Timeline for Continuum Hypothesis
Current License: CC BY-SA 4.0
11 events
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| May 19, 2022 at 9:59 | history | edited | Martin Sleziak | CC BY-SA 4.0 |
added a Wayback Machine link for the dead link
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| Jan 2, 2013 at 22:32 | history | edited | Andrés E. Caicedo | CC BY-SA 3.0 |
added 13 characters in body
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| Dec 31, 2012 at 18:42 | vote | accept | CommunityBot | ||
| Dec 30, 2012 at 23:47 | comment | added | Andrés E. Caicedo | Hi @Sally I agree. It may be in my blind spot, but I do not know of much work relating algebra and CH. There is a lot of ongoing work on consequences of CH in analysis, some of it with a strong algebraic emphasis (as in the existence of discontinuous homomorphisms, or the study of certain ideals of Banach algebras, etc) but there does not seem to be such activity on the purely algebraic front. Part of it may be the development of pcf theory, that proves useful to the study of algebraic structures of large size, but does not seem to give much information at the level of the continuum. | |
| Dec 30, 2012 at 19:08 | comment | added | Theo Buehler | The point of "naturality" is exactly what I meant by emphasizing "algebraic". Thanks a lot for all the additional pointers and all these interesting links in your answer. | |
| Dec 30, 2012 at 19:03 | comment | added | user30300 | Thanks for the comments. But it seems that there is not much well-known equivalent algebraic form for CH (compared to the long list of algebraic forms of AC). am I right !? And Happy New Year to all. | |
| Dec 30, 2012 at 18:46 | comment | added | Andrés E. Caicedo | (And yes, Fremlin's book is an excellent reference for early work on Martin's axiom and cardinal characteristics.) | |
| Dec 30, 2012 at 18:45 | comment | added | Andrés E. Caicedo | The place to look would be "Almost Free Modules: Set-theoretic methods", Revised edition, by Eklof and Mekler, North-Holland Mathematical Library vol. 65 (2002), but I am not familiar with equivalences of CH discussed there. | |
| Dec 30, 2012 at 18:43 | comment | added | Andrés E. Caicedo | Hi Theo: I do not know of "natural" algebraic characterizations, beyond Mariano's example. The third example I list using transcendence bases is not quite natural, for instance, in the sense that it is not something I would expect an algebraist would wonder about. There are a few additional natural examples from Ramsey theory (in partition calculus, and also generalizations of Schur's theorem by Jacob Fox and his collaborators), and from discrete geometry (de la Vega, and Schmerl, they have a couple of papers in Fundamenta), but purely algebraic ones seem missing. | |
| Dec 30, 2012 at 18:02 | comment | added | Theo Buehler | Hi Andres. Some minor points: 1) The question asked especially about algebraic equivalents of CH. Do you happen to know one beyond the one mentioned by Mariano in another thread? 2) I think the original article by Erdős is very readable projecteuclid.org/euclid.mmj/1028999028 and crystal clear. 3) I would add Gödel's Monthly article jstor.org/stable/2304666 to recommended reading. 4) Since you brought up MA: As a non-set theorist who happened to need to understand it I can recommend Fremlin's book books.google.com/books?id=tXVrPwAACAAJ Happy New Year! | |
| Dec 30, 2012 at 17:29 | history | answered | Andrés E. Caicedo | CC BY-SA 3.0 |