Timeline for Meta-undecidability
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Nov 25, 2016 at 14:34 | comment | added | Joel David Hamkins | My view of paraconsistent mathematics is that it is incoherent as a theory of mathematical truth. If find that it inappropriately conflates epistemic with ontological matters. The attraction of paraconsistency lies in the pragmatic concern that some of our beliefs may turn out to be wrong, but we want to reason anyway and somehow contain any contradictions that may arise. But we don't actually believe that actual mathematical truth is at bottom paraconsistent, and so it makes no sense to me to adopt paraconsistent foundations for mathematics. | |
| Nov 25, 2016 at 14:05 | comment | added | Thomas Benjamin | @JoelDavidHamkins: Also, why would you be "a little hesitant to adopt a paraconsistent meta-theory"? | |
| Nov 25, 2016 at 0:43 | comment | added | Thomas Benjamin | @JoelDavidHamkins: So what does your result imply regarding the independence results discovered thus far regarding $ZFC$? | |
| Nov 25, 2016 at 0:34 | comment | added | Joel David Hamkins | My answer also applies to KM and its strengthenings, including the one you mention, by essentially the same argument. There seems to be no way around that phenomenon. | |
| Nov 25, 2016 at 0:30 | comment | added | Thomas Benjamin | @JoelDavidHamkins: One of Olivier Esser's theorems regarding $HF_{\infty}$ is the following: "The theory $HF_{\infty}$ is mutually interpretable with $GPK^{+}_{\infty}$ which is also equiconsistent with "Kelly-Morse +_'On is weakly compact'_ " (this from Theorem 3.2 of his paper "A Strong Model of Paraconsistent Logic"). Since you have an interest in Kelly- Morse class theory, I am wondering what $KM$ + 'On is weakly compact' can show regarding Dominic's question? | |
| Nov 25, 2016 at 0:08 | comment | added | Joel David Hamkins | I've never looked closely at that theory, Thomas, so I'm not sure. While I have looked at some paraconsistent object theories, I am a little hesitant to adopt a paraconsistent meta-theory. | |
| Nov 24, 2016 at 23:48 | comment | added | Thomas Benjamin | Would the same result hold if one formulated $ZFC$ in a paraconsistent meta-theory like $HF_{\infty}$ (Hyper-Frege +_" There is an infinite well-founded set"_? | |
| Nov 24, 2016 at 7:24 | vote | accept | Dominic van der Zypen | ||
| Nov 23, 2016 at 23:42 | comment | added | Joel David Hamkins | And see also my update, which adds a conclusion for consistent strengthenings of ZFC, including the case of ZFC+Con(ZFC). | |
| Nov 23, 2016 at 23:42 | history | edited | Joel David Hamkins | CC BY-SA 3.0 |
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| Nov 23, 2016 at 23:37 | comment | added | Joel David Hamkins | My remarks also apply to that statement, which is provably true from a weak theory in any model of ZFC+$\neg$Con(ZFC), as the hypothesis will be false and provably so. | |
| Nov 23, 2016 at 22:10 | comment | added | Wojowu | This is so meta... but I think the question was actually about "if ZFC is consistent then S is independent" being independent, since this is precisely how the independence statements are proven. | |
| Nov 23, 2016 at 21:13 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |