Timeline for Is the power set axiom essential for constructing L?
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
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| Jun 18, 2023 at 5:08 | comment | added | new account | How to show that $L$ satisfies full separation without using power set in $V$ (which gives us reflection principle)? | |
| Apr 14, 2020 at 22:04 | comment | added | Zuhair Al-Johar | I mean a theory about ordinals that can interpret second order arithmetic and also stipulate further axioms asserting the existence of any ordinal that precedes the first weakly inaccessible cardinal. | |
| Apr 14, 2020 at 21:53 | comment | added | Noah Schweber | "stipulating existence of every ordinal that ZFC can define and prove to exist" And what does that mean exactly? (E.g. what does it mean in the context of second-order arithmetic to stipulate that $\beth_{73}$ exists?) | |
| Apr 14, 2020 at 21:48 | comment | added | Zuhair Al-Johar | If KP can define L within it and prove the existence of $L_\alpha$ for any ordinal $\alpha$ that it proves to exist, then this mean that second order arithmetic can do that too, since KP is a weak fragment of second order arithmetic. So can I take from that that second order arithmetic plus stipulating existence of every ordinal that $ZFC$ can define and prove to exist, would interpret ZFC? | |
| Aug 14, 2019 at 16:16 | comment | added | Zuhair Al-Johar | Sure! I think such matters are always subject to disagreement. The way how I view matters is that if we can do everything with just ordinals and sets of ordinals, I mean if we can encode all the hierarchical sets of ZFC in just a single level sets of ordinals, provided that the theory in question is not less natural than ZFC itself, then to the "less" is where I'll go, since the "more" is already covered by it. its more succinct. Thanks for the answer and the comments. | |
| Aug 14, 2019 at 13:29 | comment | added | Noah Schweber | Besides, you're only hiding the hierarchical structure: replacement is still building objects via transfinite recursion, it's just that now you lack a snappy apparatus to describe that. All in all I don't buy the foundational claim being made here. And that's fine, we can disagree on that, but it is worth pointing out that it's not universal. | |
| Aug 14, 2019 at 13:27 | comment | added | Noah Schweber | @ZuhairAl-Johar "That would be a true reduction in structure that deserves a foundational status" That's a subjective claim, and not necessarily one everyone will agree with. In particular, I disagree with it: sets of ordinals straightforwardly encode many higher-rank objects, so omitting the latter doesn't really buy anything. Moreover, the map taking a model of ZF to the structure consisting of its ordinals and sets of ordinals (however construed) loses interesting information: we can have distinct models of ZF (we need choice to fail in both) with the same ordinals and sets of ordinals. | |
| Aug 14, 2019 at 12:43 | comment | added | Zuhair Al-Johar | ..continuation: .. eliminating the hierarchical set structure altogether and collapsing it to a flat set structure of ordinals, as to get a simple theory of ordinals and sets of them that encodes relations between them as I've posted to Mathoverflow at a prior posting, which I think we can build L within it and interpret ZFC. That would be a true reduction in structure that deserves a foundational status. | |
| Aug 14, 2019 at 12:39 | comment | added | Zuhair Al-Johar | @of course the consistency strength cannot be the sole criterion for a foundational status of a theory. The theories you've mentioned already speak of ZFC, so they are not bypassing ZFC conceptually speaking. I agree with the extendability point you've mentioned. But those can be done by extracting ZFC as a byproduct of this theory, and then working on extending it. The point is that we don't need to axiomatize power set to get to ZFC. However axiom of successor cardinals seem to be necessary to get to ZFC via V=L. Actually I'm aiming at a more reductive measure, that of ,.to be continued | |
| Aug 14, 2019 at 10:33 | comment | added | Noah Schweber | Actually you seem to be doing something even more restrictive, and focusing on consistency strength. Would you consider I$\Sigma_1$ + "ZFC is consistent" - which is even stronger than ZFC in consistency strength - to be foundationally satisfying? What about I$\Sigma_1$ + "ZFC is consistent" + "I$\Sigma_1$ + "ZFC is consistent" is inconsistent"? | |
| Aug 14, 2019 at 10:32 | comment | added | Noah Schweber | @ZuhairAl-Johar Your philosophical claim has a big underlying assumption - that the "foundational satisfactoriness" of a theory is entirely determined by in its interpretability strength. Why shouldn't it hinge on what the theory actually does or does not prove outright? Also, it's not clear to me that this is a persistent phenomenon - what do we add to this theory to get something which interprets (say) ZFC + "There is a supercompact cardinal"? I think the above idea doesn't work here to show that we can more-or-less add the same sentence verbatim since we don't have a good fine structure ... | |
| Aug 14, 2019 at 10:29 | vote | accept | Zuhair Al-Johar | ||
| Aug 14, 2019 at 10:28 | comment | added | Zuhair Al-Johar | So this proves that we don't need to axiomatize power-set in order to build $L$. Axiom of Successor cardinals is weaker than axiom of power set, and yet it manages to build $L$ and so its consistency proves the consistency of ZFC, i.e. of adding power to the rest of axioms of ZFC. In nutshell ZFC-power+Successor cardinals can work as a foundational theory much as ZFC works, because simply its consistency proves the consistency of ZFC. In some sense Power set axiom is bypassed. | |
| Aug 14, 2019 at 9:22 | history | answered | Noah Schweber | CC BY-SA 4.0 |