Talk:Skolem's paradox

message: It occurred to me this morning that the paradox could be recast in an alternate form using naturals instead of reals. I don't think it should go in the article, so excuse me for telling it here.
:Start with the natural numbers. They have a certain theory T, called the theory of true arithmetic. Now, by the Lowenheim–Skolem theorem, there is an uncountable model M of the theory T of the natural numbers. But the identity map is a bijection between the natural numbers in M and the natural numbers in M, so M must be countable, a contradiction.
I think this is cute. — Carl (CBM · talk) 14:10, 11 November 2008 (UTC)
replies: text: Sounds like Non-standard arithmetic to me. I guess when a person says "think of a random number" since any finite number is just too close to 0 to be chosen at random one should really choose a non-standard infinite number :) Dmcq (talk) 18:47, 11 November 2008 (UTC)
thread_title:  Another way of telling the paradox, using natural numbers 

message: It has been admitted in this discussion that it'it[Skolems paradox] is simply a consequence of formalizing set theory if first-order logic. If higher-order logic is used to formalize set theory, then there are no nonstandard models and the paradox disappears.". I is claimed that this is due to first order logic being to weak SO SAY SO IN THE MAIN 
ARTICLE
A you can see reader they dont want to admit that the Skolem paradox is a paradox You ask WHY? Well because the paradox  means set  theory as an authoritive set of axioms is destroyed. AND THEY CANT COPE WITH THAT  —Preceding unsigned comment added by Xzungg (talk • contribs) 06:31, 12 November 2008 (UTC)
replies: text: You can see that no mathematician considers it a paradox.  At least you can't quote one published after 1935.  — Arthur Rubin  (talk) 08:27, 12 November 2008 (UTC)

text: I reverted your latest edits Xzungg, it was not Arthur Rubin. I did it because they were blatant point of view pushing. Wiki is not here to judge the merits of arguments. It is here to publish notable things in a readable way complete with citations. Whether what you say is true or not really is a much more minor consideration on wiki. What you need is to have the work peer reviewed in some journal to stand up against the other peer reviewed sources here and for people to note it. Dmcq (talk) 11:11, 12 November 2008 (UTC)

text: You put a whole lot of stuff into the introduction and removed opposing views. You have put in your views elsewhere in the article before and that could be construed as good faith edits but you changed the sense of the introduction after a lot of discussion here which has pointed out problems with what you are saying. You have pushed your point of view in an uncivil manner. Dmcq (talk) 15:27, 12 November 2008 (UTC)
thread_title:  It is admitted that the Skolem paradox is a paradox if first order logic is used 

message: Xzungg: I have edited the article to explicitly say that the paradox is a result of using first order logic. I also added a slightly longer explanation about why contemporary mathematicians don't feel the result is problematical. And I added a sourced explanation from Fraenkel et al. about why the result was so confusing in the 1920s. And I added the Fraenkel quote from 1928 describing the result as an antinomy.  At this point, all three of the quotes you have been adding to the article are already included in the article. 
I think that one reason your changes are unsatisfactory to other editors is that you just paste in a long block of quotes that doesn't fit into the structure of the article.
I don't mind explaining once or twice on the talk page why it is that contemporary mathematicians don't find the result paradoxical. However, you have been responding by repeatedly producing the same quotes from the 1920s. These are not compelling to me, as I have explained. So, without using anybody else's words, can you explain briefly what contradiction in set theory you are actually worried about? That would help move the conversation forward. — Carl (CBM · talk) 14:20, 12 November 2008 (UTC)
replies: text: Indeed, this may appear like a contradiction when phrased in informal English, but it is not an actual contradiction when investigated further. The article already explains why it is not contradictory – because it is perfectly possible for the same set to be included in two models of set theory, to be uncountable from the perspective of the first model, and to be countable from the perspective of the second model. — Carl (CBM · talk) 16:32, 12 November 2008 (UTC)

text: There are numerous references in the article that verify that the following solution is indeed accepted, and does not "gut" set theory: it is possible for the same set to appear in two models of set theory, to be countable in the first model, and to be uncountable in the second model. The references verifying that include: Skolem, Kunen, Enderton, Burgess, Kleene, Hunter, and the book by Fraenkel et al.. That's 7 references, and I could find 7 more if I needed to. So I can repeat the question: why do you argue that this solution is not accepted? — Carl (CBM · talk) 03:15, 13 November 2008 (UTC)
replies: text: "it is possible for the same set to appear in two models of set theory, to be countable in the first model, and to be uncountable in the second model." How can this be possible if Cantor's definition of uncountability meant there was no way/model for an uncountable set to be counted? Cornelius (talk) 23:05, 28 October 2016 (UTC)
replies: text:   It would be better to ask this question at WP:RD/Math.  Talk pages are for discussing what should appear in the article, not the subject matter of the article per se. --Trovatore (talk) 23:13, 28 October 2016 (UTC)
replies: text: As I understand it, the Talk page is to talk about the article's content. Clearly anything related to the article relates to what maybe should appear in it. The fact that the person I quoted explains what I'm discussing also seems to suggest this.Cornelius (talk) 23:54, 28 October 2016 (UTC)
replies: text: No, that is not correct.  Please see WP:TALK.  Talk pages are exclusively to discuss what should appear in the article.  This rule gets bent from time to time (including by me, I admit), but that is not an excuse to break it openly.  If you will post your question to the math refdesk, as I suggested above, I will be happy to address it there. --Trovatore (talk) 00:51, 29 October 2016 (UTC)
replies: text: I really don't see what guidelines one can use to differentiate between the content produced at the Talk page of an article and the WP:RD section when the question is non-specific. I'm not asking "why/is such and such set countable and why?" which would truly belong to your section. Thanks for the suggestion, but I feel that my question best belongs here, especially since the question is very tuned to this specific discussion, is pretty pivotal to the article's content and here is more of a discussion rather than a Q&A. Feel free to disagree, but I'm not inclined to agree with you on this issue here. --Cornelius (talk) 02:41, 29 October 2016 (UTC)
replies: text:  If you wish to propose an actual change to the article, of course feel free to do so.  Your question as stated has no actionable content for the article. --Trovatore (talk) 02:58, 29 October 2016 (UTC)

