Timeline for Can having no more than countably many classes, be inferred from, having every class being countable?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Feb 1, 2023 at 18:24 | history | undeleted | Zuhair Al-Johar | ||
| Jan 4, 2023 at 19:15 | history | deleted | Zuhair Al-Johar | via Vote | |
| Dec 28, 2022 at 6:43 | comment | added | Zuhair Al-Johar | @AlecRhea, pointwise models are always countable, they are the 'smallest' models of ZFC, and they are the most clear ones as far as perceiving the notion of 'set' as a subnotion of 'class', (i.e. sets as some kind of classes) is concerned. The clearest view of a class is as an extension of a predicate, and if we cannot describe a predicate by intention in a clear cut manner, then the class extending it would be dubious, since it would be an extension of an unclear intention. If we can do withtout them, then why use them? The matter is not technical, it is about the very concept of class itself | |
| Dec 28, 2022 at 6:23 | comment | added | Alec Rhea | So, based in your usage of Occam, you view countable pointwise definable models as somehow the ‘simplest’ models of ZFC? Why? | |
| Dec 28, 2022 at 6:08 | comment | added | Zuhair Al-Johar | @AlecRhea, because of the intuition of all classes to be definable in a clear cut manner, if we maintain that ALL classes must be definable in a parameter free manner, then we only have countably many parameter free formulas, and so we can only have countably many classes definable after them; and since we are speaking of pure classes, then all would be countable. IF every model of ZFC can be extended to a pointwise (i.e. parameter free) definable model, then we can do everything in those countable models, then we can Occam razor the rest, that if there is a rest. | |
| Dec 28, 2022 at 5:58 | comment | added | Alec Rhea | Does this setting satisfy your intuition for how classes ‘should’ behave? I’m curious why you’re exploring all these different formalisms for the set/class dichotomy; if you could explain why a situation where we only have countably many countable classes is appealing to you I could better judge how interesting the question is. (I am not a downvoter, but could be an upvoter) | |
| Dec 28, 2022 at 5:32 | comment | added | Zuhair Al-Johar | @AlecRhea, I'm examining the line that all classes must be extensions of predicates, and all the latter ones must be definable in a clear cut manner, i.e. definable without parameters in a finitary first order language. That said then we can only have countably many classes. And since we are speaking about pure classes then all of them are countable. | |
| Dec 28, 2022 at 2:01 | comment | added | Alec Rhea | Out of curiosity, do you have a guiding muse/heuristic you’re trying to chase down with all these different foundational approaches to proper classes? Some informal way you think proper classes ‘should’ behave that you’re trying to properly formalize? | |
| Dec 27, 2022 at 22:28 | answer | added | Holo | timeline score: 0 | |
| Dec 27, 2022 at 20:31 | history | edited | Zuhair Al-Johar | CC BY-SA 4.0 |
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| Dec 27, 2022 at 20:06 | history | asked | Zuhair Al-Johar | CC BY-SA 4.0 |