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The continuum hypothesis is known to be independent (neither provable nor disprovable) within the ZFC axioms. But as I understand it, mathematical realists (e.g. Platonists) believe that there is a single "correct" model of mathematics that corresponds to the real world, and therefore that every well-formed mathematical proposition is either "actually true" or "actually false", regardless of whether it can be proven in any particular axiom system.

The continuum hypothesis is perhaps the simplest and most intuitive claim known to be undecidable within ZFC. Do most mathematical realists believe it to be true or false?

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    $\begingroup$ To choose a model is OK, but to think that this model "corresponds to the real world" whatever this means, would be a huge overestimation of what mathematics can do.I heard that many mathematicians favour the possibility that the continuum hypothesis is false. Unfortunately , I do not remember details about the reasons. $\endgroup$ – Peter Sep 20 '17 at 17:56
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    $\begingroup$ I don't believe the truth (or not) of continuum hypothesis is a central issue for many "real world" mathematical applications. $\endgroup$ – BruceET Sep 20 '17 at 17:56
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    $\begingroup$ @hardmath "V=L" ? Never heard of this... $\endgroup$ – Peter Sep 20 '17 at 17:58
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    $\begingroup$ @Peter: Here it means that the universal class and the class of constructible sets are equal, which is how Gödel showed (inter alia) the continuum hypothesis is consistent with ZFC. $\endgroup$ – hardmath Sep 20 '17 at 18:02
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    $\begingroup$ My understanding is that amongst those mathematicians who believe CH has a definite truth value, the majority view is that it is probably false. $\endgroup$ – Noah Schweber Sep 20 '17 at 18:44
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The answer to your question may be geography-dependent. While on the West side of the Atlantic the dominant view is that CH is "probably false", on its East side and specifically in France, the opinions are heavily influenced by those of the Realist and Platonist Alain Connes who is convinced CH is true, as mentioned in this publication and also this. Connes discusses CH in detail in his A triangle of thought.

You should realize that it is not just because one calls himself a realist that one necessarily believes that CH has a definite truth value in set theory. Joel David Hamkins calls himself a realist; however he is not a realist of a set-theoretic universe but rather of a set-theoretic multiverse, where one slips effortlessly from a set-theoretic universe where CH is true to a set-theoretic universe where CH is false, at the click of a switch.

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    $\begingroup$ Maybe I'm missing something, but in the article I don't see any citation for Connes' belief that CH is true, and googling "Connes "continuum hypothesis"" doesn't pull anything up that I can tell; can you provide such? $\endgroup$ – Noah Schweber Sep 24 '17 at 22:09
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    $\begingroup$ The main source for this is his Triangle of Thought. We mentioned this in at least two articles. @NoahSchweber $\endgroup$ – Mikhail Katz Sep 25 '17 at 13:22
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    $\begingroup$ Sorry, but I'm still not convinced. Your linked articles mention that Connes uses CH, but that doesn't mean he believes it is true in a Platonic sense. I can't immediately find anything (including in your linked articles) where he asserts this stronger claim. Even a Platonist may prove theorems in a system whose truth they are not convinced of. "[Connes] is convinced CH is true" is quite a strong claim, and I don't yet see any strong evidence for it. (cont'd) $\endgroup$ – Noah Schweber Sep 25 '17 at 13:45
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    $\begingroup$ As to "A triangle of thoughts," I don't have it on hand, but amazon's book search pulls up only one instance of the phrase "continuum hypothesis," and just as an example of a ZFC-undecidable sentence. Can you provide a quote from Connes, demonstrating that he "is convinced CH is true" (rather than an extrapolation of his philosophical position from the axiom system he uses)? Or that belief in CH is widespread on the east of the Atlantic? $\endgroup$ – Noah Schweber Sep 25 '17 at 13:47
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    $\begingroup$ "f France had a strong contingent of logicians or set theorists they would probably not automatically subscribe to Connes' views" OK, but still: can you provide evidence for the claim that "specifically in France, the opinions are heavily influenced by those of the Realist and Platonist Alain Connes who is convinced CH is true"? I'm not being willfully obtuse here, I honestly don't see any evidence of this or of the broader claim that belief in CH is widespread east of the Atlantic. $\endgroup$ – Noah Schweber Sep 25 '17 at 14:06

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