Timeline for Circular, or missing, definition in set theory?

Current License: CC BY-SA 4.0

53 events

when toggle format	what		by	license	comment
May 25, 2018 at 7:17	answer	added	Zuhair Al-Johar		timeline score: 0
May 24, 2018 at 20:09	comment	added	Timothy Chow		The revised question is more clearly stated, except that Frank Quinn's insistence (in the comments to Noah's answer) that he is not asking about the logic or the metatheory are totally baffling to me. As far as I can tell, he is asking exactly for more clarification about the metatheory, and about whether using set theory for the metatheory is "circular." If he insists that he's not asking about the metatheory then I can't make any sense out of the question.
May 24, 2018 at 19:29	comment	added	Zuhair Al-Johar		I must admit that I understood nothing of this 'implementation'. Why not simply have models that satisfy the axioms?
May 24, 2018 at 19:06	comment	added	James Smith		@AndyPutman, I agree. It's often difficult to pose the right question precisely because you don't know what you're talking about! Questions like these strike me as subtle, even for the experts. It's fair if someone has a vague notion to try to clarify that notion by asking an imprecise question. Perhaps if you knew how to state the question with precision, the answer wouldn't be that far away. And, as has been pointed out, if ever there was a place to ask such questions, it's here.
May 24, 2018 at 18:42	comment	added	Noah Schweber		Incidentally, if you're interested in the idea of doing mathematics in weak (or perhaps more positively: more concrete) systems you may be interested in reverse mathematics. The connection with your question that I see is that reverse math can be thought of as a corollary of the formalist-as-defense position in my answer: if mathematical claims are ultimately only "guaranteed to be meaningful" when they are translated to statements about formal systems, we should be interested in what formal systems can prove them; (cont'd)
May 24, 2018 at 17:53	answer	added	Eric Wofsey		timeline score: 18
May 24, 2018 at 17:15	comment	added	Carl Mummert		As @Monroe Eskew mentions, there is a hierarchy of metatheories we can use to study ZFC. In PRA or PA, we can talk about provability but not models, although we can often formalize forcing arguments syntactically in these settings. In second-order arithmetic we can talk about countable models of ZFC or any other countable theory. We can also use set theories such as ZFC itself or MK as metatheories to study ZFC, allowing us to look at more general models and also to do things such as perform ultraproduct constructions on models.
May 24, 2018 at 17:05	comment	added	Monroe Eskew		I have heard many times that “most ordinary mathematics” can be formalized in second-order arithmetic. In other words, we can believe that the only genuine mathematical objects are natural numbers and real numbers, and code everything with those. If you buy this position, then an implementation of ZFC is just a real number with certain properties.
May 24, 2018 at 16:31	comment	added	Eric Wofsey		To put it another way, the axioms of set theory have a fundamentally different purpose from the axioms of group theory. We aren't trying to study models (or "implementations"); we're trying to set up formal deductive rules we can use to reason about mathematics as a whole. (Set theorists of course do also study models of set theory, but they do so already accepting some set theory as their framework. In other words, as Noah said, if you want to study models, your metatheory needs to already have set theory.)
May 24, 2018 at 16:21	comment	added	Eric Wofsey		I find somewhat strange the assumption that "Users of set theory need an implementation of the axioms". The entire point of them being axioms is that they are taken for granted, and you can reason from the axioms instead of reasoning directly about some "implementation". (This is closely related to the formalist approach that Noah discusses.)
May 24, 2018 at 16:02	comment	added	Noah Schweber		For what it's worth, I think the question is better in its current form - it's not the axiom of extensionality in particular that's the focus, and I think that detracted from the clarity of the original question. I don't think it would be appropriate for the moderators to revert it.
May 24, 2018 at 15:58	comment	added	Mikhail Katz		The new version of the question is not actually a new version but a separate follow-up question (which is less interesting than the original one IMHO). Moderators could consider reverting this substitution and encouraing the OP to post a separate question if necessary.
May 24, 2018 at 15:16	comment	added	Joel David Hamkins		Andy, I agree with you, and I don't think the question should be closed.
