Timeline for $\Sigma_n$-complete sets in the Levy hierarchy
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Sep 3, 2021 at 12:14 | comment | added | Farmer S | Assuming ZFC, the class $C$ of $V_\alpha$'s is "$\Sigma_2$-complete", in that it is $\Sigma_2$-definable, and there is a uniform reduction of $\Sigma_2$ to $\Sigma_1$-in-$C$. But I don't know if this holds only assuming ZF (this relates to @HanulJeon's question math.stackexchange.com/questions/4075153/…) | |
| Sep 3, 2021 at 12:10 | comment | added | Corey Bacal Switzer | @GabeGoldberg Yes, I wrote "set" but I guess I really meant class (e.g. satisfaction predicate)! | |
| Sep 2, 2021 at 16:08 | comment | added | Asaf Karagila♦ | (I should probably say that I had in mind the satisfaction predicate when I wrote my comments.) | |
| Sep 2, 2021 at 13:56 | comment | added | Gabe Goldberg | Are you interested in $\Sigma_n$-complete classes (e.g., satisfaction predicates) or $\Sigma_n$-complete sets of integers (e.g., truth predicates)? | |
| Sep 2, 2021 at 13:03 | history | edited | YCor | CC BY-SA 4.0 |
removed capitals from title
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| Sep 2, 2021 at 12:00 | comment | added | Asaf Karagila♦ | I'd guess that it means that for every $\Sigma_n$ predicate there is a $\Delta_1$ relation, etc. But yeah, this isn't really a studied notion here. You also bring up non-standard $\omega$, in which case what is $\Sigma_n$ for non-standard $n$? I'd imagine something like "there is a non-standard formula that defines the class on a club of $V_\alpha$s", but that sounds like an ultimately $\Sigma_3$-or-so. And in any case, you'd fare better here working in NBG or even KM (without AC, if you so desire). | |
| Sep 2, 2021 at 11:52 | comment | added | Corey Bacal Switzer | @AsafKaragila presumably yes, this is what I'm expecting, but what exactly does this mean formally? What is the analogue of many-one/Turing reduction that makes the $\Sigma_n$-truth predicate complete? I started to write in a paper that the $\Sigma_n$-truth predicate is a $\Sigma_n$-complete set and then realized I didn't know how to define this. A look through a number of standard set theory books turned up nothing on this. | |
| Sep 2, 2021 at 11:30 | comment | added | Asaf Karagila♦ | Isn't $\Sigma_n$ truth predicate the example here? | |
| Sep 2, 2021 at 9:02 | history | asked | Corey Bacal Switzer | CC BY-SA 4.0 |