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I love compatibility theory, degree theory etc and I'm astonished by the advances that have been made in the field but it often seems like computability is less cumulative than other areas of mathematics. I get the sense that it other fields it's more common to establish a result in a way that merely uses other results but doesn't require you to dig into their proofs.

Is it really true that computability theory is less cumulative in something like this sense? If so is this some essential aspect of the subject or just because it's still quite a young field? For instance, iin 100 years should we expect computability theory to look mostly the same with just more construction techniques and more kinds of notions to explore or be transformed by methods that let one indirectly prove results without directly managing ever more complexity?

Peter Gerdes
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    I don't think it is possible to predict with confidence what any field will look like in 100 years – Will Sawin Oct 14 '22 at 02:10
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    Not the only field where this feeling is present: https://mathoverflow.net/q/15292/6085 – Andrés E. Caicedo Oct 14 '22 at 02:12
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    @Ben Burns: Perhaps by "less cumulative" is meant "more hair", to use a phase that I recall used from time to time (mid 1970s to maybe mid 1990s; haven't heard it used much in more recent times, and I also can't seem to find it used this way online) for especially icky and highly technical "down and dirty" constructions and proofs. And if not, then I certainly would characterize many of the proofs as having a lot of hair, which doesn't take an expert (which I'm certainly not) to recognize. – Dave L Renfro Oct 14 '22 at 02:26
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    @BenBurns Yes, done. I mean something like what you say but you could imagine that proofs remain dirty but they somehow are able to prove more general results. For instance, lots of forcing proofs in set theory are dirty constructions but they aren't just straight building up a model to demonstrate consistency. I'd call that cumulative even though proofs are no less intricate bc they prove much more with that given lvl of detail. – Peter Gerdes Oct 14 '22 at 04:31
  • I don’t see any clear criteria here for whether proofs “require you to dig”, for what constitutes “a field of mathematics” for comparison, or for assessing 2122, so I have voted to close as “likely to be answered with opinions”. (The motivating comment about age may be similar: recursively incomparable and recursively inseparable sets were proved to exist by multiple people just over 75 and 65 years ago, so I wouldn’t call this “quite a young field”. ) –  Oct 16 '22 at 08:33
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    @DaveLRenfro, the preface to the 3d (1989) edition of Boolos and Jeffrey’s Computability and Logic mentions that kind of hair. –  Oct 16 '22 at 09:14
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    @Matt F.: I've actually read that preface (probably several times -- see "book I actually had a reading course out of in Spring 1990" in this MSE answer), but for what it's worth, the hair mention is at the end of the 1st paragraph of the preface to the 1974 1st edition, which is reprinted in the 3rd edition (I have print copies of 3rd and 5th editions). In looking at this now, I remember seeing the usage here, and this might be the strongest basis for my remembrance of the word, but I've definitely read/heard the word in many other places also. – Dave L Renfro Oct 16 '22 at 19:08

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