Timeline for Applications of Set theory vs. model theory in mathematics
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Jul 17, 2020 at 3:29 | comment | added | Haim | @TimothyChow It seems to me like an almost empirical fact that in order for a non-set theoretic statement to be susceptible to set theoretic methods, it should either be a statement involving subsets of Polish spaces with a very simple (Borel/analytic) definition (in which case methods from DST can be used) or it shouldn't be absolute (so forcing, large cardinals and inner models might be useful). It seems that non-set theoretic statements that satisfy either of the above requirements are harder to come by, whereas such requirements don't exist in the case of model theoretic applications. | |
| Jul 17, 2020 at 1:50 | comment | added | Timothy Chow | At the risk of stating the obvious: Sets have no structure. Models have structure. It shouldn't be too surprising that they would have applications to different kinds of questions, should it? | |
| Jul 16, 2020 at 19:14 | comment | added | Haim | Dynamics is another example of an area where both set theory and model theory have many applications (again, I don't know which of them satisfy your criterion for being deep), one recent example is this paper by Hrushovski, that makes an interesting connection to the Kechris-Pestov-Todorcevic correspondence: arxiv.org/pdf/1911.01129.pdf | |
| Jul 16, 2020 at 17:35 | comment | added | Haim | @MohammadGolshani What would qualify as a "deep application"? | |
| Jul 16, 2020 at 16:56 | comment | added | Mohammad Golshani | I am essentially looking for a part of mathematics outside of logic (say not a theory), where both set theory and model theory have deep applications. | |
| Jul 16, 2020 at 16:54 | comment | added | Mohammad Golshani | @Haim Yes I know that, even for example I can say Shelah's proof of independence of Ax-Kochen isomorphism theorem from CH vs. the model theoretic works on the subject. But if for example you look at the paper you sent, I think you agree that the subjects of the applications are different from what set theory has, also I think the applications stated are not as deep as those given by set theory. | |
| Jul 16, 2020 at 15:59 | comment | added | Haim | I'm not sure that your observation is correct. For example, in addition to the applications of set theory to $C^*$-algebras, there are also many applications of model theory to the field: arxiv.org/pdf/1602.08072.pdf | |
| Jul 16, 2020 at 13:15 | comment | added | nombre | Related to your second remark: I would say model theory is mostly about first order statements, whereas non elementary set theory deals with higher order statements. | |
| Jul 16, 2020 at 7:48 | history | asked | Mohammad Golshani | CC BY-SA 4.0 |