Timeline for Ontological status of some "sets" in ZFC [closed]
Current License: CC BY-SA 3.0
24 events
| when toggle format | what | by | license | comment | |
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| Jun 5, 2013 at 9:41 | vote | accept | Godot | ||
| Jun 5, 2013 at 3:41 | comment | added | Noah Schweber | I just realized that both Joel and I received downvotes on our answers without comment, presumably (?) from the same person. Could whoever downvoted please explain? | |
| Jun 5, 2013 at 2:57 | comment | added | Joel David Hamkins | I voted to re-open, since I think this is a perfectly reasonable question in the philosophy of mathematics. Indeed, I would add that this is an issue that I believe many mathematicians find confusing. | |
| Jun 5, 2013 at 2:30 | history | closed |
Qfwfq Noah Schweber Steven Landsburg Andrés E. Caicedo Goldstern |
not a real question | |
| Jun 4, 2013 at 23:50 | answer | added | Joel David Hamkins | timeline score: 13 | |
| Jun 4, 2013 at 23:12 | comment | added | Qfwfq | @Noah S: Oh I see: in that sense, sure. | |
| Jun 4, 2013 at 22:23 | history | edited | François G. Dorais |
edited tags
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| Jun 4, 2013 at 22:18 | comment | added | Andreas Blass | I agree with Noah's parenthetical comment that the problem is not ontological but epistemological. The set $X$ in the question is a perfectly good, nonempty, finite set of natural numbers, and ZF proves that. The only "problem" with this set is that we don't know as much about it as we might want. Specifically, we don't know whether 2 is an element of it and we don't know how many elements it has. But these unpleasant circumstances are entirely epistemological. | |
| Jun 4, 2013 at 21:43 | answer | added | Noah Schweber | timeline score: 6 | |
| Jun 4, 2013 at 20:06 | comment | added | The User | If you do not presuppose an “ontological meaning” of sets, you have no problems. You just have this formula defining a set and the “meaning” is different than the meaning of $\{1\}$ or $\{1,2\}$, however it is equal to one of them in the sense of your formal language. | |
| Jun 4, 2013 at 20:02 | comment | added | Noah Schweber | @Qfwfq: a description of a set corresponds to a set, but multiple descriptions may correspond to the same set, and there need not (in fact, will not) be any "effective" procedure for finding out what set a given description corresponds to. (@Zhen Lin, is this what you had in mind?) | |
| Jun 4, 2013 at 19:43 | comment | added | Qfwfq | @Zhen Lin: why not? (assuming the "description" draws elements from an ambient set, not just a class) | |
| Jun 4, 2013 at 19:35 | comment | added | Noah Schweber | The relevant axioms of set theory say, in this case, that there exists a set with such-and-such properties. They don't say - nor is there any reason they should, unless we want to leave classical logic - that we can "determine" that set in any way. The axioms are extensional, not intensional. Note that we don't need any "philosophical view" to do math: all mathematicians, despite having wildly different philosophies, can agree that (classical) ZFC proves the existence of such a set. | |
| Jun 4, 2013 at 19:34 | comment | added | Zhen Lin | To reiterate, a description of a set is not itself a set! | |
| Jun 4, 2013 at 19:29 | comment | added | Joel David Hamkins | But you can write them down, and this is easy to prove. Namely, if $\phi$ holds, then you write down both $1$ and $2$, but if $\phi$ fails, then you write down only $1$ (so the proof that you can write them down is a proof by cases). | |
| Jun 4, 2013 at 19:16 | comment | added | Godot | Thank you for your comments. I will appreciate full answer explaining modern philosophical view on such objects, together with some set-teoretic jargon, i.e., how are such "sets" named in modern set theory. To be honest I find it peculiar that I can't write down all elements of a subset of $\{1,2\}$. | |
| Jun 4, 2013 at 19:09 | comment | added | Noah Schweber | (Whoops, I didn't see Mariano's second comment before posting my own.) | |
| Jun 4, 2013 at 19:04 | comment | added | Noah Schweber | it's either just 0 or just 1. The issue is when we try to verify that our listing is correct. But this isn't a set-theoretic issue, just an issue with classical logic. If this is truly undesirable, then you can use intuitionistic logic, but there isn't really an "ontological difficulty" here of the kind (I think) you're imagining. (Also, I'd call this "epistemological" as opposed to "ontological," but maybe that's just me.) | |
| Jun 4, 2013 at 19:03 | comment | added | Noah Schweber | I'm not sure this is a real question, but: $ZFC$ does prove that there exists a set $X$ such that EITHER the continuum hypothesis (or any $\varphi$) holds and $X=\lbrace 0\rbrace$, OR the continuum hypothesis fails and $X=\lbrace 1\rbrace$. There's nothing mysterious about the set $X$: rather, the only thing behaving oddly here is the intentional nature of $X$. As to your comment, it is not always possible to list all the elements of a finite set of natural numbers and be sure that you have done so correctly; note, however, that it is possible to list the elements of the set $X$ above: | |
| Jun 4, 2013 at 18:58 | comment | added | Mariano Suárez-Álvarez | It is obviously a set, because it is constructed according to the rules that ZFC makes available to construct sets. It is a finite subset of $\mathbb N$, again obviously, because the inclusion map into $\{1,2\}$ is injective. Also obviously, it is undecidable in ZFC whether $2$ is in $X$ or not. Presumably you know all this and are trying to make some point? Remember that this is not a discussion site. | |
| Jun 4, 2013 at 18:49 | comment | added | Godot | @Mariano: is X a set? If so, is it a finite subset of $\mathbb{N}$? If yes, please list its elements. If $X$ is not a set, please explain why... | |
| Jun 4, 2013 at 18:49 | history | edited | Emil Jeřábek | CC BY-SA 3.0 |
fix markup
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| Jun 4, 2013 at 18:42 | comment | added | Mariano Suárez-Álvarez | What is the ontological status of $\{1\}$? What does that even mean? | |
| Jun 4, 2013 at 18:39 | history | asked | Godot | CC BY-SA 3.0 |