Timeline for Set theoretical multiverse and truths
Current License: CC BY-SA 3.0
24 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 16, 2017 at 21:58 | comment | added | Christopher King | It should be noted that just because we can "see" a number doesn't make it standard. For example, in a non-standard universe, there could be a creature with a non-standard number of fingers. If we told them that their "10" doesn't exist, they would laugh at us. | |
| Nov 5, 2017 at 10:27 | comment | added | Mikhail Katz | "In Joel's multiverse conception, there is no 'standard' N, so the standard natural numbers in one model could turn out to be nonstandard in another model. This means that something that appears to be a proof in PA in one model would be revealed to have nonstandard length, and not be a proof, in some other model": Weaver's first sentence is correct but his second sentence is not as Joel David's answer conclusively shows. | |
| Nov 3, 2017 at 5:18 | comment | added | Cameron Zwarich | Is the statement "all naive integers are present in all universes of the multiverse" a metamathematical statement about natural numbers that are constructed as the result of physical processes, or is it a mathematical statement subject about an unbounded finitary process that is itself subject to non-absoluteness? Taking the latter view and importing the non-absoluteness of arithmetic statements about provability while also trying to establish absolute arithmetic statements about provability seems self-defeating and ultimately incoherent. | |
| Nov 1, 2017 at 14:49 | comment | added | Mikhail Katz | "Not believing in a standard N" does not entail any of the philosophical anxieties you outlined, contrary to what you seem to assume. @NikWeaver | |
| Nov 1, 2017 at 13:13 | comment | added | Nik Weaver | @MikhailKatz: I'm afraid you really don't understand the issues here. Joel does not believe in a standard $\mathbb{N}$. So you have to be careful about what it even means to be a model of ZFC. This matter is absolutely related to finitism and possibly even to ultrafinitism. The best source on this would be Joel. Presuming that you can give a categorical answer, as you have, is quite arrogant and unsupported. | |
| Nov 1, 2017 at 12:41 | comment | added | Mikhail Katz | @MikeShulman, there could not be any debate about whether 35253586543 is "standard" or any other explicitly specified number. The universes in Hamkins' multiverse are models of ZFC and therefore are unrelated to either finitism or ultrafinitism. | |
| Oct 31, 2017 at 17:18 | comment | added | Mikhail Katz | Again, on the basis of what I wrote in the publication cited above. | |
| Oct 31, 2017 at 17:05 | comment | added | Nik Weaver | @MikhailKatz: again, my main question for you is on what basis you make this confident statement of fact about Hamkins' multiverse conception. | |
| Oct 31, 2017 at 16:58 | comment | added | Mikhail Katz | @NikWeaver, naive integers include anything you can write down in the time allotted to our civilisation using a computer as large as the universe and exploiting the fastest growing functions in our logical arsenal. Writing down 35253586543 certainly takes less effort than that. For details you can see my recent publication, All in all this question would have been more appropriate at MSE. | |
| Oct 31, 2017 at 16:31 | comment | added | Andreas Blass | @Hurkyl My comment was written not from the viewpoint of some particular universe (in the multiverse), nor from the viewpoint of a ZF model or other mathematical structure, but from the "viewpoint" of the physical world in which we live. The expressing and proving that I have in mind is not about (standard or nonstandard) formulas or Gödel numbers thereof but about what could be written with actual physical pens on actual sheets of paper. | |
| Oct 31, 2017 at 14:01 | comment | added | Nik Weaver | @MikhailKatz: on what basis do you say this? Is it just your opinion? (And what exactly is a "naive" integer?) | |
| Oct 31, 2017 at 12:17 | comment | added | Mikhail Katz | @NikWeaver, all naive integers including 35253586543 are present in each universe of the multiverse. | |
| Oct 31, 2017 at 11:51 | comment | added | Nik Weaver | @MichaelGreinecker: thank you for the question. I've added a clarifying edit. | |
| Oct 31, 2017 at 11:49 | history | edited | Nik Weaver | CC BY-SA 3.0 |
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| Oct 31, 2017 at 10:09 | comment | added | Mikhail Katz | @StevenStadnicki, one of the "points" of Hamkins' multiverse view is that the position you outlined does not hold there. Namely, in every instance of a universe in the multiverse, the integers of that particular universe will contain nonstandard integers with respect to another universe. | |
| Oct 31, 2017 at 8:04 | comment | added | Michael Greinecker | @NikWeaver Is there some paper in which you describe your views? I read your book on forcing and had the impression that you view the idea of standard natural numbers as harmless. | |
| Oct 31, 2017 at 7:39 | comment | added | Steven Stadnicki | I may have to wait for Joel's hopeful comments for this, but I'm a bit confused. I was under the impression that we can (metamathematically) take 'the' $\mathbb{N}$ as the intersection of all $\mathbb{N}$ over all theories; it's certainly true that no specific theory can talk about this $\mathbb{N}$, but there is a theory (if not more) that has this minimal $\mathbb{N}$ as 'its' $\mathbb{N}$, and its theorems can't have nonstandard lengths in any other model. What part of this (if not several) am I confused about? | |
| Oct 31, 2017 at 7:00 | comment | added | user13113 | @AndreasBlass: The point is that you're observing the multiverse from the perspective of the particular universe in which you've "expressed and proved that 35253586543 exists". Every universe in your observable multiverse will have the integer 35253586543, but you can't say anything about universes outside of it. | |
| Oct 30, 2017 at 17:53 | comment | added | Mike Shulman | @AndreasBlass I agree it's not exactly ultrafinitism, since there's no debate about whether 35253586543 exists; but if there is debate about whether it's "standard" then I think there is at least an analogy. | |
| Oct 30, 2017 at 17:34 | comment | added | Nik Weaver | @AndreasBlass: but again, if your model has a nonstandard $\mathbb{N}$ then you could "count up to" nonstandard numbers and think they were standard. | |
| Oct 30, 2017 at 17:24 | comment | added | Andreas Blass | The individual models in the multiverse are supposed to be models of ZF, and it's fairly easy to (express and) prove in an extension by definitions of ZF that 35253586543 exists. And the length of the proof, including the relevant definitions, is (unlike 35253586543 itself) short enough that one could count up to it. So 35253586543 should be "standard" in the sense of each model in the multiverse. | |
| Oct 30, 2017 at 17:23 | comment | added | Nik Weaver | @MikeShulman: exactly. Though I would be surprised to hear Joel describe himself as an ultrafinitist ... | |
| Oct 30, 2017 at 17:08 | comment | added | Mike Shulman | Cf. en.wikipedia.org/wiki/Ultrafinitism ! | |
| Oct 30, 2017 at 16:47 | history | answered | Nik Weaver | CC BY-SA 3.0 |