Timeline for Elementary + short + useful
Current License: CC BY-SA 2.5
7 events
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| Apr 5, 2011 at 18:12 | comment | added | Grant Olney Passmore | @Pete: You make a great point - One has to be very careful. But, with careful planning, I do think the ``big idea'' of compactness and an impactful impression of the power it gives one in constructing models of first-order theories can be communicated in 30 minutes. I believe this general setting of considering an arbitrary first-order theory and its models can be quite a revelation to undergraduates, as long as they have experience with reasoning about specific theories and models in the past, say after one or two abstract algebra courses. If nothing else,it leaves an inspiring impression. | |
| Apr 4, 2011 at 6:15 | comment | added | Michael Greinecker | I think the compactness theorem is useful even if you don't apply it in your work. I think it is the best way to understand what the difference between first order sentences and others is. | |
| Apr 3, 2011 at 22:58 | comment | added | Pete L. Clark | And you want to do all of this in half an hour, for undergraduates? I suppose I could compile a nonempty set of undergraduates (Qiaochu Yuan, Akhil Mathew, Zev Chonoles,...) for which this might have a chance of flying, but as a general suggestion this comes off as being much more likely to blow up in one's face. | |
| Apr 3, 2011 at 22:55 | comment | added | Pete L. Clark | Second, the course I taught consisted of eight two-hour lectures to math graduate students (who were "very good" according to at least one reasonable interpretation of the term). It was not assumed that they had any previous exposure to mathematical logic of any kind, nor any previous exposure to ultrafilters. (And in fact none of them did have any prior experience with these things.) I mentioned the Compactness Theorem in either the second or third lecture, at the time without proof. The proof came in the last lecture, after I introduced ultrafilters from scratch... | |
| Apr 3, 2011 at 22:53 | comment | added | Pete L. Clark | I have a lot of reservations about this answer, which will be more or less valid depending upon how you interpret the parameters of the question (which I also think is rather vague). First of all the OP said "100% useful". Now I happen to know and like this exact result enough to have made it the climax of a short course I taught last summer. Nevertheless I have not yet used any form of the Compactness Theorem for anything in my own work (I am an arithmetic geometer), and I think probably the majority of working mathematicians would say the same thing.... | |
| Apr 3, 2011 at 20:54 | comment | added | Anton Petrunin | Hmm, "using ultraproducts"... | |
| Apr 3, 2011 at 19:11 | history | answered | Grant Olney Passmore | CC BY-SA 2.5 |