Timeline for Is the notion of measurable cardinal definable from the perspective of set-theoretical potentialism?
Current License: CC BY-SA 4.0
21 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 11, 2019 at 19:18 | comment | added | Thomas Benjamin | @WojciechAleksanderWołoszyn: Question: Must $f$, for $f$: $\mathbb N$ $\rightarrow$ $A$ be necessarily recursive in order for the classical potentialist to answer the question implicitly using larger and larger finite initial segments of $\mathbb N$? | |
| Mar 8, 2019 at 16:52 | comment | added | Wojciech Aleksander Wołoszyn | @ThomasBenjamin I have expanded my answer by an additional argument | |
| Mar 8, 2019 at 16:52 | comment | added | Wojciech Aleksander Wołoszyn | @AlecRhea It was not my intention to suggest all this. Please let me know if the edited version raises any concerns. | |
| Mar 8, 2019 at 16:44 | history | edited | Wojciech Aleksander Wołoszyn | CC BY-SA 4.0 |
addressed comments + added an additional explanation
|
| Mar 8, 2019 at 2:07 | comment | added | Alec Rhea | As someone who takes MK seriously as a foundational theory (with a preference towards additional class collection axioms/reflection principles) I had the same reaction as Nik — a set is a class which is a member of some other class, but proper classes are objects too. | |
| Mar 7, 2019 at 16:45 | comment | added | Nik Weaver | What concerned me about the original answer was that it seemed to be making a global statement ("$V$ is not an object ... regardless of our philosophical inclinations"). | |
| Mar 7, 2019 at 16:43 | comment | added | Nik Weaver | @MonroeEskew: well, I basically agree with these points. | |
| Mar 7, 2019 at 16:37 | comment | added | Monroe Eskew | @NikWeaver I disagree. Whether something is an object depends on the context or rules, which we may change if we like. Instead of having one absolute notion of object, we can explore different notions in different formal systems. I read his assertion that “proper classes are generally not objects” as saying that in some contexts (i.e. ZF), there aren’t such things as proper classes. The post explained how talk involving definable proper classes can eliminated. | |
| Mar 7, 2019 at 14:55 | comment | added | Nik Weaver | @MonroeEskew: I think it's clear that the posted answer could be expressed in a formal metalanguage, and in that metalanguage by your (Quine's) criterion proper classes are objects. Now you may feel that the target system ZF is what really matters and assertions in the metasystem aren't to be taken seriously, but then you have the problem that the answer formulated here ("is equivalent to the assertion that for all formulas $\phi(x)$ ...") only makes sense in the metalanguage. So you're kind of trying to have your cake and eat it too. | |
| Mar 7, 2019 at 14:28 | comment | added | Monroe Eskew | @NikWeaver Now we're mixing formal and informal mathematical talk. In ordinary language, I can noun-ify anything I want, and also perform verbatimization and make adjectivity. English grammar is somewhat flexible, especially on the internet. But when some says "proper classes are not objects," I take that to be talking about what counts as an object in a formal system. | |
| Mar 7, 2019 at 14:16 | comment | added | Nik Weaver | @MonroeEskew: exactly, and statements like "a proper class, generally, is not an object" quantify over classes. | |
| Mar 7, 2019 at 12:59 | comment | added | Monroe Eskew | @NikWeaver To paraphrase a philosopher, to be is to fall under the scope of a quantifier. The question of what counts as an object is at base a grammatical question. | |
| Mar 7, 2019 at 12:13 | comment | added | Wojciech Aleksander Wołoszyn | @NikWeaver Thank you for taking an interest in my answer. I wrote that proper classes are not objects in general, because usually, people have ZF in mind when they talk about set-theory. But this is not the point here. It just happens that the problem of referring to $V$ without accepting its existence is not exclusive to potentialism. It is a ubiquitous issue that we know how to circumvent. | |
| Mar 7, 2019 at 11:37 | comment | added | Nik Weaver | @MonroeEskew: if you take NBG as your foundational theory it is wrong. This is not a serious philosophical argument. | |
| Mar 7, 2019 at 6:09 | comment | added | Monroe Eskew | @NikWeaver if you take first order ZF seriously as your foundational theory, then this follows. | |
| Mar 6, 2019 at 22:11 | comment | added | Wojciech Aleksander Wołoszyn | @ThomasBenjamin Thank you for your warm greeting :) Well, Miha points at a particular piece of $V$ where $\kappa$ reveals itself to be measurable. My answer shows a general pattern of making sense of elementary embeddings and large cardinal statements without referring to proper classes, so without referring to $V$ in particular. | |
| Mar 6, 2019 at 21:12 | history | edited | Wojciech Aleksander Wołoszyn | CC BY-SA 4.0 |
moved exceptional case to where it should be in the first place
|
| Mar 6, 2019 at 20:49 | comment | added | Thomas Benjamin | @Wojciech: Welcome to MathOverflow! How (by the way) does your answer relate to Miha's comment (if at all)? | |
| Mar 5, 2019 at 22:36 | history | edited | Wojciech Aleksander Wołoszyn | CC BY-SA 4.0 |
added 2 characters in body
|
| Mar 5, 2019 at 22:14 | comment | added | Nik Weaver | What is an "object"? Why is the set of all real numbers an "object" while the class of all ordinals is not an "object"? | |
| Mar 5, 2019 at 21:56 | history | answered | Wojciech Aleksander Wołoszyn | CC BY-SA 4.0 |