Timeline for Can this theory of classes of ordinals interpret ZFC?
Current License: CC BY-SA 4.0
8 events
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| Oct 25, 2023 at 13:03 | comment | added | Zuhair Al-Johar | @FedorPakhomov, yes the Godel pairs are elements. Though it is not necessary to use the Godel pairs, we can use what come to be the Kuratwoski pairs. The point is that we have replacement, so each pair (unordered) can be sent recursively to the limit of all images of the $\prec$ prior pairs. Thus we get unordered pairing, and from it we get to Kuratwoski pairs, and also from it to Godel pairs. | |
| Oct 25, 2023 at 12:50 | comment | added | Fedor Pakhomov | I am confused by your definition: in the beginning it appears that your sort X,Y,Z range over classes of elements, but then when talking about functions it seems that you switch to classes of pairs. With regards to the strength, pretty much sure that any reasonable clarification that would be a theory of classes of pairs of ordinals would interpret ZFC. If you restrict to monadic case (classes of elements), then it is more problematic, since monadic formulas on ordinals are known to have low expressive power, but on top you have your additional order $\prec$, which might make a difference. | |
| Oct 24, 2023 at 22:24 | comment | added | James E Hanson | I don't know how detailed you want the answer to be but I believe the answer is yes. Basically my intuition is that once you have enough machinery to code basic set theory, unrestricted replacement and the existence of cardinal successors give you that L is definable and is a model of full ZFC. I don't think you even need several of your axioms, but I'm not sure which. | |
| Oct 24, 2023 at 20:33 | comment | added | Zuhair Al-Johar | @JamesHanson, the respective axiom reads: $$\forall x \forall y \forall X \forall Y (x= \lim X \land y= \lim Y \to [x < y \to X \prec Y])$$. | |
| Oct 24, 2023 at 20:31 | history | undeleted | Zuhair Al-Johar | ||
| Oct 24, 2023 at 19:26 | history | deleted | Zuhair Al-Johar | via Vote | |
| Oct 24, 2023 at 19:05 | comment | added | James E Hanson | Isn't comprehension going to give you a class of all ordinals? This isn't intrinsically a problem, but the respective axiom seems to be written assuming $\lim X$ always exists. | |
| Oct 24, 2023 at 17:58 | history | asked | Zuhair Al-Johar | CC BY-SA 4.0 |