Timeline for Ultrainfinitism, or a step beyond the transfinite
Current License: CC BY-SA 3.0
5 events
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| Jul 1, 2012 at 5:23 | comment | added | user24527 | The problem is not forcing vs. inner models. The problem is thinking that the universe should satisfy AC, or even ZF (separation implies that Russell class is big, but it's not so in NFU). Perhaps the best way to produce natural models of the many set theories is abandoning the idea that ZFC axioms are really universal. At the moment I like type theory with typical ambiguity, plus universes (with the axiom that Tarski explicitly noted to be compatible with type theory). In summary: to realize the project of the requester, abandon either AC or even ZF for the global theory of the extension. | |
| Jun 30, 2012 at 22:48 | comment | added | Asaf Karagila♦ | I should remark that the fact that we usually end up with inner models to contradict AC is because we currently know of two good ways to produce models: forcing and inner models, however forcing preserves AC and if we start with ZFC (and since every model of ZF has $L$ inside) as we usually would we cannot go into a proper extension by forcing. However, what you said is not entirely correct too: we extend the universe, the extension itself is an inner model of a forcing extension, though. But then again, a forcing extension is an inner model of another forcing extension, so what's the problem? | |
| Jun 30, 2012 at 20:58 | comment | added | user24527 | For the very little I know, even now it might be that NF (without atoms) is inconsistent, or that on the opposite it is as weak as Zermelo (without replacement). My way to look at NF(U) is however by means of 3 level type theory with type-level pairs, plus typical ambiguity. Example: the structure of real numbers is naturally obtained "one level up", then a structure of elements must exist by typical ambiguity. Note: category theorists do not know why sets - classes - conglomerates are sufficient, but in NF one knows why. | |
| Jun 30, 2012 at 20:17 | comment | added | Mirco A. Mannucci | Thanks NN! As a matter of fact, in my comment to Joel's answer, I meant to quote NF as one alternative option, when there is an universal set. Now, long long ago, I attended a seminar by Maurice Boffa on models of NF. In those times, the problem of interpreting NF inside ZFC was open, if memory does not fail me. I have no idea where the question stands now, but if it is still open, it is all the more interesting here. The non-cantorian nature of NF would seem to be refractory to ZF reductionism... | |
| Jun 30, 2012 at 20:07 | history | answered | user24527 | CC BY-SA 3.0 |