Timeline for Can different extensions of ZF have contradictory consequences for first-order arithmetic?
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
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| Nov 6, 2013 at 12:36 | review | Suggested edits | |||
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| Aug 15, 2013 at 15:49 | comment | added | Emil Jeřábek | KM contradicts a $\Pi^0_1$ truth if it proves the negation of said truth. The negation of a $\Pi^0_1$ sentence is $\Sigma^0_1$, so in other words, KM proves a false $\Sigma^0_1$ sentence. Saying that this does not happen thus amounts to all $\Sigma^0_1$ sentences provable in KM being true, i.e., that KM is $\Sigma^0_1$-sound. | |
| Aug 15, 2013 at 14:55 | comment | added | Keshav Srinivasan | @EmilJerabek Doesn't Con(KM + all Pi_1 truths) assert that no theorems of KM contradict a Pi _1 truth? How is that different from saying that KM is Pi_1 sound? | |
| Aug 15, 2013 at 11:12 | comment | added | Emil Jeřábek | @Keshav Srinivasan: Con(KM+True $\Pi^0_1$) says that KM is $\Sigma^0_1$-sound (or equivalently, $\Pi^0_2$-sound) rather than $\Pi^0_1$-sound. | |
| Aug 15, 2013 at 5:02 | comment | added | Andrés E. Caicedo | "if @JoelDavidHamkins is right and we can't define a truth predicate within NBG" This is not a matter of opinion (and he's right). | |
| Aug 15, 2013 at 4:12 | comment | added | Keshav Srinivasan | @AndresCaicedo If it's not too much trouble, could you tell me what part to read? projecteuclid.org/euclid.pl/1235421926 | |
| Aug 15, 2013 at 3:59 | comment | added | Andrés E. Caicedo | Read the reference, it may be more productive. | |
| Aug 15, 2013 at 3:47 | comment | added | Keshav Srinivasan | @AndresCaicedo Thanks, I may read it. But could you tell me what's wrong with the reasoning in my last comment? | |
| Aug 15, 2013 at 3:32 | comment | added | Andrés E. Caicedo | Each new edit makes the question more and more confused. And you are mistaken in your last comment. You may want to read the relevant sections of Metamathematics of first-order arithmetic, where most of this is covered carefully. | |
| Aug 15, 2013 at 2:08 | comment | added | Keshav Srinivasan | @JoelDavidHamkins As far as I can tell, Con(KM + all true Pi_1 statements) is equivalent to the statement that KM is Pi_1 sound. But if KM were not Pi_1 sound, then it would prove some false Pi_1 statement, and since that statement is Pi_1, KM would also disprove it, so KM would be inconsistent. So if I'm not mistaken, Con(KM + all true Pi_1 statements) is equivalent to Con(KM), and Con(KM) is Pi_1. Where am I going wrong? | |
| Aug 15, 2013 at 1:51 | comment | added | Joel David Hamkins | As I mention in my comments on my answer, Con(KM+True $\Pi^0_1$) is such a statement. (And it seems to me that KM proves "ZF is sound".) | |
| Aug 15, 2013 at 1:44 | comment | added | Keshav Srinivasan | @JoelDavidHamkins (cont'd) Perhaps a good preliminary question would be, how would you construct a statement of first-order arithmetic that is independent of ZF + (All Pi_1 truths), and which does not follow in MK from the assumption that ZF is sound? | |
| Aug 15, 2013 at 1:39 | comment | added | Keshav Srinivasan | @JoelDavidHamkins It's fine if it's equivalent to a statement of arithmetic, I just don't want it to be one. If there's something like a large cardinal axiom which turns out to be equivalent to a statement in the language of first-order arithmetic, that would be fine with me. | |
| Aug 15, 2013 at 1:29 | comment | added | Joel David Hamkins | Regarding your third edit, how can you say that you want X to be not arithmetic, given that it must be provably equivalent to P? | |
| Aug 15, 2013 at 1:22 | comment | added | Asaf Karagila♦ | Are you sure that you know what sort of answer you are looking for? | |
| Aug 15, 2013 at 0:15 | history | edited | Keshav Srinivasan | CC BY-SA 3.0 |
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| Aug 14, 2013 at 19:48 | answer | added | Noah Schweber | timeline score: 2 | |
| Aug 14, 2013 at 18:36 | history | edited | Keshav Srinivasan | CC BY-SA 3.0 |
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| Aug 14, 2013 at 4:27 | answer | added | Andreas Blass | timeline score: 6 | |
| Aug 14, 2013 at 3:23 | answer | added | Joel David Hamkins | timeline score: 5 | |
| Aug 14, 2013 at 3:23 | history | edited | Keshav Srinivasan | CC BY-SA 3.0 |
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| Aug 14, 2013 at 3:19 | comment | added | Asaf Karagila♦ | Amazing. Two hours of waiting between cross posting your question. You should at least point out that this was posted on m.SE and provide a link. | |
| Aug 14, 2013 at 3:15 | history | asked | Keshav Srinivasan | CC BY-SA 3.0 |