Timeline for The set-theoretic multiverse as a (bi)category
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39 events
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| Oct 12, 2011 at 22:56 | comment | added | Joel David Hamkins | Michael, I agree with your main point (+1), and the discussion brings out what I find (and discuss in my paper) to be a major issue with the formalization of questions concerning the multiverse. Although many questions about the multiverse are first-order expressible within the individual universes of the multiverse, some of the troubling questions are not. And for these, we fall back on the toy model formalization. So let us take David's question here in the context of the toy model formalization. That is, in multiverse of toy models, viewed as a category, what should the morphisms be? | |
| Sep 28, 2011 at 11:14 | comment | added | Not Mike | I apologize for any ambiguity in what I've said. Just wanted to add: the multiverse view is designed to be the anti-pure platonist position (which in this case is the universe or ultimate-L view.) The underlying premise here is that the multiverse cannot be positioned in a way which allows it to be captured by any single universe (otherwise it would make a horrible counter-point to the ultimate-L view.) The list of principle Joel outlines at the end accomplish this rather nicely by proposing it be viewed as some kind of backward reflection scheme (where the new univ contains the old.) | |
| Sep 28, 2011 at 9:57 | comment | added | Todd Trimble | Michael, you kept asking "which V are you going to use?"; I assumed that that V meant some background in which to speak of a multiverse. That background is what I was calling a super-universe (faute de mieux). Sorry if I misunderstood. More importantly, while I wasn't able to extract an axiomatic theory of multiverse from Joel's paper, the proofs (without scare quotes) would prefer to proofs which proceed from those axioms. (Philosophically, I don't have faith in "the multiverse" or "a multiverse" as a Platonic entity, but I do put stock in mathematical proofs which proceed from axioms.) | |
| Sep 28, 2011 at 2:04 | comment | added | Not Mike | The question asked about the multiverse, not about some toy model of universes you call the multiverse. This is the view I've taken, and what I've said is coherent with that assumption. And yes, Other notions of arrow between universes make sense, and can tell you set-theoretic information, but these things are distinct from, and not in keeping with the intent outlined in that paper. | |
| Sep 28, 2011 at 2:00 | comment | added | Not Mike | In regards to your first comment: in order to maintain the intent of the multiverse and remain coherent with its proposed axioms (which only make use of elementary equivalence/submodel and elementary embedding.) In regards to your second comment: There is no "super-universe" which can capture the multiverse. And finally, non-identity universal elementary embedding cannot be "proven" to exist without strong higher order background assumptions. | |
| Sep 28, 2011 at 0:21 | comment | added | Todd Trimble | With regard to the fact that existence of (some posited notion of) arrow between some pair of universes may depend on the properties of the "super-universe" in which a multiverse is regarded as living, my reaction is: so what? My expectation is that one will be primarily interested in arrows (whatever they are chosen to be) that can be proved to exist based on the multiverse axioms (whatever they are exactly), i.e., in those that exist independent of super-background. But no matter: this issue is completely orthogonal to David's question, which is a conceptual question. | |
| Sep 27, 2011 at 23:51 | comment | added | Todd Trimble | "Finally, the only notion of arrow that makes any sense is that defined by elementary embedding" -- what makes you so sure? Perhaps we should hear from Joel at this point (instead of dragging out an endless discussion in comments), so I'll bow out in just a moment, but: it probably depends on what you want to do. E.g., in view of a nice paper by Andreas Blass, one might be interested in exact functors between categories of sets. What sorts of exact functors exist indeed reflects interesting set-theoretic properties, but far less restrictive than if one chose elementary embeddings as maps. | |
| Sep 27, 2011 at 20:11 | comment | added | Not Mike | Finally, the only notion of arrow that makes any sense is that defined by elementary embedding, which as I've stated, is not going to capture relationships between forcing extensions, and has its own difficulties when you note that as stated in the proposed axioms, the existence of such an embedding must be witnessed from the perspective of a particular universe, containing the ones under discussion. | |
| Sep 27, 2011 at 20:06 | history | edited | Not Mike | CC BY-SA 3.0 |
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| Sep 27, 2011 at 20:04 | comment | added | Not Mike | (feel free to replace V with model of ETCS because the same problem exists) | |
| Sep 27, 2011 at 19:55 | comment | added | Not Mike | So my question to the both of you is: Which $V$ are you going to use to decide there is an arrow between two universes? | |
| Sep 27, 2011 at 19:48 | comment | added | Not Mike | The best example of this: Suppose $\kappa$ is supercompact in $V$, then $\kappa$ is not supercompact in $V$'s associated inner model $L$. Putting the combinatorics aside, this means that $L$ fails to capture the most of the embeddings ("arrows") which witness $\kappa$ is supercompact. | |
| Sep 27, 2011 at 19:42 | comment | added | Not Mike | What I'm saying is that: the existence of the "arrows" changes depending on the ground model, and makes the notion ill-defined and useless in a global setting. | |
| Sep 27, 2011 at 13:00 | comment | added | David Corfield | Yes, David R. is expressing my position. I'm simply asking what we should think of what Joel said about commutative diagrams. He's talking about arrows and an identity between compositions of arrows. I want to now what kinds of arrow we could usefully consider, and whether we can relate arrows between a pair of objects other than by identity, i.e., by 2-arrows. | |
| Sep 27, 2011 at 0:23 | comment | added | David Roberts♦ | "Usable formulations of the naive notion of Category are presented as internally definable objects, in the sense that you must be in some V or have a particular V in mind before you can actually make use of the notions and tools of the theory" this is not true, else ETCS as foundation would be impossible. | |
