Timeline for Theorems that are 'obvious' but hard to prove
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Sep 8, 2012 at 18:17 | comment | added | Henry Cohn | I agree that the completeness theorem sounds very intuitive, but I think this is misleading. It takes some serious thought to convince yourself that a particular definition of formal proofs captures mathematical practice (even if you believe intuitively that some definition should work, it's much less obvious that a given deductive system really is complete). Furthermore, I'd bet that many mathematicians would find it equally intuitive that there should be a complete proof system for second-order logic, and of course incompleteness tells us there isn't. So completeness is pretty subtle... | |
| Oct 1, 2011 at 15:24 | comment | added | James D. Taylor | Will, I disagree. The incompleteness theorem talks about something quite different. About whether there is a small set of axioms that imply every true statement about the integers (or other models). I dislike the comparison people make between Godel's completeness theorem and his incompleteness theorem. They talk about very different things. Emil, you are correct. The proof is not very hard. But it is much easier to state the theorem and believe it than to actually prove it. | |
| Sep 30, 2011 at 11:36 | comment | added | Emil Jeřábek | The completeness theorem is not really hard to prove. The propositional part may be a bit messy (which can be alleviated by a wise choice of a proof system), but the main Henkin construction is quite straightforward. | |
| Sep 30, 2011 at 3:59 | comment | added | Will Sawin | among people trained on the incompleteness theorem, this is not obvious. | |
| Sep 30, 2011 at 2:02 | history | answered | James D. Taylor | CC BY-SA 3.0 |