May 24, 2018 at 15:03	comment	added	Andy Putman		anywhere in my work or reading. I learned a lot from the various answers to this question. I would hate it if MO became just a place where specialists can ask other specialists their super-technical questions (after all, don't all of us mostly know the experts in our own special fields, and thus can more efficiently just email them?)
May 24, 2018 at 15:00	comment	added	Andy Putman		People's pedigree is important in the sense that if someone has made important contributions to mathematics, then they have earned the right to be given the benefit of the doubt. One of the important functions of MO is that it is supposed to be a place where professional mathematicians can ask questions that interest them but are outside their speciality. Sometimes these questions might sound stupid to a specialist, but I can assure you that as someone whose education probably resembles Frank's more than that of a logician, the distinction between theory and meta-theory has never came up
May 24, 2018 at 14:47	comment	added	Joel David Hamkins		To my way of thinking, the answer to the question is the theory/meta-theory distinction, discussed at length in any good treatment of set theory, including a typical good undergraduate or graduate set theory course. A deep felicity with this distinction underlies the set-theoretic advances with the independence phenomenon. Set theorists build new models (or interpretations) from old, thereby showing the relative consistency of various set-theoretic principles, often in terms of the large cardinal hierarchy. In particular, this treatment is deeply understood and definitely not "missing."
May 24, 2018 at 14:33	history	edited	Frank Quinn	CC BY-SA 4.0	added 48 characters in body
May 24, 2018 at 14:20	history	edited	Frank Quinn	CC BY-SA 4.0	Revision in response to early comments
May 24, 2018 at 10:56	comment	added	Carl Mummert		@Jim Conant: rather than thinking of math as already there in Platonist-like terms, one can also see math as already there in practice, as a semi-formal theory. I think a larger issue lurking behind this kind of question is that the "foundationalist" mentality that there will be a single formal theory for mathematics (or that this is desirable) is much less common among logicians now than it was in the past - but the literature hasn't caught up, so questions like this might seem to ask about a particular kind of foundational role for ZFC that few are still trying to defend.
May 24, 2018 at 10:17	comment	added	Andrés E. Caicedo		@arsmath In case the previous to last comment was (partly) meant for me, the convincing bit was "as someone who has basically never paid attention to foundational issues, I found the answers useful and clarifying".
May 24, 2018 at 6:47	comment	added	arsmath		It's fine to find the question interesting if you've never thought of it. But that's equally true if the question was asked by a college sophomore who stayed up all night smoking pot and thinking about their philosophy and logic class.
May 24, 2018 at 6:38	comment	added	arsmath		So people are now just openly treating people differently because of their pedigree? No smart person can ever ask a dumb or lazy question?
May 24, 2018 at 5:36	answer	added	Greg S		timeline score: -1
May 24, 2018 at 5:09	comment	added	Jim Conant		@TimothyChow: thanks for the link to your forcing paper. There seems to be a very clear point that ZFC encodes mathematics but is not the ontological basis for mathematics, and that makes perfect sense to me. (And of course the whole point of forcing is to use ZFC as an object itself to be studied mathematically.) But then, I don't see how one is not forced into a sort of naive Platonism that the math is already there, even if in some multi-verse form.
May 24, 2018 at 0:15	history	reopened	Jim Conant Andy Putman user6976 Timothy Chow Andrés E. Caicedo
May 24, 2018 at 0:14	comment	added	Andrés E. Caicedo		Andy's comment convinced me.
May 23, 2018 at 22:37	comment	added	Timothy Chow		@arsmath : I disagree that the question "shows absolutely no effort to learn anything about first-order logic." I'll grant that the question isn't clearly phrased, but it's not a "technical" question of the sort that can be answered by just picking up a textbook. It's a question about whether set theory can be used as a foundation for mathematics and is thus at least a partly philosophical question about a point that is often not clearly addressed in textbooks.
May 23, 2018 at 18:17	review	Reopen votes
May 23, 2018 at 21:55
May 23, 2018 at 17:56	history	closed	Qfwfq arsmath Carl Mummert Ben McKay Peter LeFanu Lumsdaine		Not suitable for this site
May 23, 2018 at 17:51	comment	added	Andy Putman		Frank's work in geometric topology is fundamental and deep. Given his contributions to the subject, he has in my opinion earned the right to ask questions here even if specialists find them naive (and what is more, as someone who has basically never paid attention to foundational issues, I found the answers useful and clarifying). I would be opposed to closing this question.
