Timeline for Examples of set theory problems which are solved using methods outside of logic
Current License: CC BY-SA 4.0
20 events
| when toggle format | what | by | license | comment | |
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| Aug 25, 2018 at 8:55 | answer | added | Danielle Ulrich | timeline score: 8 | |
| Aug 21, 2018 at 5:23 | answer | added | none | timeline score: 6 | |
| Aug 19, 2018 at 2:53 | comment | added | user44143 | I downvoted because answering this just leads to asking whether we call the methods "inside logic" or "outside logic", without any good way to draw the boundary. But I don't mind distinguishing particular formulations as more set-theoretic or analytic or algebraic. So I'd rather see, and I would upvote, an alternative question like: "What set theory problems stated without much algebra have been solved using algebraic methods?" | |
| Aug 18, 2018 at 16:33 | comment | added | Dave L Renfro | Possibly my two comments here might be an example. Also, Frederick Bagemihl and some of his students have obtained results linking the continuum hypothesis to ambiguous points of planar functions, especially Harvey Stanley Fox's 1972 Ph.D. dissertation The Continuum Hypothesis and Planar Functions (not published). | |
| Aug 18, 2018 at 14:40 | comment | added | Uri Bader | there are applications of ergodic theory both to discrepetive set theory and to additive combinatorics, though I guess these are not in the scope of the intended question. | |
| Aug 18, 2018 at 11:48 | comment | added | Erfan Khaniki | @AndrejBauer, Could you give some examples or references for such problems? | |
| Aug 18, 2018 at 9:54 | comment | added | Andrej Bauer | I agree with @JoelDavidHamkins that the boundaries are not clear. Does computability theory count as "logic"? If not, there are problems in intuitionistic set theory that are solved using computability theory. | |
| Aug 18, 2018 at 5:18 | comment | added | Mohammad Golshani | I found the paper Dimension theory and forcing by Zapletal interesting. As it is stated in the paper, the solution is somewhat unusual in that the forcing is concisely defined and analysed in terms of infinite-dimensional topology; however, its combinatorial description is not readily available. | |
| Aug 17, 2018 at 22:43 | comment | added | Lee Mosher | If you had asked for logic problems outside of logic, rather than set theory problems outside of set theory, then maybe Tarski's problems about the first order of free groups theory fit the bill, having been solved by methods of geometric group theory. | |
| Aug 17, 2018 at 22:38 | comment | added | Joel David Hamkins | I don't think logic or set theory have such sharp boundaries that it is possible to provide a definitive answer. Huge parts of set theory, such as Borel equivalence relation theory or set-theoretic topology, are deeply connected with other related areas, and it could sometimes be difficult to describe a method as existing in only set theory or the companion area. | |
| Aug 17, 2018 at 22:24 | comment | added | none | Does Cantor's diagonal proof that the reals are uncountable count as "outside logic"? | |
| Aug 15, 2018 at 20:01 | comment | added | Haim | (cont) I'm not familiar with the details of Talagrand's proof, but it seems that his proof is purely measure theoretic. It is conceivable that a better understanding of Talagrand's ideas will lead to a solution of the above forcing theoretic problem (e.g. by showing that his algebra doesn't add random reals). | |
| Aug 15, 2018 at 19:55 | comment | added | Haim | More of a speculation than an answer: It's conjectured that every Suslin ccc forcing adds a Cohen real or a random real. Shelah showed that if such a forcing adds an unbounded real then it adds a Cohen real. Farah and Zapletal showed that if $\mathbb P$ is Suslin ccc and $\omega^{\omega}$-bounding then $RO(\mathbb P)$ is a Maharam algebra. Therefore, the above problem reduces to the following: Does every Maharam algebra adds a random real? Talagrand showed that there is a Maharam algebra which is not a measure algebra, solving an old problem by Von Neumann... | |
| Aug 15, 2018 at 18:31 | comment | added | Taras Banakh | There are some equivalences of CH in Analysis (like the existence of a Peano curve $(x(t),y(t))$ such for each $t$ one of the coordinate functions $x$ or $y$ is differentiable at $t$). This a theorem of Morayne. The problems about near coherence of ultrafilters can be reformulated in the language of composants of $\beta \mathbb R_+$ (this is also a language outside of Set Theory). | |
| Aug 15, 2018 at 15:36 | comment | added | Timothy Chow | There are theorems in what is commonly referred to as "combinatorial set theory" or "infinitary combinatorics," such as Ramsey theory on infinite sets, that are proved using what one might call "combinatorial" arguments rather than "techniques of mathematical logic." For example there's the Erdos-Rado theorem ams.org/journals/bull/1956-62-05/S0002-9904-1956-10036-0 I'm not sure if this is the sort of thing you're looking for, though. | |
| Aug 15, 2018 at 8:35 | comment | added | Monroe Eskew | Can you give an example problem for which this could conceivably happen? I have a feeling that if someone asks, for example, a consistency question about Banach spaces, and then it turns out to have a ZFC answer via methods internal to functional analysis, then we would just say it wasn't a set-theoretical problem after all. | |
| Aug 15, 2018 at 8:08 | comment | added | Asaf Karagila♦ | Getting tenure? :) | |
| Aug 15, 2018 at 6:13 | comment | added | Carlo Beenakker | not sure whether any (major) problems have been solved in this way, but algebraic set theory might qualify as a technique. | |
| Aug 15, 2018 at 5:49 | history | edited | Martin Sleziak |
added (examples) tag - feel free to revert the edit if you think it is not a good fit
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| Aug 15, 2018 at 4:46 | history | asked | Mohammad Golshani | CC BY-SA 4.0 |