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Why do mathematicians care about the Continuum Hypothesis? Does it have philosophical implications? If it was true or false, would it have had some sort of implications in mathematics?

Does the average mathematician ever need to care about this result?

Asaf Karagila
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user109871
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    I think that nowadays those who care about it are those who expect every statement to have a truth value. – Asaf Karagila Oct 08 '18 at 18:40
  • I've heard of some proofs of conjectures that are thought to be true independent of the Continuum Hypothesis but nonetheless make essential use of the Continuum Hypothesis in their proof. I can't recall what those conjectures were off the top of my head, but they were in topology. – Hayden Oct 08 '18 at 18:44
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    Your post would do better without the final question, which is entirely subjective. Pick your (least) favorite topic $X$ in any branch of mathematics and you can ask the same question: Does the average mathematician ever need to care about $X$? – Lee Mosher Oct 08 '18 at 18:45
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    @Lee: Since $X$ is one of the most used topological spaces, I imagine that the answer is yes. :P – Asaf Karagila Oct 08 '18 at 18:46
  • Some results are dependent on CH, e.g. global dimension of $\prod\mathbb{C}$: https://mathoverflow.net/questions/68436/what-the-heck-is-the-continuum-hypothesis-doing-in-weibels-homological-algebra – freakish Oct 08 '18 at 18:47
  • https://math.stackexchange.com/questions/79346/proofs-given-in-undergrad-degree-that-need-continuum-hypothesis https://math.stackexchange.com/questions/627098/what-would-a-world-where-mathsfch-is-false-look-like https://math.stackexchange.com/questions/648550/statement-that-is-provable-in-zfcch-yet-unprovable-in-zfc-lnot-ch https://math.stackexchange.com/questions/675400/continuum-hypothesis-iff https://math.stackexchange.com/questions/2811629/has-a-counterexample-to-the-continuum-hypothesis-ever-turned-out-to-be-useful – Asaf Karagila Oct 08 '18 at 18:52
  • @Rushabh: I don't see how the axiom of choice is related to this. – Asaf Karagila Oct 08 '18 at 18:56
  • Useful references: Sierpinski's 1934 well-known book Hypothèse du Continu and Steprans' recent survey paper History of the continuum in the twentieth century. – Dave L. Renfro Oct 08 '18 at 19:04
  • @Asaf I think that is a bit of an oversimplification. It turns out to be a very useful assumption in certain settings. Even if one is agnostic or worst :-) about truth and such, surely one is still interested in its usefulness. (And, of course, in some cases, it is versions of its negation that do the same.) – Andrés E. Caicedo Oct 08 '18 at 20:32
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    @user109871 What is an "average mathematician"? A mediocre one? if you mean "a typical one", then rather than saying that you need to explain what you mean, what sort of mathematicians you are trying to exclude in your description, and why you feel they need to be excluded (and why their exclusion is a meaningful thing to do). Part of what I mean is that if you intend to exclude people that actively think about blah, and then ask those that remain to say something insightful about blah, it is reasonable to find whatever is said suspect. – Andrés E. Caicedo Oct 08 '18 at 20:36
  • @Andrés: Sure, but it's not in the sense that it's a deeply philosophical important thing. It's an interesting axiom, with interesting consequences. (Even if I like to say it's the most hated axiom, whenever you prove something under CH, the first question is "How do we get rid of CH?") – Asaf Karagila Oct 08 '18 at 20:45
  • @Asaf But it seems to me that there is a bit of a gap between "does it have philosophical implications?" and "it's a deeply philosophical important thing". As it happens, I think the question of the truth-value of CH is indeed of deep philosophical significance, regardless of its mathematical relevance. I also think that it leads to a rich mathematical theory. Either one of those two aspects may be valuable (of course, to different audiences, for different reasons) even if not related. – Andrés E. Caicedo Oct 08 '18 at 20:54
  • @Andrés: Well, as you may well know, I have an issue with "truth value" in an absolute sense. :-) – Asaf Karagila Oct 08 '18 at 22:30
  • @Asaf Yes, and what I meant by "the question of the truth-value of CH" is whether it makes sense, just as much as what it is, if it does. – Andrés E. Caicedo Oct 08 '18 at 22:51
  • @Andrés: Simple answer to that one "doesn't make sense, moving on." :-P – Asaf Karagila Oct 08 '18 at 22:52
  • @Asaf Pretty much what Feferman said... – Andrés E. Caicedo Oct 08 '18 at 22:57
  • @Asaf But I think your position is even more extreme. – Andrés E. Caicedo Oct 08 '18 at 22:57
  • @Andrés: I have an issue with Feferman's stand. He claims the question is nonsensical. I think the question makes a lot of sense, as a question, the answer is the truth value doesn't make sense. But because all truth values make no sense. In any case, I think we're drifting to my opinion here, which is a bit off topic. So let's stop. :-) – Asaf Karagila Oct 08 '18 at 22:59
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    @Asaf We definitely need to get together for a drink or three. – Andrés E. Caicedo Oct 08 '18 at 23:00

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