Timeline for A Question Regarding Boolean-valued Models
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Feb 9, 2013 at 0:55 | history | edited | user9072 |
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| Dec 2, 2012 at 13:09 | comment | added | Thomas Benjamin | 'multiverse' do they seem to choose (I am asking for the "sociological fact"....)? | |
| Dec 2, 2012 at 13:05 | comment | added | Thomas Benjamin | @Emil: The reason that Skolem believed that ZF (ZFC) was not a "privileged logical theory" was because "Even the notions of 'finite','infinite', 'simply infinite sequence',and so forth turn out to be merely relative within axiomatic set theory [in this case, first-order ZF (ZFC)--my comment]." Do most mathematicians use ZFC as a foundational theory or merely the 'language of set theory' as a convenient (and useful) language in which to define mathematics. If in fact they use ZFC as a foundational theory, do they implicitly choose the 'set-theoretic multiverse'? Which universe in the | |
| Nov 26, 2012 at 14:50 | comment | added | Emil Jeřábek | (Ideas for alternative foundations that have nonnegligible following are categorial approaches which intend to base mathematics on category theory instead of set theory, and constructivist approaches which often use some sort of intuitionistic theory of types. [There are nontrivial connections between these two.] Neither group is going to be impressed by a proposal of another set theory.) | |
| Nov 26, 2012 at 12:56 | comment | added | Emil Jeřábek | That most mathematicians use (or are willing to accept if someone presses them to consider the issue, which is often irrelevant to their everyday research) ZFC as the foundational theory is a sociological fact concerning a pragmatic choice. As much as I would agree that ZFC is hardly a “logically privileged theory” even without reading Skolem’s paper, this has no relevance in it. | |
| Nov 24, 2012 at 5:02 | comment | added | Thomas Benjamin | Also, the title of the paper I mentioned to Andres is Timothy Chow's "A Beginner's guide to forcing". I am also interested in getting the title and cite for the Hajek paper so I can look it up. | |
| Nov 24, 2012 at 1:01 | comment | added | Thomas Benjamin | Actually, the Skolem quote should be "privileged logical theory". Sorry. | |
| Nov 24, 2012 at 0:54 | comment | added | Thomas Benjamin | @Emil: Thanks also for your comments. Very helpful. Regarding the portion of your comment that ZFC has been successfully adopted by the mainstream mathematical community as the foundation of mathematics (is this in fact a misquote?), I refer you to Skolem's paper "Some remarks on axiomatized set theory" (van Heijenoort, pp 290-301). If Skolem's critique of Zermelo's set theory (and by extension ZFC) is correct then first-order ZFC cannot deemed a "logically privileged theory" which is necessary for ZFC to be an adequate foundation for mathematics. | |
| Nov 24, 2012 at 0:34 | comment | added | Thomas Benjamin | Do you know of any papers on Boolean-valued models proper that attempt to do this (unless you hold that Boolean algebras with values between 0 and 1 are a type of 'fuzzy logic', in which case a paper like "The Beginner's Guide to Forcing" might be adequate)? If you do please let me know. | |
| Nov 24, 2012 at 0:18 | comment | added | Thomas Benjamin | @ 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. | |
| Nov 23, 2012 at 12:08 | comment | added | Emil Jeřábek | ... 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. | |
| Nov 23, 2012 at 12:00 | comment | added | Emil Jeřábek | 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), ... | |
| Nov 23, 2012 at 7:16 | comment | added | Andrés E. Caicedo | 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) | |
| Nov 23, 2012 at 6:09 | comment | added | David Roberts♦ | Topos theory should probably be mentioned here. | |
| Nov 23, 2012 at 5:52 | history | asked | Thomas Benjamin | CC BY-SA 3.0 |