Timeline for Starting Hilbert's Program on the other end
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Mar 4, 2014 at 21:04 | comment | added | Lucas K. | @Christoph, there is indeed no precise description of "simple". For that mathematics and logic are an art as it has ever been. However, adding an additional axiom saying that ZFC is consistent, sounds simple in English, but formally it isn't. You have to formalize ZFC completely in the finitary system. The axiom will be a huge axiom. That can not be considered simple. | |
| Mar 3, 2014 at 17:02 | comment | added | Thomas Klimpel | @EmilJeřábek Neither the question nor the answer indicate to me that your interpretation was the intended one. As many people are confused about the true meaning of Gentzen's consistency proof of PA, already the point that "T" and "U" need not be the same theory seems to be missed. The statement might simply be assumed to have some meaning, and an independence result is taken to be a proof of equiconsistency. (There might be a theorem that this actually works in case of ZF like theories, but independence results are also available for theories that can't even talk about consistency.) | |
| Mar 3, 2014 at 14:41 | comment | added | Emil Jeřábek | @ThomasKlimpel: The phrase “$T$ proves the consistency of $S$ relative to $U$” means $T\vdash\mathrm{Con}_U\to\mathrm{Con}_S$. I don’t understand what else could it be mistaken for. | |
| Mar 3, 2014 at 8:21 | comment | added | Thomas Klimpel | I'm not sure whether PA is really able to prove the consistency of PA relative to PA. You probably mean "relative to the consistency of X" instead of "relative to X". Or maybe you didn't even think about what it means to talk about the consistency of ZFC within X. Note that you probably need to be able to talk about the consistency of ZFC within X if you want to prove it within X. | |
| Mar 2, 2014 at 21:53 | comment | added | Christoph-Simon Senjak | What do you mean by "Simple"? You can drop the axiom of choice, you can drop classical logic, as others have already said. On the other hand, why are these systems "simpler"? In theory, you could just use omega-Heiting-Arithmetic with the additional axiom that in some reasonable syntactic embedding, ZFC is consistent. | |
| Mar 2, 2014 at 21:15 | answer | added | user44143 | timeline score: 10 | |
| Mar 2, 2014 at 21:09 | comment | added | Emil Jeřábek | A related question: mathoverflow.net/questions/48365/… | |
| Mar 2, 2014 at 21:01 | history | asked | Lucas K. | CC BY-SA 3.0 |