Timeline for Independence of P = NP?
Current License: CC BY-SA 2.5
8 events
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Dec 21, 2010 at 12:10 | comment | added | user5810 | er... yeah, I suppose you can make an important distinction between ZFC and its models. | |
| Dec 21, 2010 at 12:05 | comment | added | Ed Dean | Sure, you can say that CH is "more canonical" in some sense, or you can, say, argue for -CH along the lines of Koellner, working from a network of results by Woodin. In each of these cases, you are no longer letting ZFC and what it proves be the sole arbiter. All I wrote before is that " As long as ZFC is the arbiter of such matters, there's nothing more to say ..." | |
| Dec 21, 2010 at 11:56 | comment | added | user5810 | What more that can be said is that CH is 'more canonical' than -CH, because (I remember reading somewhere that) you can go from either to the other by forcing extensions, while minimal models satisfy CH. | |
| Dec 21, 2010 at 11:53 | comment | added | Ed Dean | As long as ZFC is the arbiter of such matters, there's nothing more to say than CH is independent of our axioms. So if you want to make a definite case one way or the other, you're asking people to change what is currently a pretty well entrenched convention. You'd need to offer some good reasons, of some sort or another. I recommend looking up "intrinsic" and "extrinsic" justification of axioms in relation to Godel, and perhaps some of the writings of Peter Koellner if you're interested in such matters. | |
| Dec 21, 2010 at 11:48 | comment | added | Ed Dean | I don't. My point is only that any statement which "merely" holds in some particular model of ZFC, rather than all of them, holds a less privileged pedigree than statements which hold in all of them (i.e. are provable in ZFC). Most non-logician mathematicians don't give much thought to foundations, nor do they need to. If you demanded an answer as to their foundations, they'll most likely say something like ZFC (and hope you go away :). If you point at a model of ZFC + -CH and argue that settles CH in the negative, anyone can point at a model of ZFC + CH and say "What about that?" .... | |
| Dec 21, 2010 at 11:34 | comment | added | Zirui Wang | Why do you consider a model in which ZFC hold outside the universe in which mathematicians work? (Model theory.) | |
| Dec 21, 2010 at 10:10 | history | answered | Ed Dean | CC BY-SA 2.5 |