Recently I am writing a book about making a bound for my DNA ordinal. I am almost finished with the book (which currently have 7800 words written into 20 pages), but I need some serious help regarding set theory for making the bound and a following proof. My best result to this point is that my DNA ordinal is between PTO(Pi^1_3-CA0) and PTO(Pi^1_3-CA+BI), which is on the realm of 2-stable ordinals. Noted that I generally like to refer to SLCON (see here ) as a side tool to help me to determine the size of DNA ordinal.
Here's some important results from SLCON, as in the page, plus some minor modifications based on some chat and some professional study of ordinal hierarchies: 1st ω-ple 2-stable ordinal; limit of first n 2-nonprojectables = S[SS_ω[σσ_ω]]
(Noted that the SLCON page is actually wrong here for technical reasons because 2-nonprojectables and ω-ple 2-stables are not exactly the same conditions. The former one is actually a stronger condition than the former one. More precisely, the supremum of the first n ω-ple 2-stable ordinals is the first nonprojectable ordinal)
1st (ω+1)-ple 2-stable ordinal; 1st 2-nonprojectable and (+1)-2-stable: S[SS_ω[σσ_ω+1]]
I had done some works to refine the hierarchy on papers. More specifically, to make the hierarchy of fine structures between the two ordinals indicated above. Below is my work:
So any help here would be appreciated, thankyou