Timeline for Russell's paradox as understood by current set theorists
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Mar 29, 2023 at 20:45 | history | edited | LSpice | CC BY-SA 4.0 |
Name of question, and link to answer, while this is on the front page
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| Sep 17, 2020 at 15:36 | comment | added | S. Carnahan♦ | @user253751 This sort of unbounded hierarchy of classes can be axiomatized using universes: en.wikipedia.org/wiki/Grothendieck_universe | |
| Sep 17, 2020 at 1:13 | comment | added | Pace Nielsen | I think I now understand some of the confusion. Some readers took "Russell's paradox" as the formal statement that ZFC+Comprehension is inconsistent. I understand Russell's paradox as a philosophical problem with naive set theory. I'll add a comment to the question. | |
| Sep 17, 2020 at 1:04 | comment | added | Alec Rhea | @PaceNielsen I'm not sure about set theorists, good question. (although it doesn't really match the title of the question or the main body question from before anymore, which is what I was addressing). | |
| Sep 17, 2020 at 1:03 | comment | added | Alec Rhea | @PaceNielsen It is true that from a consistency strength point of view, adding classes adds less consistency strength than even a single inaccessible cardinal, and so the classes of MK class theory (for example) can be viewed as the sets of a universe of ZFC+there is an inaccessible cardinal. It does seem that modern category theorists, at least, view sets as formalizing 'small' collections since this is exactly the terminology they use. (a category is small if all its homs form a set, locally small if each individual hom is a set, etc.) | |
| Sep 17, 2020 at 0:51 | comment | added | Pace Nielsen | Alec, I hope that the edit to my question clarifies what I was asking. Your answer seems to suggest that sets are no longer thought of (by set theorists) as formalizations of arbitrary collections, but rather as formalizations of small collections, among a chain of different types. But isn't that chain of types just another version of set theory (perhaps with larger cardinals)? Couldn't your so-called classes just be the sets past the first inaccessible cardinal? Or, from another angle, what is it (fundamentally) that distinguishes a class from a set?--only the language? | |
| Sep 17, 2020 at 0:29 | history | answered | Alec Rhea | CC BY-SA 4.0 |