Timeline for Russell's paradox as understood by current set theorists
Current License: CC BY-SA 4.0
44 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 30, 2023 at 0:29 | comment | added | Daniel Asimov | The solution I am naïvely familiar with — to call any kind of "set" a "class" instead of a set in case it created paradoxes — strikes me as being amazingly simple compared to the depth and subtlety of Russell's Paradox. Too simple, in fact. It didn't even require analyzing which kinds of situations lead to Russell's-type paradoxes! What we really want is to find a resolution of the paradox that includes everything that ought to be a set and excludes everything that ought not to be one. Is the standard resolution the optimal one? | |
| Mar 29, 2023 at 18:13 | answer | added | Kristian Berry | timeline score: 1 | |
| Sep 21, 2020 at 4:05 | comment | added | Alec Rhea | Possible partial duplicate, in light of the new/correct question interpretation? I think Joel's answer in particular is relevant. mathoverflow.net/questions/45037/… | |
| Sep 17, 2020 at 22:27 | comment | added | Pace Nielsen | " It seems to me almost completely unrelated to the original version of the question!" Ah, the joys of communication over the internet. | |
| Sep 17, 2020 at 22:20 | comment | added | Pace Nielsen | Note #2: Nor were my comments meant to be evidence against set theory, or ZFC in particular. They were just meant to raise questions about how to interpret ZFC, proper classes, etc... from a meta-theoretic/philosophical perspective. | |
| Sep 17, 2020 at 22:14 | comment | added | Pace Nielsen | @AlexKruckman Thanks! I think you are right that "Does ZFC axiomatize what it's supposed to axiomatize?" is a good rephrasing of my concern. Perhaps "Can any first order theory appropriately axiomatize the naive notion of collection, in light of Russell's paradox?" would be even more precise. My comments about models were meant to be evidence towards a negative answer to these questions, but never as evidence against model theory itself! Note: There appear to be no such issues with respect to arithmetic. The models of, say, Peano arithmetic, include what we would consider "true" arithmetic. | |
| Sep 17, 2020 at 22:08 | comment | added | Alex Kruckman | All that said, I've changed my downvote to an upvote, because I think the most recent revision of the question is actually an interesting philosophical question about the role of proper classes in set theory. It seems to me almost completely unrelated to the original version of the question! | |
| Sep 17, 2020 at 22:05 | comment | added | Alex Kruckman | You wrote "I would want a model to actually 'model' the thing it is supposed to 'model'". To be tautological: models of ZFC are indeed models of ZFC, which is what they're supposed to model. It seems bizarre to take issue with the models here, when your actual concern is with the theory: Does ZFC axiomatize what it's supposed to axiomatize? | |
| Sep 17, 2020 at 22:04 | comment | added | Alex Kruckman | You wrote: "Consider the following question: Could any model of set theory actually be what we would consider the "true universe" of set theory?" Of course not - but that's not the point of models of set theory. The point of models of set theory is to provide set-sized objects that we can manipulate to do meta-mathematics about the theory. | |
| Sep 17, 2020 at 18:24 | comment | added | Pace Nielsen | @Wojowu I would agree, but would modify that to say: set theory aims to capture the iterative construction of sets through any potentially plausible construction. This allows reasoning about strongly inaccessible cardinals, for instance. But, as per Rodrigo's answer, I think there are some serious issues with reasoning about the "true" $V$ as if it were a completed object. I'd love to hear from set theorists on that front. | |
| Sep 17, 2020 at 14:40 | comment | added | Pace Nielsen | @NateEldredge I'm sorry if the tone came across as antogonistic. I was writing it with a smile on my face, but I'll try to edit. I'll also try to adress the "syntactic sugar" comment (which is only one way to view classes, btw), Sam's comment, and Matt's comment. | |
| Sep 17, 2020 at 14:40 | history | edited | Pace Nielsen | CC BY-SA 4.0 |
deleted 695 characters in body
|
| Sep 17, 2020 at 13:51 | comment | added | Wojowu | I am far from well acquainted with philosophy, hence just a comment, but to reply tk your final question, of what sets in modern set theory formalize: in my eyes, the answer is essentially that they aim to capture the iterative construction of sets. We start with the empty set. Then we consider sets built out of it, and then sets built out of what we have so far, and so on. Rodrigo's answer essentially discusses how we view this conception as being open-ended, hence without letting us consider the totality of all sets | |
| Sep 17, 2020 at 7:51 | history | became hot network question | |||
| Sep 17, 2020 at 3:02 | answer | added | Rodrigo Freire | timeline score: 13 | |
| Sep 17, 2020 at 2:57 | comment | added | Sam Hopkins | To add to Nate Eldredge's last point, the question could be improved by quoting specific language where some set theorists put forward the picture you're suggesting. | |
| Sep 17, 2020 at 2:51 | comment | added | Nate Eldredge | I haven't downvoted, but one possible explanation is that some of the question gives the impression of attacking straw men. For instance, "we are never told why we can't collect classes into bigger classes". I think many people would say there are perfectly clear reasons why you can't; e.g., if you think of classes as "syntactic sugar" for formulas, then you can't have a class of classes because classes (i.e. formulas) are simply not objects in set theory's universe of discourse. The tone seems needlessly antagonistic, as if you're accusing set theorists of some sort of deceit. | |
| Sep 17, 2020 at 2:45 | comment | added | Zhen Lin | I don’t actually believe there are small collections and large collections, though. For one thing, I permit myself to change what I mean by small; for another, when I am feeling Platonistic my view would be that every collection is small in some context. I think I am a formalist, in practice. | |
| Sep 17, 2020 at 2:44 | comment | added | Nate Eldredge | So with your edits, I think I'm unclear as to what exactly you mean when you speak of "arbitrary collections". Evidently there are certain properties that you think a theory of "arbitrary collections" ought to have, and which standard set theory doesn't have, but it's not clear to me what they are. It sounds, for instance, like you want there to be a "collection of all collections", but do you also think that the theory of collections ought to include comprehension? If yes, why is it reasonable to "want" an inconsistent theory? If no, then what if anything do you want in its place? | |
| Sep 17, 2020 at 2:20 | comment | added | Pace Nielsen | @ZhenLin Aren't you a Platonist in this thing? Apparently you believe there is a way to separate collections into two types---one large and one small---even though that perspective is not a priori forced upon you. | |
| Sep 17, 2020 at 2:05 | comment | added | Zhen Lin | I think of proper classes (= large collections) as being different from sets (= small collections), yes. I am aware that there is a (very useful) perspective that sees the NBG universe contained in the MK universe and the MK universe contained in, say, the universe of Mac Lane set theory with one Grothendieck universe. At the same time I am aware that axioms about what goes on high up in the cumulative hierarchy can imply things about arithmetic so I am wary of believing that this picture is “reality”. I suppose I’m not a Platonist. | |
| Sep 17, 2020 at 1:55 | comment | added | Pace Nielsen | @ZhenLin Thanks! Do you then view proper classes as something fundamentally different than sets? (If so, what makes them fundamentally different?) Or do you secretly view NBG as contained in MK, which is contained in a version of ZFC with universes? | |
| Sep 17, 2020 at 1:52 | comment | added | Zhen Lin | @PaceNielsen I (try to) work with NBG if I use proper classes. (But I don’t try very hard – I just remember to avoid the obvious pitfalls like trying to form a collection of proper classes; so it’s possible I accidentally stray into MK territory sometimes.) | |
| Sep 17, 2020 at 1:51 | comment | added | Pace Nielsen | And for those downvoting, I would appreciate comments letting me know why you are downvoting. I've tried to clarify the question quite a bit. | |
| Sep 17, 2020 at 1:50 | comment | added | Pace Nielsen | @AlexanderWoo Russell's paradox certain says that the notion of "the collection of all collections" is nonsense. But I don't see how it undercuts the idea of an "arbitrary collection". Do you have a reference to back up that stronger claim? | |
| Sep 17, 2020 at 1:44 | comment | added | Alexander Woo | Russell's Paradox says that the notion of an arbitrary collection is nonsense. So sets can't formalize arbitrary collections; that would be formalizing nonsense. | |
| Sep 17, 2020 at 1:43 | comment | added | Pace Nielsen | @ZhenLin It sounds like, yes, when push comes to shove, you allow that some categories are not inside universes. If so, how do you formalize "proper classes"? | |
| Sep 17, 2020 at 1:32 | comment | added | Zhen Lin | It depends on what I am doing. If complicated constructions are not required I could work with categories that have a proper class of objects and morphisms. Sometimes complicated constructions are needed but I only need to consider small categories, or at worst essentially small categories. It isn’t always necessary to assume a universe. | |
| Sep 17, 2020 at 1:23 | comment | added | Pace Nielsen | @ZhenLin So you don't take it as given that any category you ever work with lives inside some (small) universe? | |
| Sep 17, 2020 at 1:21 | comment | added | Zhen Lin | Assuming a proper class of strongly inaccessible cardinals doesn’t really change my perspective. It just creates a tower of smallness notions. | |
| Sep 17, 2020 at 1:18 | history | edited | Pace Nielsen | CC BY-SA 4.0 |
added 543 characters in body
|
| Sep 17, 2020 at 1:07 | comment | added | Pace Nielsen | @AlexKruckman To answer your question about my comment on models, consider the following question: Could any model of set theory actually be what we would consider the "true universe" of set theory? (The answer is no, the domain of a model is a set.) By the way, it's not that I would necessarily want a model to contain all collections---its that I would want a model to actually 'model' the thing it is supposed to 'model'. | |
| Sep 17, 2020 at 0:56 | comment | added | Pace Nielsen | @ZhenLin My impression of category theory (from the few texts I've read) is that it usually assumes some sort of axiom of universes, where any given universe is contained in a larger universe. So there are sets in universe 0, then bigger sets in universe 1, etc... Would you agree with that? If not, please clarify. If so, that's really just (normal) set theory with (lots of) strongly inaccessible cardinals. | |
| Sep 17, 2020 at 0:52 | comment | added | Zhen Lin | I think the answer you get will depend on the kind of mathematician you ask. For me, as a category theorist, I think of set theory as formalising small collections. And perhaps set theorists would say set theory formalises inductive/well-founded constructions. | |
| Sep 17, 2020 at 0:44 | comment | added | Pace Nielsen | @NateEldredge I've tried to clarify my question. Feel free to ask additional questions, if that doesn't clear up what you are asking about. | |
| Sep 17, 2020 at 0:43 | history | edited | Pace Nielsen | CC BY-SA 4.0 |
added 1174 characters in body
|
| Sep 17, 2020 at 0:36 | comment | added | Nate Eldredge | Are you saying that from some philosophical perspective, we really "need" a set-like theory that does include naive comprehension, and that standard set theory is unsatisfactory because it doesn't? I would guess that current set theorists would simply disagree with you. | |
| Sep 17, 2020 at 0:32 | comment | added | Pace Nielsen | @AlexKruckman My experience is different. I would agree that most texts "avoid" the paradox, by limiting the comprehension axioms, but that undercuts the purpose of sets (as a formalization of arbitrary collections). It also introduces extra layers of complexity (by introducing "classes" as collections that are too large to be sets). | |
| Sep 17, 2020 at 0:29 | comment | added | Pace Nielsen | @NateEldredge I never said classes are arbitrary collections. I just said they were collections. | |
| Sep 17, 2020 at 0:29 | answer | added | Alec Rhea | timeline score: 4 | |
| Sep 17, 2020 at 0:26 | comment | added | Nate Eldredge | Classes aren't "arbitrary" collections; they can't have classes as elements. And thus no problem for Russell. I'm a novice, but I really don't see Russell's paradox as saying anything deeper for modern set theory than "don't try to adopt the axiom schema of naive comprehension". | |
| Sep 16, 2020 at 22:58 | comment | added | Alex Kruckman | I'm also confused by "if sets were meant to formalize the general notion of a "collection", then a true/Platonic set theory should have no "models" at all". A model is just that, a model. There's no reason to expect a model of a first-order set theory to contain all collections, any more than we would expect a countable model of the first-order theory of the real field to contain every real number. | |
| Sep 16, 2020 at 22:54 | comment | added | Alex Kruckman | I'm confused as to why you think standard texts in set theory do not "resolve" Russell's paradox. It seems to me that most introductory set theory texts contain clear explanations of why Russell's paradox is not an issue when we work in first-order ZFC. | |
| Sep 16, 2020 at 21:59 | history | asked | Pace Nielsen | CC BY-SA 4.0 |