1))2 )) has growth rate \(\psi(I_\omega+\psi_{\Omega_{I+1}}(I_\omega+\psi_{\Omega_{I+1}}(I_\omega)))\)
  89. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3))+1))2+1 )) has growth rate \(\psi(I_\omega+\psi_{\Omega_{I+1}}(I_\omega+\psi_{\Omega_{I+1}}(I_\omega)+1))\)
  90. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3))+1))3 )) has growth rate \(\psi(I_\omega+\psi_{\Omega_{I+1}}(I_\omega+\psi_{\Omega_{I+1}}(I_\omega)2))\)
  91. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3))+1)+1) )) has growth rate \(\psi(I_\omega+\psi_{\Omega_{I+1}}(I_\omega+\psi_{\Omega_{I+1}}(I_\omega+1)))\)
  92. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3))+1)+S(S(T,2)+S(S(T,3)+S(S(T,3))))) )) has growth rate \(\psi(I_\omega+\psi_{\Omega_{I+1}}(I_\omega+\psi_{\Omega_{I+1}}(I_\omega+I)))\)

In summary, S(S(T,2)+S(S(T,3)+S(S(T,3))+1)+S(S(T,2)+S(S(T,3)+S(S(T,3))))) has growth rate \(\psi(I_\omega+\psi_{\Omega_{I+1}}(I_\omega+\psi_{\Omega_{I+1}}(I_\omega+I)))\). There are one more layer of \(\psi\) than the S(S(T,2)+S(S(T,3)+1)+S(S(T,2)+S(S(T,3))n)) series.

  1. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))) )) has growth rate \(\psi(I_\omega)\)
  2. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))) )+ S(S(T,3))) has growth rate \(\psi(I_\omega+\Omega_{\psi_I(I_\omega)+1}3)\)
  3. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))) )+ S(S(T,3)+S(S(T,3),1))) has growth rate \(\psi(I_\omega+\psi_I(I_\omega+1))\)
  4. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))) )+ S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))) )) has growth rate \(\psi(I_\omega+I)\)
  5. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))) )+ S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3))))+1 )) has growth rate \(\psi(I_\omega+I\omega)\)
  6. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))) )+ S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3),1)) )) has growth rate \(\psi(I_\omega+\psi_{\Omega_{I+1}}(\Omega_{I+1}))\)
  7. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))) )+ S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3))) )) has growth rate \(\psi(I_\omega+\psi_{\Omega_{I+1}}(\Omega_{I+1}3))\)
  8. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))) )+ S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1))) )) has growth rate \(\psi(I_\omega+\psi_{\Omega_{I+1}}(\psi_{I_2}(0)))\)
  9. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))) )+ S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))2))) )) has growth rate \(\psi(I_\omega+\psi_{\Omega_{I+1}}(I_22))\)
  10. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))) )+ S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))2+S(S(T,3))))) )) has growth rate \(\psi(I_\omega+\psi_{\Omega_{I+1}}(\Omega_{I_2+1}3))\)
  11. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))) )+ S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))2+S(S(T,3)+S(S(T,3),1))))) )) has growth rate \(\psi(I_\omega+\psi_{\Omega_{I+1}}(\psi_{I_3}(0)))\)
  12. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))) )+ S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3)))3))) )) has growth rate \(\psi(I_\omega+\psi_{\Omega_{I+1}}(I_32))\)
  13. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))) )2) has growth rate \(\psi(I_\omega+\psi_{\Omega_{I+1}}(I_\omega))\)
  14. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1))))+1 )) has growth rate \(\psi(I_\omega+\psi_{\Omega_{I+1}}(I_\omega+1))\)
  15. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1))))2 )) has growth rate \(\psi(I_\omega+\psi_{\Omega_{I+1}}(I_\omega+\psi_{\Omega_{I+1}}(I_\omega)))\)
  16. S(S(T,2)+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+1) )) has growth rate \(\psi(I_\omega+\psi_{\Omega_{I+1}}(I_\omega+\psi_{\Omega_{I+1}}(I_\omega+1)))\)
  17. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))) ) has growth rate \(\psi(I_\omega+\psi_{\Omega_{I+1}}(I_\omega+\psi_{\Omega_{I+1}}(I_\omega+\psi_{\Omega_{I+1}}(I_\omega))))\)
  18. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3),1)) has growth rate \(\psi(I_\omega+\Omega_{I+1})\)
  19. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3))) has growth rate \(\psi(I_\omega+\Omega_{I+1}3)\)
  20. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+1)) has growth rate \(\psi(I_\omega+\Omega_{I+\omega})\)
  21. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))) )) has growth rate \(\psi(I_\omega+\Omega_{I2})\)
  22. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3),1)) )) has growth rate \(\psi(I_\omega+\Omega_{\psi_{\Omega_{I+1}}(\Omega_{I+1})})\)
  23. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3))) )) has growth rate \(\psi(I_\omega+\Omega_{\Omega_{I+1}})\)
  24. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+1)) )) has growth rate \(\psi(I_\omega+\Omega_{\Omega_{I+\omega}})\)
  25. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))) )) )) has growth rate \(\psi(I_\omega+\Omega_{\Omega_{I2}})\)
  26. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1))) )) has growth rate \(\psi(I_\omega+\psi_{I_2}(0))\)
  27. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1))) )+S(S(T,3))) has growth rate \(\psi(I_\omega+\Omega_{\psi_{I_2}(0)+1}3)\)
  28. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1))) )+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))) )) has growth rate \(\psi(I_\omega+\Omega_{\psi_{I_2}(0)+I})\)
  29. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1))) )2) has growth rate \(\psi(I_\omega+\Omega_{\psi_{I_2}(0)2})\)
  30. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)))+1 )) has growth rate \(\psi(I_\omega+\Omega_{\psi_{I_2}(0)\omega})\)
  31. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1))+S(S(T,3),1)) )) has growth rate \(\psi(I_\omega+\Omega_{\psi_{\Omega_{\psi_{I_2}(0)+1}}(\Omega_{\psi_{I_2}(0)+1})})\)
  32. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1))+S(S(T,3))) )) has growth rate \(\psi(I_\omega+\Omega_{\Omega_{\psi_{I_2}(0)+1}})\)
  33. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1))) )) )) has growth rate \(\psi(I_\omega+\Omega_{\Omega_{\psi_{I_2}(0)2}})\)
  34. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1))2) )) has growth rate \(\psi(I_\omega+\psi_{I_2}(1))\)
  35. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+1)) )) has growth rate \(\psi(I_\omega+\psi_{I_2}(\omega))\)
  36. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),2)),0) )) )) has growth rate \(\psi(I_\omega+\psi_{I_2}(I_2))\)
  37. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))2) )) )) has growth rate \(\psi(I_\omega+\psi_{I_2}(I_22))\)
  38. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))2+S(S(T,3))) )) )) has growth rate \(\psi(I_\omega+\psi_{I_2}(\Omega_{I_2+1}3))\)
  39. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))3) )) )) has growth rate \(\psi(I_\omega+\psi_{I_2}(I_32))\)
  40. S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))+S(S(T,3)+ S(S(T,2)+S(S(T,3)+S(S(T,3)))+S(S(T,3)+S(S(T,3),1)+S(S(T,2)+S(S(T,3)+S(S(T,3))+1)))) )) has growth rate \(\psi(I_\omega+\psi_{I_2}(I_\omega))\)
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