Timeline for Is any true sentence in the second-order Peano Axioms provable
Current License: CC BY-SA 2.5
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 18, 2011 at 19:33 | vote | accept | Mohamed Alaa El Behairy | ||
| Feb 8, 2010 at 2:45 | answer | added | Neil Fitzgerald | timeline score: 4 | |
| Feb 8, 2010 at 0:08 | comment | added | François G. Dorais | @Charles: Any complete proof theory for second-order logic is necessarily (1) infinitary, since compactness fails for second-order logic, and (2) almost indescribable, so as to avoid diagonalization arguments similar to Gödel's Theorem. The requirements are so bad that it is often claimed that there is no such system, though this claim is technically wrong without some implicit assumptions on what constitutes a deductive system. A trivial complete deduction system is the semantic consequence relation itself, but that doesn't help much... | |
| Feb 7, 2010 at 23:37 | comment | added | aorq | @François G. Dorais: Can you provide some more detail for your comment, either in another comment or in an answer? Specifically, can you describe the options as to what I can give up and get in exchange? Thanks in advance. Incidentally, I appreciate your contributions to this site very much. | |
| Feb 7, 2010 at 23:23 | answer | added | Ulrik Buchholtz | timeline score: 8 | |
| Feb 7, 2010 at 15:49 | comment | added | François G. Dorais | For a clear answer, you would need to explain your 'suitable proof system' or, at least, your requirements for 'suitable'. Indeed, one of the principal difficulties with second-order logic is that every proof system is unsuitable in one way or another. The answer to your question depends on what you're willing to give up for completeness. | |
| Feb 7, 2010 at 15:32 | comment | added | Mohamed Alaa El Behairy | To clarify: I am asking about syntactic completeness in the sense that there exists a proof for every true statment, using a suitable proof system. | |
| Feb 7, 2010 at 15:11 | answer | added | Joel David Hamkins | timeline score: 3 | |
| Feb 7, 2010 at 14:49 | history | asked | Mohamed Alaa El Behairy | CC BY-SA 2.5 |