May 23, 2018 at 17:50	comment	added	Carl Mummert		My challenge in writing an answer to the question is that I can't tell if the question is literally "what is the domain of $\in$", or whether it is something deeper. Clarification would be very welcome.
May 23, 2018 at 17:29	comment	added	arsmath		It's a trivial issue, despite the pedigree of the person asking it. It shows absolutely no effort to learn anything about first-order logic. It is a more elementary question than 50% of the questions at Math Stack Exchange.
May 23, 2018 at 16:35	comment	added	Mikhail Katz		I don't think the attempts to shut down this question are reasonable. The issue may be trivial for editors trained in logic and set theory, but notice that the OP is a leading specialist in 4-manifold topology. If this issue bothers him it is a sure sign that it is not a trivial issue for traditionally trained mathematicians.
May 23, 2018 at 7:23	comment	added	Asaf Karagila♦		Many words have been minced on this over at Mathematics.
May 22, 2018 at 22:39	comment	added	Carl Mummert		We use ZFC to help us understand sets. We use sets to formalize first-order logic, including ZFC. This is an example of a kind of hermeneutical circle - en.wikipedia.org/wiki/Hermeneutic_circle . To break the circle, the most common recourse is to not formalize ZFC using sets, for example by viewing it as solely a syntactic theory that can be studied in weak systems like PRA. The downside of that approach, of course, is that it eliminates our ability to talk about semantics - models of set theory - so we can only talk about provability. All of this is well known, in any case.
May 22, 2018 at 17:24	comment	added	Qfwfq		That one is not a function (within the theory) hence it doesn't make sense to ask for it's domain.
May 22, 2018 at 16:30	review	Close votes
May 23, 2018 at 18:01
May 22, 2018 at 15:34	comment	added	Monroe Eskew		You will always have some primitives. One might question your notions of operators and functions and ask how you define them noncircularly.
May 22, 2018 at 13:50	history	edited	Qfwfq	CC BY-SA 4.0	added 3 characters in body
May 22, 2018 at 13:24	answer	added	Burak		timeline score: 20
May 22, 2018 at 12:41	comment	added	Mikhail Katz		Reopened within less than 25 minutes after closing! This must be a record.
May 22, 2018 at 12:38	history	reopened	Mikhail Katz Carlo Beenakker paul garrett HenrikRüping Francois Ziegler
May 22, 2018 at 12:16	history	closed	Andrés E. Caicedo Gro-Tsen Mike Shulman Neil Strickland Andreas Blass		Not suitable for this site
May 22, 2018 at 11:52	answer	added	Mikhail Katz		timeline score: 10
May 22, 2018 at 11:08	answer	added	Zuhair Al-Johar		timeline score: 1
May 22, 2018 at 3:14	history	edited	Mike Shulman		edited tags
May 22, 2018 at 2:46	answer	added	Timothy Chow		timeline score: 15
May 21, 2018 at 21:17	answer	added	Giorgio Mossa		timeline score: 5
May 21, 2018 at 21:15	review	Close votes
May 22, 2018 at 12:17
May 21, 2018 at 20:59	answer	added	Noah Schweber		timeline score: 35
May 21, 2018 at 20:22	comment	added	Alec Rhea		In ZFC set theoy it is implicitly understood that the domain of discourse we quantify over is the universe of sets $\{x:x=x\}$, although this question would be more appropriate over at math SE.
May 21, 2018 at 20:19	comment	added	Zuhair Al-Johar		$x \in A$ is a "propositional function", it returns the values of truth or false for every two objects substituting the symbols x and the symbol A, I'm using the terms of Russell in his introduction to mathematical philosophy. The domain of that function is a set of all those sets that the theory is speaking about, i.e. of all those sets that can substitute those symbols mentioned above. I don't see any circularity here? it is called as the domain of discourse.
May 21, 2018 at 20:06	history	asked	Frank Quinn	CC BY-SA 4.0

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