## Devlin’s “Constructibility” as a resource

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It is fairly well-known among set-theorists that Keith Devlin's 1984 book "Constructibility" has flaws in its initial development of fine structure theory. (See Lee Stanley's review 1 of the text for the Journal of Symbolic Logic, for example.) I've had the book on my shelf for twenty years now, and although there is much in there that I find interesting, the fact that I know there are some errors in it means that I've been reluctant to invest a lot of time working through it. So, this brings me to my questions for the experts:

How badly do these flaws mar the rest of the book? Is the damage localized to the initial development of properties of the the J-hierarchy, or is it much more widespread?

Of particular interest to me are the following questions:

1) Is Devlin's treatment of the Covering Lemma for L on solid ground?

2) What about his treatment of morasses?

I know that there are other sources for this material, but I've always appreciated Devlin's writing style.

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1
Having studied Magidor's covering lemma whose proof can be slightly modified to prove the Jensen covering lemma, I can suggest this as an alternative (which is interesting on its own). (Magidor M. Representing Sets of Ordinals as Countable Unions of Sets in the Core Model.** Trans. of the Amer. Math. Soc., Vol. 317, No. 1 (Jan., 1990), pp. 91-126) - link: jstor.org/stable/2001455 – Asaf Karagila Oct 10 2011 at 20:19
Thanks Asaf. Devlin's book is allegedly giving Magidor's proof of the Covering Lemma (which is also the proof outlined in Jech's book) so in theory he shouldn't be using fine structure. That's why I was hoping his treatment of it was solid. I'd like to be able to tell a grad student who wants to see a proof of the covering lemma to "Just go look at V.5 in Devlin's book". (Of course, Magidor is an excellent writer as well, though) – Todd Eisworth Oct 10 2011 at 20:32
I was told by a rather reliable source that Jech skips the hard work of the proof and just gives an outline as though it is simple. Magidor is an avid writer, and he teaches even better. :-) – Asaf Karagila Oct 10 2011 at 21:05
You may also find some slightly relevant help in the comments to this question: mathoverflow.net/questions/38573 – Asaf Karagila Oct 11 2011 at 15:22

1
My summary of the review: consider BS to be RUD (sorry for the bad joke.) – Trevor Wilson Jul 26 at 16:59

## 1 Answer

Mathias has a paper where he corrects the flaws that occur in Devlin's theory BS (= Basic Set Theory). The theory has to be only slightly strengthened to be correct. (It is more than sufficent to add an axiom that asserts for any set and any $n \in \omega$ there is a set of all its sized $n$-subsets.) It is really only in dealing with syntax and showing that certain straightforward concepts are $\Delta_1$ that BS comes unstuck.

I think the book can be safely read beyond a certain point. It is true that Devlin has not correctly proved that the satisfaction relation for $\Delta_0$ formulae is $\Delta_1$, but one can just take the attitude that the result is correct (as Jensen showed), it is just that that particular development in BS failed. BS needs another axiom and all would be well.

Thus, the constructions of the $J$-hierarchy, the fine structural concepts of projecta, mastercodes these are all fine, as are the constructions of trees, $\Box$, Morasses etc, and the Covering Lemma can all be safely read since there one is past the point where these delicate matters are being considered.

Mathias: Weak Systems of Gandy, Jensen and Devlin, Centre de Recerca Matem`atica, Barcelona, 2003--2004, Birkh\"auser Verlag, 2006, Eds: Bagaria, Joan and Todorcevic, Stevo, Trends in mathematics Series.

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Thanks Philip! This is perfect! – Todd Eisworth Nov 17 2011 at 14:32
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