`  == A scan of the article == The following words I flagged as could be linked (or not), as about 1/2 I would have had to look up to get the jist (more the X's the less I knew about the topic): :XRelative computability > XXTuring reduction :XXReducibility,  :XXdegree structure :Thue, XThue system, there is a link to semi-Thue system :Normal form :Decidable :Undecidable, :identity element :XXXXsimplicial complex :XXXhomeomorphic space :Diophantine equation :XXX ``local strutural properties, global structural properties`` :XXXalpha-recursion theory :Xfirst order formula =? first order arithmetic > Peano axioms  In one place you used the abbreviation ``r.e. Turing degrees`` without defining what ``r.e.`` stands for  you do define it later in the article.  Some background/history might be useful (here or in another article with a recursion context):  The word recursion: First use (1616), 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.``)    Peano's 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 Peano-Dedekind, and ``primitive recursive functions`` (who first defined those?) and ``recursive functions`` of Gödel-Turing-Post-Herbrand-Church-Péter-Ackermann-Kleene etc? Wasn't it the antinomies/paradoxes of Russell ca 1900 and his crowd? and Cantor's method on top of it  a kink in the Logicist/Finitist program of Hilbert? So was it Hilbert's 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 1950. We have Kleene's book (1952) (+ 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 die-hard recursion theorists. I loose the ``recursion theorist`` thread from 1960 on. Hilbert's tenth is solved. Gödel was on his last legs, Chaitin was a student. From 1960 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 (historically-inclined) book  ``The history of recursion theory``?    On a more humorous take on the matter: the random wiki-student 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 3rd 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 `