Timeline for What governs our "perception?" about the platonic realm of sets?
Current License: CC BY-SA 4.0
12 events
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| May 8 at 14:44 | comment | added | Randall Holmes | Sheer numbers are not the determiner of importance. It is the sets which have structure that is useful for mathematics. This discussion is veering off on a tangent which I think is not useful... I do have more to say about this, ask me on email. | |
| May 7 at 19:13 | comment | added | Zuhair Al-Johar | @RandallHolmes, The Infinite Boffa model construction leads to: $$n=0,1,2,3,\ldots; \exists x_1, \cdots, \exists x_n: \\ x_1,\cdots, x_n \text { are sets of urelements } \land \\ |{\sf Set}| < |x_1| < |x_2| < \cdots < |x_n|$$. Accordingly we have "extremely more urelements than sets", actually "there are mostly urelements". So, $\sf NFU$ is much about those urelements much as it about sets. Actually, seeing that sets are an extreme minority makes of NFU essentially an Urelement Theory, rather than a Set theory. | |
| May 7 at 13:15 | comment | added | Randall Holmes | Naturality is something precise, which does require that there be lots of urelements. "Natural" here is a technical term. These are basically the models which are built from an initial segment of the cumulative hierarchy with an external automorphism. | |
| May 7 at 6:49 | comment | added | Zuhair Al-Johar | @RandallHolmes, Is having more Ur-elements than sets essential for NATURALITY? I mean as far as it relates to models of NFU. Current experience suggests that, I assume. Do you think it's possible to construct a natural model otherwise? | |
| May 6 at 20:29 | comment | added | Randall Holmes | Zuhair, I'm thinking of the theory of the natural models of NFU, which indeed have lots of urelements. | |
| May 6 at 18:54 | comment | added | Zuhair Al-Johar | @RandallHolmes, NFU by itself is not more natural than NF. But "NFU + there are more Ur-elements than Sets", this is nearer in interpretability to the usual set theory of ZFC. The proof of consistency of NFU with Ur-elements being incomparable or sub-numerous to sets, is as hard as that of NF. | |
| May 6 at 16:22 | comment | added | Randall Holmes | I do think that fascinating questions remain about NF, and I hope that someone answers them! The fact that an ill-posed proposal works does tell us that there is an unexpected interesting corner of the world which merits investigation. That is not the same as thinking we should all work in NF :-) | |
| May 6 at 16:06 | comment | added | Randall Holmes | The history of Quine's proposal of NF needs analysis. He made mistakes in the course of it, but it was much less ad hoc than it appeared, and Specker's 1962 results and Jensen's 1969 results, if they had been digested properly, show that the proposal actually did make sense. But NF itself is to say the least an ill-posed proposal. | |
| May 6 at 15:53 | comment | added | Joel David Hamkins | Yes, of course. I am glad that you have greatly clarified the nature of your views. | |
| May 6 at 15:51 | comment | added | Randall Holmes | @JoelDavidHamkins There are some points of disagreement, too, please note. But there is a hazard both for you and for other witnesses to this conversation of ascribing to me beliefs and purposes which I do not have. That is a lot of the reason that I posted the longer reply. | |
| May 6 at 15:42 | comment | added | Joel David Hamkins | +1. I am very glad that you have responded to my post, which I had worried was too critical, and I am very pleasantly surprised to find so many points of agreement. | |
| May 6 at 15:35 | history | answered | Randall Holmes | CC BY-SA 4.0 |