| Sep 27, 2011 at 0:21 | comment | added | David Roberts♦ | What one can ask, I would hope, is that given two universes (which are 'objects' of the multiverse), what sort of maps between them are there, that reflect the fact they are universes? We don't need to consider the whole multiverse for this exercise. | |
| Sep 27, 2011 at 0:18 | comment | added | David Roberts♦ | @Michael - I think you don't understand what David C is asking. He is not asking if one can model the multiverse in a universe, and as I pointed out, the definition of a 2-category is first order, so we don't need an ambient set theory in which to 'place' it. From a ct point of view one shouldn't ask that elements of different models live in a single $V$, and this is the position I'm guessing David is taking. | |
| Sep 26, 2011 at 13:26 | comment | added | Not Mike | @Todd, What is actually happening here is akin to the following observation: Provided $ZFC$ is consistent there can be no model for $ZFC+\neg Con(ZFC)$, yet $ZFC+\neg Con(ZFC)$ is consistent. Its a catch 22 that you cannot get around. | |
| Sep 26, 2011 at 13:09 | comment | added | Not Mike | @Todd, (continued) In particular, this applies to objects like categories, and their extensions. | |
| Sep 26, 2011 at 13:03 | comment | added | Not Mike | @Todd, What I've said is that the closure properties of the multiverse are incompatible with any object which can be fully positioned inside a single $V$. This is why I took the time to talk about the first-order properties Joel lists and point out that the model he produced for them, is in his own words "a toy." In other words: The actual intented interpretation of the multiverse transcends naive notions like set and class. Which implies that any model which you can exhibit and claim is the multiverse, is by default not the multiverse. | |
| Sep 26, 2011 at 12:49 | comment | added | Not Mike | @Todd @David, First, the multiverse is presented in a manner dealing only with elementary equivalence (this is clearly stated in the axioms.) As such the only type of morphism consistent with capturing the intent of the multiverse is one which describes elementary equivalence, and is not element based (which discounts the use of a geometric morphism.) Second, the elements of the particular models in the multiverse do not and cannot exist simultaneously within one single $V$ (each $V$ sees only a small fragment of the mv) which implies that you cannot faithfully represent the entire mv n a $V$. | |
| Sep 26, 2011 at 12:19 | comment | added | Todd Trimble | Michael, I'll second David C.'s comment: that discussion merely shows that logical morphisms might not be the best thing to consider. Your point 2 in the addendum seemingly attempts to say something much stronger, that in principle, based on "ontology", category theory can have nothing useful to say about a notion of multiverse. That point didn't make a lot of sense to me: considering Grothendieck toposes as universes (which is in line with Joel's ideas), it would also predict that there is no useful notion of morphism between Grothendieck toposes. Which is false! (Geometric morphisms) | |
| Sep 26, 2011 at 10:22 | history | edited | Not Mike | CC BY-SA 3.0 |
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| Sep 26, 2011 at 10:01 | comment | added | Not Mike | @David, the forcing construction is centered on establishing the consistency of logical statements, and in a sense, we do not care about particular generic extensions beyond there logical properties. That having been said, the logical properties a particular extension possess does not tell us how it was constructed. Which given the intent of the multiverse (as a collection of distinguished members of equivalence classes defined by elementary equivalence) implies that there is no way to reconcile the two notions favorably. | |
| Sep 26, 2011 at 9:37 | history | edited | Not Mike | CC BY-SA 3.0 |
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| Sep 26, 2011 at 9:17 | history | edited | Not Mike | CC BY-SA 3.0 |
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| Sep 26, 2011 at 9:01 | history | edited | Not Mike | CC BY-SA 3.0 |
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| Sep 26, 2011 at 8:53 | history | edited | Not Mike | CC BY-SA 3.0 |
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| Sep 26, 2011 at 8:08 | comment | added | David Corfield | @Michael, the discussion you mention in your comment gives reasons for thinking that requiring arrows to be logical functors is too restrictive. But that still leaves my question unanswered. | |
| Sep 26, 2011 at 5:44 | comment | added | Mariano Suárez-Álvarez | I also can't see in what way this is an answer to the question. | |
| Sep 26, 2011 at 4:16 | history | edited | Not Mike | CC BY-SA 3.0 |
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| Sep 26, 2011 at 3:51 | comment | added | Not Mike | @Todd, the discussion relating to Joel's answers to this question mathoverflow.net/questions/42710/… does a much better job explaining the difficulties, even under the restricted notion of ETCS, than I personally can. | |
| Sep 26, 2011 at 0:02 | comment | added | Todd Trimble | I'm not understanding how any of this is addressing David Corfield's question: what, conceptually, is an appropriate notion of morphism between universes in a multiverse (and, going further, what is the appropriate notion of 2-morphism between morphisms)? For example, one could posit logical functors between the underlying toposes of universes of sets as morphisms, but how appropriate would this be for capturing salient aspects of Joel's program? (Despite David C.'s use of the word "capture", I didn't think he was hoping to characterize "the" multiverse by asking such a question.) | |
| Sep 25, 2011 at 22:58 | comment | added | François G. Dorais | David, the problem is the other way around. The set-theoretic multiverse is usually not first-order. For example, set theorists generally prefer well-founded universes, which is not a first-order concept. | |
| Sep 25, 2011 at 22:00 | comment | added | David Roberts♦ | 2-categories are first-order as well...And since a ZFC-category is a model of ETCS (also a first-order theory), everything David C is talking about is first order. I would like to be told where my naive idea is going wrong. | |
| Sep 25, 2011 at 20:34 | history | rollback | Scott Aaronson |
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| Sep 25, 2011 at 16:50 | history | edited | Not Mike | CC BY-SA 3.0 |
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| Sep 25, 2011 at 16:45 | history | edited | Not Mike | CC BY-SA 3.0 |
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| Sep 25, 2011 at 16:27 | history | answered | Not Mike | CC BY-SA 3.0 |