Timeline for What's the benefit of adding a well-ordering over all classes to $\textsf{MK}$?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 29, 2023 at 19:54 | comment | added | Joel David Hamkins | Yes, for example, KM+CC is bi-interpretable with ZFC- + there is a largest cardinal, which is inaccessible. | |
| Oct 29, 2023 at 19:47 | comment | added | Zuhair Al-Johar | For the record, of course I wanted $\prec$ to be allowed to appear in comprehension. So, here there is a clear benefit gained in that it would enable CC. And you said it enables many bi-interpretation results. | |
| Oct 29, 2023 at 19:45 | vote | accept | Zuhair Al-Johar | ||
| Oct 29, 2023 at 19:40 | comment | added | Joel David Hamkins | No, because every model of KM can be reduced to the L of the model, which has a definable well order for its classes. | |
| Oct 29, 2023 at 19:39 | comment | added | Zuhair Al-Johar | Would it prove the consistency of KM? | |
| Oct 29, 2023 at 19:37 | comment | added | Joel David Hamkins | If the expanded language was allowed to appear in the comprehension scheme, then such a well order relation would enable the proof of CC. So the theory would be strictly stronger than KM and not conservative over it. | |
| Oct 29, 2023 at 19:34 | comment | added | Zuhair Al-Johar | Nice! This answer greatly enrich the subject. What I had in mind is the second option, that is we add a new binary relation symbol $\prec$ to the signature of $\sf MK$, then add the axiom schema that $\prec$ is a well order over classes, i.e. an areflexic, transitive, connected and well founded relation over all classes. $\prec$ need not be definable. From this answer a definable well order would be beneficial, but would an indefinable one be too? | |
| Oct 29, 2023 at 18:56 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
added 188 characters in body
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| Oct 29, 2023 at 18:44 | history | edited | Joel David Hamkins | CC BY-SA 4.0 |
added 433 characters in body
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| Oct 29, 2023 at 18:30 | history | answered | Joel David Hamkins | CC BY-SA 4.0 |