User blog comment:Ikosarakt1/Fast-growing hierarchy/@comment-157.193.53.9-20130620180435/@comment-78.20.192.139-20130628205224

"Q: Rule 6 requires   ω ≤ t p ( a ) ≤   Ω  s       , and rule 7 requires that     Ω  s   < t p ( a ) =   Ω  u + 1        , so there doesn't seem to be a rule where     Ω  s   < t p ( a ) =   Ω  u        , where u is a limit ordinal. So, for example, there doesn't seem to be a rule for  <span style="position: absolute; clip: rect(1.74em, 1000em, 3.047em, -0.557em); top: -2.606em; left: 0em;"> <span style="position: absolute; clip: rect(1.975em, 1000em, 3.002em, -0.557em); top: -2.606em; left: 0em;"> ψ   0   ( <span style="position: absolute; clip: rect(1.74em, 1000em, 2.797em, -0.543em); top: -2.606em; left: 0em;"> Ω  <span class="mi" id="MathJax-Span-387" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">ω   ). "

A: This case does not appear since cofinalities will be either successor cardinals or omega

since we are below the first fixed point of the Omega function.

Thus as one would guess

<span style="position: absolute; clip: rect(1.74em, 1000em, 3.047em, -0.557em); top: -2.606em; left: 0em;"> <span style="position: absolute; clip: rect(1.975em, 1000em, 3.002em, -0.557em); top: -2.606em; left: 0em;"> ψ  0   ( <span style="position: absolute; clip: rect(1.74em, 1000em, 2.797em, -0.543em); top: -2.606em; left: 0em;"> Ω  <span class="mi" id="MathJax-Span-387" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">ω   )     [n]=   <span style="position: absolute; clip: rect(1.74em, 1000em, 3.047em, -0.557em); top: -2.606em; left: 0em;"> <span style="position: absolute; clip: rect(1.975em, 1000em, 3.002em, -0.557em); top: -2.606em; left: 0em;"> ψ   0   (       <span style="position: absolute; clip: rect(1.74em, 1000em, 3.047em, -0.557em); top: -2.606em; left: 0em;"> <span style="position: absolute; clip: rect(1.975em, 1000em, 3.002em, -0.557em); top: -2.606em; left: 0em;"> ψ   0   ( <span style="position: absolute; clip: rect(1.74em, 1000em, 2.797em, -0.543em); top: -2.606em; left: 0em;"> Ω  <span class="mi" id="MathJax-Span-387" style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;">ω   n). .

In fact the assignment is taken from Buchholz Schütte.

By the way lower bounds for Pi^1_1-TR_0 can be found in

http://wwwmath.uni-muenster.de/logik/Personen/rds/festschrift.pdf

Upper bounds can be found in the PhD thesis of Rathjen but the sorce is in German.

Best,

Andreas