text: We agree that, during the 1920s, mathematicians found Skolem's result troubling. However, they no longer find it troubling, because now we understand first-order logic much better. I have already read the quotes you keep repeating; they don't show what you think they show. They only show that at the time, people found the result troubling. I don't know what else to say here; you seem to be ignoring the temporal aspect here. — Carl (CBM · talk) 13:54, 13 November 2008 (UTC)
thread_title:  2008-11-12 

message: No contemporary mathematicians believe that Skolem's paradox is a real paradox. Xzungg, please stop adding that claim to the lede. I thought it had been left there by accident, only to find it had been inserted again. Several different people have commented on this talk page, all with the opinion that Skolem's paradox is not an actual paradox in set theory. — Carl (CBM · talk) 14:40, 12 November 2008 (UTC)
replies: text: The result is almost always called "Skolem's paradox" but this does not mean it represents an actual contradiction or antinomy. The term "paradox" is just used by convention. In the same way, the Banach-Tarski paradox is named a paradox even though it is not actually contradictory. When you say it is a "real paradox" that sounds to me like you are saying there is an actual mathematical contradiction in the result - even Skolem's original paper explained why there is no contradiction.
You say "The result is almost always called "Skolem's paradox" but this does not mean it represents an actual contradiction or antinomy" You only say that based on the unaccepted Skolem solution. At the time Abraham Fraenkel noted that Skolems relativism did not satisfactorly disprove the antinomy and that there was no agreement as to his relativist solution
"Neither have the books yet been closed on the antinomy, nor has agreement on its significance and possible solution yet been reached." – ([[Abraham Fraenkel] in "Einleitung in die Mengenlehre" 3rd ed p. 333, 1928, quoted in "The Bulletin of symbolic logic"" Vol.6, no 2. June 2000, pp. 147 http://www.math.ucla.edu/~asl/bsl/0602/0602-001.ps 
Dirk Van Dalen commenting on the above remarks of Fraenkel noted that most mathematician at the time accepted Faenkels views that the paradox was an antinomy and Skolems solution was unacceptable
The majority of mathematicians followed Faenkels scepticism http://www.math.ucla.edu/~asl/bsl/0602/0602-001.ps
 
Contemporary mathematicians only say it is not a paradox based on Skolems unaccepted solution
Dont say The Banach-Tarski paradox is not a paradox because it is avoided by only abandoning the naive concept of volume. You say the Banach-Tarski paradox is not a real paradox you are wrong.The Banach-Tarski paradox is a paradox if we stick to naive concept of volume.The paradox arises because of the axiom of choice. Mathematicians prefer to abandon the naive concept of volume than drop the axiom of choice.You would not say the Russell paradox is not a real paradox just because it is avoided [by abandoning naive set theory] in set theory due to the ad hocAxiom schema of specification  
3. Axiom schema of specification (also called the axiom schema of separation or of restricted comprehension): If z is a set, and \phi\! is any property which may characterize the elements x of z, then there is a subset y of z containing those x in z which satisfy the property. The "restriction" to z is necessary to avoid Russell's paradox and its variants
You have stated in the article that Russells paradox is a real paradox to avoid this paradox mathematician abandon naive set theory. So same must apply for The Banach-Tarski paradox it is a real paradox only avoided by abandoning the naive concept of volume.
So dont mislead readers by saying  The Banach-Tarski paradox  is not a paradox  —Preceding unsigned comment added by Xzungg (talk • contribs) 02:49, 13 November 2008 (UTC)
replies: text: Also, please avoid using terms like "rubbish" when responding to other editors. — Carl (CBM · talk) 16:30, 12 November 2008 (UTC)
replies: text: I wouldn't say it's a convention — one meaning of "paradox" is "apparent contradiction".  In this sense SP is a paradox:  It appears to be a contradiction, to many newcomers.  Once they learn a bit more about it it no longer appears that way. --Trovatore (talk) 19:00, 12 November 2008 (UTC)
replies: text: Sure, you're right, but there are lots of equally surprising results, that defy untrained intuition, that are not called "paradoxes". — Carl (CBM · talk) 03:11, 13 November 2008 (UTC)

text: Of course in relation to the Banach-Tarski paradox if we had a 'naive volume theory' which it conflicted with then there would have been a contradiction in that theory. Naive set theory did have to be amended as it led to a contradiction, nowadays set theory doesn't (we hope!) suffer from a problem like that. Are you saying maths does have an actual contradiction in one of its theories? As to cardinality not being absolute, personally I no more expect an axiomatic model to capture everything about my intuitions than I expect a Barbie doll should actually feel love for Ken, frankly I'm amazed they do so well. Dmcq (talk) 06:47, 13 November 2008 (UTC)
replies: text: This is a bit off-topic, but there's a lot of misunderstanding about this "naive set theory" thing.  The perception that Russell's paradox necessarily applies to unformalized set theory, and that comprehension had to be limited ad hoc to avoid it, is basically wrong.  Russell's paradox applies to set theory in which the intensional and extensional notions are conflated, as Frege definitely did; whether Cantor did it or not is a matter of debate.  The axiom of separation is quite a natural outgrowth of an informal view of set theory based on the idea that extensional sets are unordered lists of pre-existing objects chosen lawlessly, with this operation iterated into the transfinite.  Granted that this was not so clear at the time—the notion of the von Neumann universe was formulated later, and the picture is clear only in retrospect. --Trovatore (talk) 07:03, 13 November 2008 (UTC)
thread_title:  "real  paradox" 

message: In any countable model of set theory, the proposition that asserts the non-existence of an enumeration of some set is true if there is no enumeration within the model.  That is why there can be countable models without contradiction: no such "internal" enumeration exists, but "external" enumerations do exist. Michael Hardy (talk) 23:41, 13 November 2008 (UTC)
thread_title:  Internal vs. external countability 

message: From Wikipedia:Talk_page_guidelines: "Talk pages are for discussing the article, not for general conversation about the article's subject (much less other subjects). Keep discussions on the topic of how to improve the associated article. Irrelevant discussions are subject to removal." (my emphasis)  See also Wikipedia:NOT#FORUM: "... bear in mind that talk pages exist for the purpose of discussing how to improve articles; they are not mere general discussion pages about the subject of the article, nor are they a helpdesk for obtaining instructions or technical assistance. If you wish to ask a specific question on a topic, Wikipedia has a Reference Desk, and questions should be asked there rather than on talk pages."
Xzungg, if you have trouble understanding undergraduate mathematical logic, this is not the place to ask for help.  This is not the place to promote your own theories and research when they are clearly not suitable for inclusion into the article by Wikipedia policies.  Do so in your own web space.  Any further irrelevant discussion, whether by you or others, will be removed.  --C S (talk) 02:53, 15 November 2008 (UTC)
thread_title: Removal of irrelevant forum-like discussions

message: Trovatore wrote "I actually I like a lot of Joshtrimble's reformulation, but it's too chatty. Please discuss on talk page".
I like it too though not as a complete replacement. I think the article at the moment deals mainly from a mathematical perspective, perhaps he could contribute to giving it a philosophical bent and make it more accessible to a wider audience? Dmcq (talk) 11:51, 22 January 2009 (UTC)
replies: text: I liked it very much.  It's what I would expect from the Stanford Encyclopedia of Philosophy, from which it was lifted word for word.  --C S (talk) 12:12, 22 January 2009 (UTC)
replies: text: I thought it was a little too chatty. I don't mind reworking the lede to make it more accessible, though. The thing that I do want to avoid is going into too much depth in the lede. I thought that the main difficulty in the proposed version was the number of sentences phrased as questions. I think that rephrasing these things into declarative sentences makes the tone more encyclopedic. I'll see what I can do the the lede. — Carl (CBM · talk) 14:17, 22 January 2009 (UTC)
thread_title:  Joshtrimble's reformulation 

message: It may be helpful to include such a sentence in the page.  Tkuvho (talk) 19:53, 5 February 2011 (UTC)
thread_title:  first-order sentence asserting non-countability 

message: The first sentence of the second paragraph of the article, the sentence that says what the paradox is, uses (if I have understood it correctly) the word "countable" to mean "countably infinite". This encrypted use of language makes the article even harder to understand than it needs to be. Maproom (talk) 19:06, 5 March 2015 (UTC)
replies: text: Hmm?  How do you figure it's using it to mean "countably infinite"?  Is there something in the sentence that would be false in the finite case? --Trovatore (talk) 20:00, 5 March 2015 (UTC)
replies: text: So it does just mean countable?  Ok, I am still having trouble understanding the sentence, but that's my fault. Maproom (talk) 21:18, 5 March 2015 (UTC)
replies: text: Well, but if you're having trouble, other people might be as well.  Can you explain what's giving you trouble? --Trovatore (talk) 01:52, 6 March 2015 (UTC)
replies: text: It is clear from the debates above that other people have trouble understanding the article ;-)  . I suspect that part of my problem is my failure to grasp what "model" means. An actual example might help. Maproom (talk) 07:49, 6 March 2015 (UTC)
thread_title:  Countable