Timeline for What is the spectrum of possible cofinality types for cuts in an ordered field? Or in a model of the hyperreals? Or in a nonstandard model of arithmetic?
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| Oct 29, 2020 at 9:30 | history | edited | YCor |
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Sep 5, 2014 at 7:47 | answer | added | Andrej Bauer | timeline score: 1 | |
| Aug 11, 2011 at 20:57 | answer | added | Dave Marker | timeline score: 13 | |
| Aug 11, 2011 at 5:33 | answer | added | Andreas Blass | timeline score: 8 | |
| Aug 11, 2011 at 5:33 | answer | added | Ali Enayat | timeline score: 12 | |
| Aug 11, 2011 at 2:17 | answer | added | boumol | timeline score: 10 | |
| Aug 11, 2011 at 2:10 | comment | added | Joel David Hamkins | Shelah's arxiv article: arxiv.org/PS_cache/math/pdf/0112/0112212v4.pdf | |
| Aug 11, 2011 at 2:10 | comment | added | Joel David Hamkins | Shelah's "symmetrically closed" fields are precisely those in which all unfilled cuts have mismatched cofinality types, in my terminology. He describes a "symmetric closure" process by which any field can be symmetrically closed, by systematically filling the symmetric cuts, and proving that this process terminates transfinitely. | |
| Aug 11, 2011 at 2:04 | comment | added | Joel David Hamkins | Yes, and I noticed your answer to the other question. Shelah's paper is great! And it definitely answers this question also. Indeed, the point of that paper seems to be to produce fields exactly affirming the property in my question 1. Please post as an answer. | |
| Aug 11, 2011 at 1:48 | comment | added | boumol | As far as I understand you are also interested on "an ordered field that is not complete, but which nevertheless exhibits the nested interval property". I will suggest you take a look at the paper "Quite Complete real closed fields" by Shelah. | |
| Aug 10, 2011 at 20:59 | comment | added | Joel David Hamkins | Amit, yes that is what I mean. (And all downvotes have evidently been retracted.) Gerhard, thanks for the idea, which may help. I know that in the context of a general linear order, one can arrange any kind of crazy behavior for the cofinality types of the cuts; my question is to what extent you can do this inside an ordered field or nonstandard model of the reals or of arithmetic. | |
| Aug 10, 2011 at 19:48 | comment | added | Gerhard Paseman | Here is a suggestion which may be off-the-wall, or it may be useful. Dedekind-Macneille completion works for posets to produce a lattice; some of the issues you bring up here may have been addressed in the literature on such completions. Gerhard "Ask Me About System Design" Paseman, 2011.08.10 | |
| Aug 10, 2011 at 19:48 | comment | added | Amit Kumar Gupta | Never mind, that's clearly what you mean. | |
| Aug 10, 2011 at 19:47 | comment | added | Amit Kumar Gupta | I'm guessing whoever down-voted meant to up-vote and hit down accidentally. Anyways, by "upper cofinality" and "lower cofinality" are you referring to $\kappa$ and $\lambda$, respectively, in a cut of cofinality $(\kappa,\lambda)$? | |
| Aug 10, 2011 at 19:19 | history | asked | Joel David Hamkins | CC BY-SA 3.0 |