A scan of the article  The following words I flagged as could be linked or not as about  I would have had to look up to get the jist more the Xs the less I knew about the topic XRelative computability  XXTuring reduction XXReducibility  XXdegree structure Thue XThue system there is a link to semiThue system Normal form Decidable Undecidable identity element XXXXsimplicial complex XXXhomeomorphic space Diophantine equation XXX local strutural properties global structural properties XXXalpharecursion theory Xfirst order formula  first order arithmetic  Peano axioms  In one place you used the abbreviation re Turing degrees without defining what re stands for  you do define it later in the article  Some backgroundhistory might be useful here or in another article with a recursion context  The word recursion First use  comes from recur to run again L re  currere as in current To repeat Recursion RETURN the determination of a succession of operations on one or more preceeding elements according to a rule or formula involving a finite number of steps or not  recursive a procedure that can repeat itself indefinitely or until a specified condition is met    Peanos definitions that followed the Peano axioms are recursive in nature and these came from Dedekind correct What possessed Dedekind and Peano to investigate this Then what happened in the time between PeanoDedekind and primitive recursive functions who first defined those and recursive functions of GdelTuringPostHerbrandChurchPterAckermannKleene etc Wasnt it the antinomiesparadoxes of Russell ca  and his crowd and Cantors method on top of it  a kink in the LogicistFinitist program of Hilbert So was it Hilberts problems that focused their work on Foundations and Metamathematics Did Brouwer and his band of merry men play a role This brings us up to ca  We have Kleenes book   his professorship which seems to me to be the magnus opus that set off Wang Davis Minsky Melzak Lambek and  and all the host that was to split off to became the theoretical computer scientists and some Russians too isolated by the cold war That left the diehard recursion theorists I loose the recursion theorist thread from  on Hilberts tenth is solved Gdel was on his last legs Chaitin was a student From  point on  who were the players Who or what schools of thought are the major players now I know only the name Chaitin You mentioned a few others You do hint at some of the big problems  Maybe you know of a good historicallyinclined book  The history of recursion theory    On a more humorous take on the matter the random wikistudent who bumps into recursion theory might ask When I grow up choose a college pick a field of study declare my major become more theoretical than computer science I want to be a mathematician  why would I want to be a recursion theorist I think the answer is Because you get to work on some really cool unsolved problems The kid asks Yeah such as Explain My dad was an actuary What the hell is an actuary I used to wonder In rd grade I told people he sold insurance Then when I got in high school he showed me his books of mortality tables he definitely did not ignite in his son the fiery actuarial spark How do you ignite the fire in the mathematics student Hope some of this helps wvbailey