Timeline for Is the notion of measurable cardinal definable from the perspective of set-theoretical potentialism?
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| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
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| Apr 20, 2019 at 22:40 | review | Close votes | |||
| Apr 29, 2019 at 3:05 | |||||
| Mar 8, 2019 at 15:44 | comment | added | Thomas Benjamin | @NoahSchweber: Thank you. | |
| Mar 8, 2019 at 15:42 | comment | added | Noah Schweber | @ThomasBenjamin Yes. Again, bringing the term "potentialist" (etc.) into the picture just confuses things further; you're just asking whether the relevant notions are definable in ZFC (as opposed to NBG or similar). For some of these this is immediate (e.g. I1 and I3 are already formulated directly in terms of sets alone); for others it takes work. I'm not a large cardinal expert, and it's possible that there are large cardinal notions which really don't have any known ZFC formulation, but my impression is that this isn't the case. | |
| Mar 8, 2019 at 15:41 | vote | accept | Thomas Benjamin | ||
| Mar 8, 2019 at 15:39 | comment | added | Thomas Benjamin | @NoahSchweber: I have just read your answer and to answer the question in your above comment, it does! Am I to presume that your method of translating supercompactness into the potentialist language will work for large cardinal axioms of greater consistency strength than supercompactness? Even for Rank-into-Rank axioms? | |
| Mar 8, 2019 at 15:26 | vote | accept | Thomas Benjamin | ||
| Mar 8, 2019 at 15:40 | |||||
| Mar 7, 2019 at 17:14 | answer | added | Noah Schweber | timeline score: 3 | |
| Mar 7, 2019 at 16:49 | comment | added | Noah Schweber | @ThomasBenjamin Why doesn't the (original!) definition "$\kappa$ is measurable iff there is a $\kappa$-complete nonprincipal ultrafilter on $\kappa$" count as "local enough" for your purposes? It's completely first-order, and pays attention only to a very bounded amount of the universe (namely, whether $\kappa$ is measurable or not is determined entirely by $V_{\kappa+2}$). Indeed, these sort of "small generators" for elementary embeddings are crucial tools for understanding these more complicated (and "class-flavored") large cardinal notions ... | |
| Mar 6, 2019 at 22:44 | comment | added | Asaf Karagila♦ | Thomas, I think that what Dr. Eskew is trying to suggest is that all large cardinals in ZFC are expected to stay internal to the universe in their definition, because ZFC is a first order theory, and second-order definitions are not particularly useful when you try to stay within the bounds of first order theory. | |
| Mar 6, 2019 at 20:39 | comment | added | Thomas Benjamin | @MonroeEskew: Regarding the the first-order expressibility of large cardinals, are there papers you could suggest I read regarding the notion that all large cardinals should have a first-order formulation? | |
| Mar 5, 2019 at 21:56 | answer | added | Wojciech Aleksander Wołoszyn | timeline score: 2 | |
| Feb 28, 2019 at 16:03 | comment | added | Thomas Benjamin | @MonroeEskew: Interesting. What I mean by "deriving the notion of measurable cardinal" is whether convergent potentialism and truth-in-the-limit-structure amounting to truth-in-the-approximating-structures would allow one to be able to define measurable cardinals as the critical point of the elementary embedding $j$: $V$ $\rightarrow$ $M$ (if at all). Hope that helps. | |
| Feb 28, 2019 at 15:55 | comment | added | Monroe Eskew | But I don’t understand what you mean by “derive the notion of measurable cardinal.” | |
| Feb 28, 2019 at 15:54 | comment | added | Thomas Benjamin | @NoahSchweber: Miha correctly understood my question. However, your point is well taken and I will attempt to do better. | |
| Feb 28, 2019 at 15:52 | comment | added | Monroe Eskew | @ThomasBenjamin— Far predating Hamkins’ notion of potentialism, there have been concerns about the first-order expressibility of large cardinal axioms. Measurable cardinals were first characterized in combinatorial terms, motivated by questions from the early days of analysis. Then came the useful elementary embedding characterization. A general stance is that large cardinals should have a first-order formulation, which I would say is a stricter version of potentialism. | |
| Feb 28, 2019 at 15:50 | comment | added | Thomas Benjamin | @YCor: Actually, one does not get rid of philosophical speculations when working with axioms (as the existence of the set-theoretic muliverse shows), rather, they allow one to study philosophical speculations more precisely. | |
| Feb 28, 2019 at 15:45 | comment | added | Thomas Benjamin | (cont.) muddied.... | |
| Feb 28, 2019 at 15:44 | comment | added | Thomas Benjamin | @MihaHabič: You have essentially understood the point I was trying to make with my question. Given that $V$ is a limit structure for a convergent potentialist system $\mathcal W$, does the potentialist translation for $V$ (according to the theorem Prof. Hamkins states in his slide presentation which I quoted) and the fact that $V$ as the limit structure does not exist cause problems (so to speak) for defining measurability in terms of non-trivial elementary embeddings from $V$ to transitive inner models of $V$ (which would be the elements of $\mathcal W$)? Hope the waters are not further | |
| Feb 28, 2019 at 10:17 | comment | added | YCor | What does this have to do with math philosophy? philosophy can be a motivation for setting up axioms, but the interest of working with axioms is that one gets rid of philosophical speculations. | |
| Feb 28, 2019 at 8:59 | comment | added | Miha Habič | If you restrict your pieces of the universe to have size at most $\kappa$, then I suspect you won't be able to express measurability, but will get stuck somewhere in the region of a Ramsey cardinal. | |
| Feb 28, 2019 at 8:57 | comment | added | Miha Habič | I agree that the question is not particularly clear, but one way of making sense of it would be to ask whether one necessarily needs to talk about all of $V$ in order to define measurability, or do bounded pieces of the universe suffice (I suppose "bounded" can be interpreted in different ways, depending on the kind of potentialism one is interested in). If you allow pieces of the from $V_\alpha$, then the answer is yes, as the measurability of $\kappa$ is witnessed by objects somewhere in the neighbourhood of $V_{\kappa+3}$ (either the measure, or just an embedding defined on $V_{\kappa+1}$). | |
| Feb 27, 2019 at 23:20 | comment | added | Noah Schweber | I honestly don't understand what this is asking. The text you've cited describes a way to translate actualist accounts into potentialist accounts; what does that have to do with a measurable cardinal specifically? Are you asking whether a particular actualist argument for measurables can be translated into a potentialist argument for measurables? If so, what argument do you have in mind? | |
| Feb 27, 2019 at 22:50 | review | Close votes | |||
| Mar 7, 2019 at 3:05 | |||||
| Feb 27, 2019 at 19:37 | history | asked | Thomas Benjamin | CC BY-SA 4.0 |