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I just want to make sure that I have the correct reference for the original theorem of shooting a club of singular cardinals to make a Mahlo cardinal become a non-Mahlo inaccessible cardinal. I can't access the paper right now, and wondering if can verify that this theorem is in this paper? Is this just attributed to just Jensen or Jensen and Solovay?

R. B. Jensen and R. M. Solovay, Definable sets of minimal degree, in: Mathematical Logic and Foundations of Set Theory: Proceedings of an International Colloquium Under the Auspices of the Israel Academy of Sciences and Humanities, Jerusalem, 11-14 November 1968, edited by Y. Bar-Hillel, Studies in Logic and the Foundations of Mathematics Vol. 59 (North-Holland, 1970), pp. 122–128.

Thank you for any help.

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    $\begingroup$ Hi, Erin! This is not the correct reference, the theorem seems unrelated to the content of the paper, and Solovay is not one of the authors. The word Mahlo does not appear in the paper (which is striking since Jensen really likes to use this word). Jech does list this as exercise 21.4, which is attributed to Jensen in the notes at the end of the chapter, but without any reference. I wonder if it is unpublished. But Jensen did invent club shooting forcing, so it seems likely the attribution to him is correct. $\endgroup$
    – Gabe Goldberg
    Commented Aug 19 at 0:00
  • $\begingroup$ Thanks Gabe! I have a paper book version of my thesis and because of my fault all of the references are printed ?? so I was taking a guess since I thought it was due to Jensen. I will attribute it to him, I am just going to mention it. I also tried "Forcing Closed Unbounded Sets" and "Adding a closed unbounded set" by other authors but I believe you that Jensen invented club shooting. Thank you!! $\endgroup$
    – Erin Carmody
    Commented Aug 19 at 0:24
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    $\begingroup$ Actually, I was wrong! I guess shooting a club is due to Baumgartner, Harrington, Kleinberg in the paper "Adding a closed unbounded set" you mentioned. But Jech says the Mahlo-killing thing is due to Jensen, so it must be the case. $\endgroup$
    – Gabe Goldberg
    Commented Aug 19 at 7:41
  • $\begingroup$ That makes sense! Thank you! $\endgroup$
    – Erin Carmody
    Commented Aug 19 at 11:25

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UPDATE/CORRECTION: The original reference for the result asked about is the following 1968 paper of Jensen and Solovay (and the credit goes to Jensen, as explained below).

Jensen, R. B and Solovay, R. M., Some applications of almost disjoint sets. Mathematical Logic and Foundations of Set Theory (Proc. Internat. Colloq., Jerusalem, 1968), pp. 84–104.

MORE DETAIL: The method of eliminating the Mahlo property while preserving strong inaccessibility invented in the above paper was generalized in the following paper of William Boos (the paper is based on his doctoral dissertation at the University of Wisconsin, written under the direction of Kenneth Kunen).

William Boos, Boolean Extensions which Efface the Mahlo Property, The Journal of Symbolic Logic, Vol. 39, No. 2 (Jun., 1974), pp. 254-268.

The first paragraph of the MathSciNet review (by Frank Drake) of the above paper of Boos reads as follows (the bold font is my own enhancement).

The author presents four forcing constructions (equivalently: complete Boolean algebras), all concerned with altering the Mahlo properties of a cardinal, and stemming in the first place from a construction of Jensen [see R. B. Jensen and R. M. Solovay, Some applications of almost disjoint sets, Mathematical logic and foundations of set theory/ (Proc. Internat. Colloq., Jerusalem, 1968), pp. 84–104, North Holland, Amsterdam, 1970; MR0289291]. Jensen's construction took a strongly Mahlo cardinal and (in the generic extension) destroyed the Mahlo property while leaving the cardinal still strongly inaccessible. The first of the author's constructions is a generalization of this, and takes a cardinal $\kappa$ that is strongly $\kappa$-Mahlo, and an ordinal $\alpha<\kappa$, and makes $\kappa$ strongly $\alpha$-Mahlo but not $\alpha + 1$-Mahlo in the extension. The second combines the first with an Easton-type argument to make $\kappa$ the first strongly $\alpha + 1$-Mahlo cardinal.

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  • $\begingroup$ Thank you! I do remember that Boos did a lot with Mahlo cardinals. $\endgroup$
    – Erin Carmody
    Commented Aug 21 at 1:06
  • $\begingroup$ Thank you very much this is great to hear! $\endgroup$
    – Erin Carmody
    Commented Aug 21 at 14:28
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    $\begingroup$ @ErinCarmody You are most welcome, I certainly fine-tuned my knowledge of history of the subject by answering your question. $\endgroup$
    – Ali Enayat
    Commented Aug 21 at 15:27

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