Sometime around 1975, Leo Harrington wrote a set of notes, apparently 13 pages long, entitled Kolmogorov's $R$-operator and the first nonprojectible ordinal. I do not know how widely they were circulated (or if they were ever available from the UCB library or anywhere).
From what I understand, these notes relate recursion on the first stable ordinal in the first nonprojectible to recursion in a certain explicitly defined type-3 functional. A more precise statement is given in Thomas John's paper “Recursion in Kolmogorov's $R$-operator and the ordinal $\sigma_3$” (J. Symbol. Logic 51 (1986) 1–11), but the proof is not reproduced there. A related but slightly different result of Harrington's, with a different type-3 operator, is quoted (as example 4.10) by Stephen Simpson's “Short course on Admissible Recursion Theory” (355–390 in: Fenstad &al. eds., Generalized Recursion Theory II (Oslo 1977), North-Holland 1978).
I'd very much like to see a copy of these notes, or a proof of any closely related result (e.g., the one quoted by Simpson's paper mentioned above).
The author has been kind enough to see if he can find them, but he isn't too optimistic. I've also written to a number of people who worked in the subject around that time (Sacks, Shore, Simpson and Soare), but without success. So I now turn to MO in the hope that someone has heard of these notes or knows where a copy might be found.
[Xref: link to meta thread]
PS: While I'm aware that offering prizes other than reputation points is frowned upon on MO, if someone should go through the (real-world!) trouble of copying, scanning or mailing these notes for me, I think it would be appropriate that I should respond with a small (real-world!) token of thanks, like an Amazon gift card or something. 🙂
Best Answer
I asked Alekos Kechris. He had a copy and made a scan of it. Here is a link to it.
http://dl.dropbox.com/u/2566697/Harrington.pdf
Regards, Ted