Timeline for Is this theory synonymous with ZF + Global Choice?
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| 6 hours ago | answer | added | Jade Vanadium | timeline score: 0 | |
| 10 hours ago | comment | added | Zuhair Al-Johar | @JadeVanadium, hmmm... from Eliot's proof of bi-interpretability of global choice and global well order presented to your other thread, it appears that a positive answer to this question is possible? | |
| yesterday | comment | added | Jade Vanadium | That said, I've started to think you can actually find the desired synonymity, but it's much more complicated than what you probably had in mind for your question. Basically, you can show synonymity of your theory with that modified theory I mentioned, and that theory (I think) can then be proven synonymous with $\sf{ZF}+\text{"Global Choice"}$ using a separate mechanism. I'm working on writing a proof of that secondary synonymity, and will post it as a separate question when done, and then link to that as part of an answer to yours. | |
| yesterday | comment | added | Jade Vanadium | I think that might make it harder. To prove synonymity of two theories, you basically need (among other things) to find a bijection between the class of all models of one theory to the class of all models of the other. Your theory has fewer ($\subseteq$) models than $\sf{ZF}+\text{"global wellorder"}$, so finding a synonymity is nontrivial. I think you can prove synonymity to the theory $\sf{ZF}+\text{"global wellorder"}$ plus the axiom $\sup\{x+1 : x\in S\}<\sup\{x+1 : x\in Z\}\implies S<Z$, where $x+1$ denotes the order-theoretic successor defined in terms of $<$ | |
| yesterday | comment | added | Zuhair Al-Johar | what about if we ask about ZF+ global well ordering and this theory. Both theories are in the same language, so perhaps the question about their synonymy is easier? | |
| 2 days ago | comment | added | Jade Vanadium | I think this question is much deeper than it was intended to be. The theory $\sf{ZF}+\text{"global choice"}$ doesn't have an obvious synonymity with the theory $\sf{ZF}+\text{"global wellordering"}$. The usual method for constructing a wellorder using a Choice function is not invertible, as many different choice functions lead to the same wellorder, so proving synonymity seems unusually difficult. Would I be correct in saying that the intended question is: can we find a mutual interpretation where the membership relation is standard in both cases? | |
| Dec 7 at 19:08 | history | asked | Zuhair Al-Johar | CC BY-SA 4.0 |