Scott-Solovay unpublished paper on ``Boolean valued models of set theory''

Question

I have read some papers from 1970$^{th}$, and in some of them, the paper of Scott and Solovay on ``Boolean valued models of set theory'' is given as a main reference, with many references to the results from it. Unfortunately the paper never published.

Question 1. Does anyone know the historical reasons for not publishing the paper?

Of course I know there are some papers and books covering the topic, in particular:

1) Boolean-valued set theory and forcing by Richard Mansfield, John Dawson.

2) Set Theory: Boolean-Valued Models and Independence Proofs by John Bell.

The second reference gives some historical points about the creation of Boolean valued method. Though the above references are very good for learning the method, I am mainly interested in the original paper.

Question 2. Is there any typed or scanned version of the Scott-Solovay paper available? how can I find a version of the paper?

Of course, maybe the simplest answer is: send an email to one of the authors, and ask them about the paper. But I would rather first try the Mathoverflow.

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As it is stated in the answer below (and I was aware of it), the paper by Scott ``A proof of the independence of the continuum hypothesis'' presents some aspects of the theory. But it does not give answer to my questions.

Answer 1

Scott was editor of the Oxford logic guides and was involved in the preparation of Set Theory: Boolean-Valued Models and Independence Proofs (Oxford Logic Guides). He wrote a foreword to it and in this passage he discusses the reasons it was not published:

"There are many references in the literature to the Scott–Solovay paper, which was to be published as an expanded version of the 1967 notes. This paper does not exist, and it is my own personal failing for not putting it together from the materials I had at hand. I discussed it several times with Robert Solovay, but we were not at the same institution and could not work very closely together. He drafted parts of certain sections, but he was working on so many papers at the same time that he did not have the opportunity to draft the whole paper. The present book essentially supplants the projected Scott–Solovay paper. Part of my own difficulty about writing the Scott–Solovay paper was the fantastic growth of the field and the speed with which it changed. During the winter of 1968–1969 I became profoundly discouraged because I felt unable to make any original contributions: any ideas I had were either wrong or already known. It is easy enough to say now that I should have been content to be a reporter and expositor, but, at the very moment when one is being left behind, things seem less pleasant. I put these remarks forward not as an excuse but simply as an explanation of why I could not complete what I set out to do."

Answer 2

It is worth reading the following paragraphs from the G. Moore's paper "The origins of forcing". The paper may lead you to other useful resources.

In his model Solovay had used Borel sets of positive measure as forcing conditions. Solovay pointed out to Scott, at Stanford in September 1965, that Cohen's definition of forcing could be regarded as a way of assigning Boolean values to formulas [Scott 1967, 108; 1978, xiv]. Moreover, Solovay noticed that if he associated with each formula the pair of sets consisting first of those conditions forcing the formula and, second, of those forcing the negation of the formula, then he obtained a complete Boolean algebra. Solovay did not think of the Boolean algebra topologically, and when he showed it to Scott at Stanford, Scott pointed out that it consisted of regular open sets [Solovay 1982].

Scott reversed the process: Since it was possible to start with forcing and obtain a Boolean algebra, then it was possible to start with Boolean-valued logic and obtain forcing from it. Meanwhile, Solovay had independently come to the same conclusion [Scott 1967, 109; 1978, xv]. At that point there occurred a dispute between Scott and Solovay over Boolean-valued models. Solovay was afraid that Scott, whom he then still regarded as a "big name", was going to steal his result from him. Then their differences were resolved, and they agreed to prepare a joint paper, which was partially written but never completed [Solovay 1981]. Instead, Scott's lecture notes on Boolean-valued models, given in 1967 at the U.C.LA. set theory conference, circulated widely among the cognoscenti.

Answer 3

This is just a long comment and it is possible that nothing is new here for you.

"D. Scott, A proof of the independence of the continuum hypothesis", is an interesting prequel to the then "in preparation" Scott-Solovay paper (see reference 8 here) that never appeared.

It also appears that Scott lectured on this topic at the Fourteenth annual summer research institute on axiomatic set theory at UCLA in July-August 1967. The lecture notes were supposed to appear in volume two of these proceedings (see Shoenfield's article in volume one) but it never appeared in print.