Timeline for Extensionality, regularity, NBG--
Current License: CC BY-SA 4.0
38 events
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| Nov 11 at 6:01 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Jul 14 at 5:04 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Mar 16 at 4:09 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Nov 17, 2023 at 4:02 | history | bumped | CommunityBot | This question has answers that may be good or bad; the system has marked it active so that they can be reviewed. | |
| Oct 19, 2023 at 19:42 | comment | added | Frode Alfson Bjørdal | @DavidRoberts I share the opinion that one, while communicating with others, may, and, in cases, even should, adjust the popularizing efforts so as to be appropriate for the audience addressed. There are various matters which may have weight, in such considerations, as e.g. expediency. Some times one may miss the target, which is unfortunate. At any rate, communicating is not very easy, and it requires efforts from all participants who want to understand. | |
| Oct 19, 2023 at 16:34 | comment | added | Frode Alfson Bjørdal | @DavidRoberts Thanks for the review! Scott's result is indeed very rarely appreciated in the literature; even in the following otherwise good answer to a query, on the relative consistency of extensionality, the answer is not satisfactory as it does not relate Scott's result on the consistency of extensionality relative to his replacement based (\mathbbmss{ZF}(\dot{=}))-Extensionality. math.stackexchange.com/questions/4246921/… | |
| Oct 18, 2023 at 20:12 | comment | added | David Roberts♦ | Aha, I see. I could not for the life of me find, at that time, a page confirming the existence of the NH edition. I was indeed misled by the "date" entry on 1961 on the Magnus Press page, and the scraped date in Google Books. I found a review of the first edition cambridge.org/core/journals/… | |
| Oct 18, 2023 at 17:12 | comment | added | Frode Alfson Bjørdal | @DavidRoberts It was new to me that Magnes Press published a second edition in 1967. I have been using the first edition of the book, and that was published by North-Holland in 1962; here is a link to a photo of its title page public.pdf-archive.com/js37HjFR. | |
| Oct 18, 2023 at 4:00 | history | edited | Frode Alfson Bjørdal | CC BY-SA 4.0 |
Style: Minus "induced"
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| Oct 18, 2023 at 2:27 | comment | added | David Roberts♦ | I also think that the book is not published by North-Holland, but is rather: Essays on the foundations of mathematics, dedicated to A. A. Fraenkel on his seventieth anniversary. Y. Bar-Hillel, E. I. J. Poznanski, M. O. Rabin and A. Robinson, eds. Jerusalem, the Hebrew University: Magnes Press, 1961 magnespress.co.il/en/book/… (cf google.com.au/books/edition/… for the paper in question) (EDIT this is a second edition!) | |
| Oct 18, 2023 at 0:13 | comment | added | David Roberts♦ | @FrodeAlfsonBjørdal regardless of the other parts of the discussion between yourself and James, I share the opinion that pasting two raw BibTeX entries is not helping the reader. Most preferable is something that looks like a proper bibliographic entry as you'd see in a paper, and, if possible, a link (and best of all a stable link). This book does not seem easy to find, so obfuscating its information inside BibTeX code makes matters even harder for people trying to answer and help here. | |
| Oct 17, 2023 at 21:02 | history | edited | Frode Alfson Bjørdal | CC BY-SA 4.0 |
Made E an initial class in the inductively given least class of classes specified.
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| Oct 17, 2023 at 18:53 | answer | added | Frode Alfson Bjørdal | timeline score: 0 | |
| Oct 17, 2023 at 11:44 | comment | added | Frode Alfson Bjørdal | @JamesHanson I suggest, rather, that you here have revealed a rather too smirk attitude, that is not conducing to find the truth . So I flagged you last two comments. | |
| Oct 17, 2023 at 7:21 | comment | added | James E Hanson | @FrodeAlfsonBjørdal Frankly, your original question is also just poorly written. You don't specify which version of replacement you are using (which as we have just established, can significantly change the character of the theory in question), and you expected the reader to just know what 'closure under E' and 'circular permutation' mean. In general, you seem to be expecting others to do the lion's share of the work of interpreting your question, rather than putting the effort in yourself to communicate an unambiguous question. | |
| Oct 17, 2023 at 7:20 | comment | added | James E Hanson | @FrodeAlfsonBjørdal I'm not going to deny that I could have spent more time looking at the paper and trying to figure out how it supported your statement, but that doesn't change the fact that's it's pretty basic academic courtesy in situations like this (two seemingly similar definitions with notably distinct consequences) to spell out what you mean as clearly as possible, rather than just spending two seconds pasting some raw BibTeX code and expecting the other person to spend ten minutes poring over a paper. | |
| Oct 17, 2023 at 6:49 | comment | added | Frode Alfson Bjørdal | @JamesHanson Moreover, one should read the last sentence of Scott's article: "Hence our conclusion is that ZF($\dot{=}$) is weakened by dropping extensionality, but ZF$(\overset{=}{.}$) is not." | |
| Oct 17, 2023 at 6:28 | comment | added | Frode Alfson Bjørdal | @JamesHanson I think that you should as well have paid attention to Scott's paragraph three, where he compares his article with Gandy, R. On the Axiom of Extensionality II, Journal of Symbolic Logic, 24.4, 287-300, 1959. Gandy 1959 showed that NBG minus extensionality interprets NBG. Scott in that paragraph clearly points to $4, and states that a change in replacement leads to the result that the induced version of ZF is interpretable by the same ZF minus extensionality. | |
| Oct 17, 2023 at 6:02 | comment | added | James E Hanson | @FrodeAlfsonBjørdal I'm sorry but I just feel like you are not doing a great job of communicating clearly if you say 'Scott proved X' and point to a paper which leads with saying that it's going to prove not X. | |
| Oct 17, 2023 at 5:24 | comment | added | Frode Alfson Bjørdal | @JamesHanson I can claim what I claim. Scott has, as you will have noticed, several results in the paper, and it is true that some of his versions of ZF are weakened by deleting extensionality. But there is also the crucial result that one of his versions of ZF is interpretable by the same version of ZF minus extensionality. I also noticed that some accounts of Scott only took the former results into account. But I take Scott's article to be the canonical account, and his account is not given in the second paragraph. | |
| Oct 17, 2023 at 5:05 | comment | added | James E Hanson | @FrodeAlfsonBjørdal I don't think you can really claim that your statement is correct in an unqualified sense when the paper you're citing uses a materially different definition of 'ZF minus extensionality and foundation' and literally states in the second paragraph that ZF minus extensionality and foundation is strictly weaker than ZF minus foundation. | |
| Oct 17, 2023 at 4:59 | comment | added | Frode Alfson Bjørdal | @JamesHanson The statement I made, which you are characterizing, was incidental, and it is correct. Scott's results are not relevant to any presuppositions I make as regards U. | |
| Oct 17, 2023 at 4:22 | comment | added | James E Hanson | @FrodeAlfsonBjørdal Okay well then the statement depends in a strong way on the precise formulation of replacement, so I feel like you should specify that when making this kind of statement, especially when the commonly cited result in this area (given here for instance) is more or less the opposite. | |
| Oct 17, 2023 at 3:42 | comment | added | Frode Alfson Bjørdal | @JamesHanson Yes, I am sure. The result is concisely formulated in the penultimate paragraph of page 130. But the preceding paragraphs are needed to unriddle it. | |
| Oct 17, 2023 at 3:32 | comment | added | James E Hanson | @FrodeAlfsonBjørdal Are you sure that that's what's shown in More on the Axiom of Extensionality? I'm looking at a copy of it right now and it looks like Scott is proving that ZF minus extensionality and foundation is strictly weaker than ZF minus foundations. | |
| Oct 17, 2023 at 1:16 | comment | added | Frode Alfson Bjørdal | @JamesHanson @Book{Bar-Hillel1961, ALTauthor = {author}, editor = {Bar-Hillel, Y. and others}, title = {Essays on the Foundations of Mathematics}, booksubtitle = {Dedicated to A. A. Fraenkel on his seventieth anniversary}, publisher = {North-Holland Publishing Company}, location = {Amsterdam}, year = {1961}, OPTkey = {key}, OPTvolume = {volume}, OPTnumber = {number}, OPTseries = {series}, address = {Amsterdam}, OPTedition = {First}, OPTmonth = {month}, OPTnote = {note}, OPTannote = {annote}, } | |
| Oct 17, 2023 at 1:14 | comment | added | Frode Alfson Bjørdal | @JamesHanson Incidentally, Scott showed that ZF minus extensionality and foundation interprets ZF minus foundation: @InBook{Scott1961, author = {Scott, D.}, ALTeditor = {editor}, title = {More on the Axiom of Extensionality}, OPTchapter = {chapter}, OPTpublisher = {publisher}, year = {1961}, OPTkey = {key}, OPTvolume = {volume}, OPTnumber = {number}, OPTseries = {series}, OPTtype = {type}, OPTaddress = {address}, OPTedition = {edition}, OPTmonth = {month}, OPTpages = {207-214}, note = {In: Bar-Hillel, pp. 115-131}, OPTannote = {annote}, } | |
| Oct 17, 2023 at 1:05 | comment | added | Frode Alfson Bjørdal | @JamesHanson I have a proof, by means of induction, that all members of U are wellfounded. | |
| Oct 17, 2023 at 0:27 | comment | added | James E Hanson | It's not really clear to me that regularity is going to hold in $U$. In my experience when you try to throw away both extensionality and foundation you start to really lose the ability to form inductive constructions (like the class of hereditarily well-founded sets). Do you have a proof of this statement? | |
| Oct 16, 2023 at 23:47 | history | edited | Frode Alfson Bjørdal | CC BY-SA 4.0 |
Style: minus a "the"
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| Oct 16, 2023 at 23:44 | comment | added | Frode Alfson Bjørdal | @HanulJeon "Also, do you allow "re-interpreting" the equality symbol over 𝑉 to a different equivalence relation?" Equality is taken à la Leibniz, with the improvement by Whitehead & Russell in PM, as \forall u(a\in u\to b\in u). | |
| Oct 16, 2023 at 23:39 | history | edited | Frode Alfson Bjørdal | CC BY-SA 4.0 |
Edit 2: I changed the original V to U, so as to avoid the confusion with the cumulative hierarchy.
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| Oct 16, 2023 at 23:36 | comment | added | Frode Alfson Bjørdal | @HanulJeon "How can we construct 𝑉 without Extensionality?" Notice that the V is not the V as in the cumulative hierarchy. I edit and substitute with U. The details are rather too complex to divulge her. But notice that the axioms apart from extensionality are all, roughly, of the form \forall x(x\in U\to \Gamma(x)\in U); extensionality and regularity cannot be brought to that form. | |
| Oct 16, 2023 at 23:22 | history | edited | Frode Alfson Bjørdal | CC BY-SA 4.0 |
Edit: The question(s) presuppose that NBG is consistent.
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| Oct 16, 2023 at 20:36 | comment | added | Hanul Jeon | How can we construct $V$ without Extensionality? Also, do you allow "re-interpreting" the equality symbol over $V$ to a different equivalence relation? | |
| Oct 16, 2023 at 19:14 | comment | added | Frode Alfson Bjørdal | @JamesHanson E={(x,y)|(x,y)\in V\wedge x\in y} | |
| Oct 16, 2023 at 18:53 | comment | added | James E Hanson | What is closure under E? | |
| Oct 16, 2023 at 17:55 | history | asked | Frode Alfson Bjørdal | CC BY-SA 4.0 |