Timeline for Is there any danger far from home? (Edited & Revised Version) [closed]
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22 events
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| Dec 1, 2013 at 7:51 | comment | added | user42090 | @NoahS: Thanks for your useful comments. Actually my main purpose in this question was just sharing my intuition about a possible correspondence between provable statements from a theory (which is a syntactical notion) and near statements to a theory (which is a geometric intuition related to the notion of distance). I should confess that the case is much more complicated than what I described above but I feel it worth to send it as a post in MO even if it is not a very exact question. Because perhaps it could be interesting and useful for some people with different approaches. | |
| Dec 1, 2013 at 4:05 | comment | added | Noah Schweber | Tl;dr: when talking about logics, a good rule of thumb is to take slightly more care than feels necessary in making your ideas precise. (I have been guilty of breaking this rule many, many times.) | |
| Dec 1, 2013 at 4:02 | comment | added | Noah Schweber | - or else, that negation doesn't behave classically. Now, these situations would not necessarily be uninteresting, although I'm not really sure what it would be "doing," but it's certainly strange enough that some motivation is needed (what exactly our notion of proof is, why it is what it is, etc.). And this is the work that I think is missing from this question. | |
| Dec 1, 2013 at 4:00 | comment | added | Noah Schweber | In fact, here's an easy argument that for any notion of infinitary proof, either no "eventual inconsistencies" such as you describe arise, or the proof notion is very far from classical logic: Suppose $s$ is a sentence such that $ZFC+s$ is consistent but not "$L_{\alpha\alpha}$-consistent," i.e., $ZFC$ proves $\neg s$ at "level" $\alpha$. By the completeness theorem, $ZFC+s$ has a model $M$; but since $s$ holds in $M$, $\neg s$ does not hold in $M$. This means that our notion of infinitary proof is strictly stronger than satisfiability, which is odd - (cont'd) | |
| Nov 30, 2013 at 16:08 | comment | added | Noah Schweber | The paper "Barwise: Infinitary logic and admissible sets" has some stuff explaining the difficulty of "proof" in infinitary logics; in particular, see page 14, where a completeness theorem is described for fragments of infinitary logic, but does not extend to all infinitary logics. | |
| Nov 30, 2013 at 16:04 | comment | added | Noah Schweber | Andrej, that's a good point. I hope I didn't come off as aggressive above. @Georg: The notions of "proof" and "contradiction" for infinitary logics are a bit more subtle than you seem to be giving them credit for. In particular - at least, in all systems I know about - there are unsatisfiable $L_{\infty, \infty}$-sentences which do not "prove" a contradiction in any nice sense of the word "prove." The point is: you need to actually define what you mean by "proof," etc. | |
| Nov 30, 2013 at 9:33 | comment | added | Andrej Bauer | The question is flawed in the sense that a competent researcher in logic and set theory would be aware of the problems it contains. But I see completely trivial questions about constructive math asked all the time and nobody seems to mind. To my mind, there is a strong bias here. If I downvoted the trivialities asked about cosntructive math everyone would get on my case. So please the set-theory posse, show some mercy. | |
| Nov 30, 2013 at 5:52 | comment | added | user42090 | @NoahS: What is the problem exactly? The notions of "proof" and "contradiction" are well-defined in each infinitary logic $L_{\alpha,\beta}$ and so we can define a theory to be "consistent" if "it cannot prove a contradiction in the logic $L_{\alpha,\beta}$". | |
| Nov 30, 2013 at 5:05 | comment | added | Noah Schweber | Already at Definition 1, there's a problem - what does "consistent" mean for infinitary logics? | |
| Nov 30, 2013 at 3:13 | comment | added | user42090 | @FrançoisG.Dorais@Stefan Kohl@Steven Landsburg@Andres Caicedo@Andreas Blass@Dima Pasechnik: It seems there was a problem in my system. The previous edit was incomplete in appearing. I added a complete revision. Please reopen the question. | |
| Nov 30, 2013 at 3:11 | history | edited | user42090 | CC BY-SA 3.0 |
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| Nov 30, 2013 at 0:01 | review | Reopen votes | |||
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| Nov 30, 2013 at 0:01 | history | edited | user42090 | CC BY-SA 3.0 |
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| Nov 29, 2013 at 23:43 | history | edited | user42090 | CC BY-SA 3.0 |
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| Nov 29, 2013 at 17:34 | history | closed |
Stefan Kohl♦ Steven Landsburg Andrés E. Caicedo Andreas Blass Dima Pasechnik |
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| Nov 29, 2013 at 15:15 | comment | added | François G. Dorais | @Johannes: I didn't downvote but there are two reasons, in addition to the uninformative title, why this question is not well thought out. 1.- The Compactness Theorem trivializes the notions defined in the question. 2.- A Wikipedia search for "infinitary logic" or related terms would direct to the correct definitions. | |
| Nov 29, 2013 at 14:33 | comment | added | Johannes Hahn | Why the downvotes? Even though there is an obvious gap in the question (see Simon Henry's answer) it is not unfixable and given that St Georg has obviously put some effort into making this a good MO question, we should give him the chance to close the gap and provide the missing information. | |
| Nov 29, 2013 at 10:21 | review | Close votes | |||
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| Nov 29, 2013 at 10:12 | answer | added | Simon Henry | timeline score: 7 | |
| Nov 29, 2013 at 9:41 | history | edited | user42090 | CC BY-SA 3.0 |
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| Nov 29, 2013 at 9:29 | history | edited | user42090 | CC BY-SA 3.0 |
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| Nov 29, 2013 at 9:15 | history | asked | user42090 | CC BY-SA 3.0 |