Timeline for Completeness vs Compactness in logic
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Dec 2 at 14:35 | comment | added | Alexander Pruss | When I teach logic, I say that soundness and completeness are checks on our system: soundness checks that we haven't put in too many deductions and completeness that we haven't put in too few. I think it is a beautiful achievement of the human mind that we have provably found a sound and complete first-order logic, thereby provably completing a project started by Aristotle. Completeness is important to know for a system as it means we don't have to collective look for more first-order deductions. But once we have completeness, we don't build on it per se much, while we do build on compactness. | |
| Jun 30, 2011 at 18:45 | comment | added | Mike Shulman | Joel: Well, maybe if one subscribes to a Platonic view that mathematical objects "exist" somewhere out there, independently of what we can prove about them. Personally I find that kind of questionable, but many mathematicians seem to adhere to something like it. | |
| Jun 27, 2011 at 21:41 | comment | added | Joel David Hamkins | Pete, I might also add that the methods of local isomorphism and back-and-forth were invented by Cantor in his proof on the uniqueness of the countable dense endless linear order, and logicians perhaps like to regard them as among the fundamental contributions of logic. | |
| Jun 27, 2011 at 21:25 | comment | added | Joel David Hamkins | Pete, you are very kind, but I think you are being modest. Like many maturing subjects, model theory increasingly touches other mathematical areas, and while the majority of model theorists I know continue to self-identify as logicians, I also know a number of model theorists who don't quite know what label to apply to their work. Of course, it is often work that crosses established boundaries that is the most valuable in mathematics, and perhaps the most difficult. Perhaps a similar situation has arisen in set theory, which has become vast, now touching many areas of mathematics. | |
| Jun 27, 2011 at 21:14 | comment | added | Joel David Hamkins | Mike, yes indeed, and perhaps I stated the view more carefully in my answer to the other question. Of course the proof theorists are undertaking a fascinating and important foundational study. And although mathematicians generally strive to prove their results, wouldn't you say that this arises mostly from a concern with the mathematical objects themselves rather than with a concern specifically with the proof objects? (And one might view proof theory as a case where the proof objects become the mathematical object of study.) | |
| Jun 27, 2011 at 4:33 | comment | added | Pete L. Clark | @Joel: I agree completely with your answer (for what that's worth -- you are 1000 times the expert I am on this subject). When I taught introductory model theory last summer and spoke to some colleagues who had had encounters with it, I came to the idea that perhaps model theory could be recast so as not to be part of mathematical logic at all. I found some sympathy for this in Poizat's introductory text, which phrases things in terms of "local isomorphisms" and "back and forth" but I didn't have the time (and perhaps not the expertise) to really follow up on this. What about you? | |
| Jun 26, 2011 at 19:19 | comment | added | Mike Shulman | Of course, as has been brought up elsewhere (and as you mentioned in your answer to the other question) the assertion that "we" care about the models, rather than the proofs, depends upon who "we" refers to. One might argue that at a foundational level, what we really care about is always the proofs. For instance, even when studying model theory, I would venture to assert that model theorists reach their conclusions by proving them. (-: | |
| Jun 25, 2011 at 14:42 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |