Around 1974, Leo Harringto wrote an unpublished note entitled "The constructible reals can be anything", in which he proved that it is consistent that being $\Delta^1_n$ is the same as being constructible.

Harrington proved his theorem using a version of almost disjoint coding of Jensen and Solovay.

Is there any reference in which a proof of the above theorem is given which is similar to the Harrington's original proof.

Remark. There are proofs of Harrington's theorem by Kanovei, using different methods, see for example

(1) Kanovei. The independence of some propositions of descriptive set theory and second order arithmetic. Soviet Math. Dokl . 1975, 16, 4, pp. 937 – 940.

(2) Kanovei. On the nonemptiness of classes in axiomatic set theory. Math. USSR Izv. 1978, 12, pp. 507 – 535.

  • $\begingroup$ This is the second time I've seen a request for Harrington's notes on MO (a different set of notes). You may try to ask some of the people who answered this question: mathoverflow.net/questions/122888/… $\endgroup$ – Jason Rute Jul 10 '14 at 13:05
  • $\begingroup$ @Mohammad Golshani: The actual title of the preprint is, "The constructible reals can be [almost] anything" (as the scan of the preprint shows). It seems an interesting result, but what is its significance (I don't doubt its significance, but am interested in how it relates to the results of Solovay and Jensen regarding nonconstructible sets of integers)? $\endgroup$ – Thomas Benjamin Aug 9 '14 at 16:32

It seems the Harrington's preprint is here http://logic-library.berkeley.edu/catalog/detail/2135 but to access it requires CalNet login. Maybe Harrington himself could help if you write to him: http://math.berkeley.edu/~leo/index.html

P.S. Prof. Harrington kindly sent me a scan of the preprint. It can be found here http://www.inp.nsk.su/~silagadz/Harrington.pdf

  • $\begingroup$ Thank you for the link. I remember I sent an email to Prof. Harrington about his paper some time ago, but I did not receive any answer. $\endgroup$ – Mohammad Golshani Jul 10 '14 at 9:50
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    $\begingroup$ You could write an email to the Logic Group of Berkley, asking for a reprint. logic.berkeley.edu/contact.html $\endgroup$ – Francesco Polizzi Jul 10 '14 at 9:59
  • $\begingroup$ The ms at inp.nsk.su/~silagadz/Harrington.pdf has two handwritten addendums dated 4/75 and 5/75, entitled resp. "Separation without Reduction" and "Addendum, Part II, to The constructible reals can be (almost) anything". I don't know if those are downloadable anywhere though $\endgroup$ – Vladimir Kanovei Sep 28 '14 at 14:33

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