What were the intuitions motivating the creation (or discovery, if you will) of Boolean-valued models? I have searched for the Scott-Solovay paper on the subject, but to no avail. There also seems to be no survey paper dealing with their history. This is a follow-up question to a previous question: "Boolean-valued models vs. the Infinite-valued Logic of Lukasiewicz and set theory". Given Chang's result regarding the consistency of the Axiom of Comprehension (at least a version of it) in infinite-valued logic, what motivated the community of set theorists to take the Boolean-valued model route vs. the infinite-valued logic route (though Boolean-valued models might be classified as a type of infinite-valued logic, there are differences)?
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asked Nov 23, 2012 at 5:52
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$\begingroup$ Topos theory should probably be mentioned here. $\endgroup$– David Roberts ♦Commented Nov 23, 2012 at 6:09
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2$\begingroup$ There is no Scott-Solovay paper, though there are some notes that were informally circulated and were supposed to be turned into it. The material appears in several sources, but the main reference, both for the mathematics, and for historical comments, is the book by John Bell on "Boolean-valued models". The third edition appeared in 2005: Bell, John L. Set theory. Boolean-valued models and independence proofs, third edition. Oxford Logic Guides, 47. The Clarendon Press, Oxford University Press, Oxford, 2005. MR2257858 (2007d:03087) $\endgroup$– Andrés E. CaicedoCommented Nov 23, 2012 at 7:16
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3$\begingroup$ This is a false dichotomy. The right question is why set theorist did not abandon ZFC in favour of Łukasiewicz logic with full comprehension as the foundations of mathematics, and the answer to that should be obvious: ZFC has been extensively developed for half a century (at that point) and it was successfully adopted by the mainstream mathematical community, whereas Chang’s system was based on an unfamiliar logic which is quite awkward to work in (e.g., you have to constantly keep track of how many times you used an assumption of a theorem, due to the lack of contraction), ... $\endgroup$– Emil JeřábekCommented Nov 23, 2012 at 12:00
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3$\begingroup$ ... and it was not at all clear whether the theory is capable of developing common mathematical tools needed to serve as the foundation of mathematics. (After another half a century, not much has changed, except there are some results indicating that the theory is rather inadequate to develop mathematics, namely Hájek proved that it is inconsistent with induction.) Though if you are asking what motivated the set-theoretical community, I’d guess that most of the community never even heard of Chang’s result. $\endgroup$– Emil JeřábekCommented Nov 23, 2012 at 12:08
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$\begingroup$ @ Andres: Thanks for the comment--very helpful. I took a look at the first chapter of Bell's book, hoping it would give an intuitive motivation for generalizing from the {0,1} Boolean algebra to a Boolean algebra with more than two elements but no, all it did (correctly, of course) was show that you can consistently make the generalization. I guess what I am hoping to do is to understand how to interpret the 'intermediate values' between 0 and 1 when the Boolean algebra contains elements other than 0 or 1. $\endgroup$– Thomas BenjaminCommented Nov 24, 2012 at 0:18
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