Timeline for Are there any important geometric consequences of the Generalised Continuum Hypothesis?
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22 events
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| Feb 4, 2021 at 12:22 | comment | added | Tim Campion | I'm actually surprised I can't immediately find a MO question specifically collecting "non-pure-set-theoretic" applications of CH. It's worth linking here and there are quite a number of applications of CH discussed in the answers here. Gosh, neither has the continuum-hypothesis tag! Plenty more here on MO, but I don't immediately see anything else not covered there. Not sure about GCH statements per se, beyond the top answer on the second question, where OP has actually left a comment already. | |
| Feb 4, 2021 at 4:47 | comment | added | Noah Schweber | @MoziburUllah Tim wasn't insulting your intelligence at all, he was asking an important question: "Are you interested in both types of theorem, or are you interested only in theorems which follow from GCH but not from CH?" The fact that GCH trivially implies CH was narrative lead-in to that. | |
| Feb 4, 2021 at 4:27 | comment | added | Tim Campion | @MoziburUllah To reiterate the question about GCH: an important consequence of GCH is CH. There are probably more theorems in the literature which state "Assuming CH, the following is true..." than theorems of the form "Assuming GCH, the following is true..", however both types of theorem are consequences of GCH. Are you interested in both types of theorem, or are you interested only in theorems which follow from GCH but not from CH? | |
| Feb 4, 2021 at 4:24 | comment | added | Mozibur Ullah | @Noah Schweber: Thanks for pointing this out, but to be honest, I'm distinctly underwhelmed by cones. If they were talking about convexity, well that would be something else. | |
| Feb 4, 2021 at 4:21 | comment | added | Mozibur Ullah | @Tim Campion: Your subtleties are lost on me. To me, geometry has a very vivid and concrete existence. GCH has a very well known characterisation as the equality of the alpha and beth cardinality hierchies. As for importance, I let that be understood as however the poster wants to answer it - within the limits of that ever fructable quantity known as mathematical maturity. | |
| Feb 4, 2021 at 4:12 | comment | added | Tim Campion | In the question, the words "important" and "geometric" are both extremely subjective. And we haven't clarified what "GCH" means. Are applications of GCH which are already applications of CH of interest? If so then there are any number of results in general topology to talk about, which are just as "geometric" as the long line! To me, it sounds like OP is unfamiliar with any applications of GCH outside of pure set theory. So I suggest the most appropriate interpretation of the question is "What are some applications of GCH outside of pure set theory?". | |
| Feb 4, 2021 at 1:07 | comment | added | Noah Schweber | @MoziburUllah Googling the authors gets this, which seems right. | |
| Feb 4, 2021 at 1:03 | comment | added | Mozibur Ullah | @Ycor: Do you have a title for the paper? | |
| Feb 4, 2021 at 1:01 | comment | added | YCor | @MoziburUllah Look for a paper by Kramer, Shelah, Tent, Thomas. | |
| Feb 4, 2021 at 0:32 | comment | added | Mozibur Ullah | @YCor: Do you have a reference for that claim? | |
| Feb 4, 2021 at 0:32 | comment | added | Mozibur Ullah | @Dave L Renfro: If not, then it's not really of any relevance to my question. One reason why I find set theory uninteresting is its narrowness. Unlike the other great theories of mathematics like geometry and algebra it stands on a strange isolated limb on its own whilst claiming to represent it's foundations. Large claims. My guess, is that as mathematicians move beyond the doctrinaire positions of second countable manifolds, they will find a wealth of examples to illuminate the concepts of set theory in a much more vivid way and can only benefit mathematics as a whole. | |
| Feb 4, 2021 at 0:14 | comment | added | YCor | CH is equivalent to the statement that all asymptotic cones of the symmetric space of $\mathrm{SL}_3(\mathbf{R})$ are isometric. | |
| Feb 3, 2021 at 23:13 | comment | added | Mozibur Ullah | @Dave L Renfro: Do you know whether any of their work on "implications to CH" was of geometric interest? | |
| Feb 3, 2021 at 22:31 | comment | added | Dave L Renfro | Frederick Otto Bagemihl (1920-2002) and a few of his students studied some implications of CH and variations of CH in cluster set theory. For a short survey, see Bagemihl's paper The present state of ambiguity theory (pp. 85-90 in Infinite and Finite Sets To Paul Erdős on His 60th Birthday, Volume I, 1975). See also his 1967 paper The hypothesis $2^{\aleph_0}\leq {\aleph}_n$ and ambiguous points of planar functions. However, this may not be geometric enough for you. | |
| Feb 3, 2021 at 21:42 | comment | added | Noah Schweber | @MoziburUllah Is this the paper you have in mind? If so, I believe only $\mathsf{CH}$ is relevant there. | |
| Feb 3, 2021 at 21:36 | comment | added | Mozibur Ullah | @Michael Greiner: Please have a look at Rafael Dahmen's paper to see a view to the contrary. He equips the long line with a $C^r$ differentiable structure with r=1 to infinity. These, unlike the usual xase, aten'tbunique. But then again we'be become accustomed to that with Milnors exotic spheres. My personal belief is that geometrising set theory will make set theoretic concepts such as the GCH and large cardinal axioms more vivid. It certainly has that for me with GCH. | |
| Feb 3, 2021 at 21:36 | comment | added | Noah Schweber | @ElliotGlazer You're quite right, I'd forgotten about that - there's no need to go to Schoenfield or large cardinals, that's a much stronger result. (Reference.) | |
| Feb 3, 2021 at 21:30 | comment | added | Elliot Glazer | @NoahSchweber I'm not sure what large cardinals have to do with this. GCH is $\Pi^2_1$-conservative over ZF+DC. Unless you're referring to GCH being projectively conservative over ZF + large cardinals? | |
| Feb 3, 2021 at 21:27 | comment | added | Noah Schweber | @MichaelGreinecker And even more so if we want a $\mathsf{ZFC}$-provable relationship: e.g. under an appropriate large cardinal assumption, $\mathsf{GCH}$ won't affect projective sentences, so you can't hope for a projective "test" for $\mathsf{GCH}$ without proving the inconsistency of those large cardinals. | |
| Feb 3, 2021 at 21:13 | comment | added | Noah Schweber | @MichaelGreinecker Per Schoenfield absoluteness, things will have to get fairly set-theoretic for GCH to be relevant in the first place. | |
| Feb 3, 2021 at 21:11 | comment | added | Michael Greinecker | The long line strikes me as more set-theoretic than geometric, but this might reflect my ignorance. | |
| Feb 3, 2021 at 21:03 | history | asked | Mozibur Ullah | CC BY-SA 4.0 |