Talk:Skolem's paradox

Current status of the paradox, and work by Peter Suber

Another editor is occasionally adding text that seems to push the viewpoint that Skolem's paradox actually destroys the possibility of set-theoretic foundations. This is not at all the current viewpoint on the result among mathematical logicians. Two references are provided late in the article to more recent authors who comment on the paradox, and many more references could be found that express the same viewpoint: that there really is no paradox at all, and that the issue is simply the non-absoluteness of countability. It's true that this was not as well understood when Skolem first published, and the Zermelo apparently never accepted it. But the viewpoint from 1935 is not the contemporary viewpoint, and we shouldn't treat authors from the 1930s as if they represent the current consensus. And Zermelo's views are covered in detail in the article already.

What rubbish Skolem said his relativist solution destroyed set theory so did von Neuman just because contemporary mathematician dont say it is a paradox that dose not invalidate what earlier mathematician said was a paradox and as such you must present both accounts and stop distorting the facts just to save set theory from being invalidated. You say "But the viewpoint from 1935 is not the contemporary viewpoint" So what you must present both accounts and stop distorting the facts just to save set theory from being invalidated. —Preceding unsigned comment added by Xzungg (talk • contribs) 2008-11-09T14:38:27

A second point that is raised is the work of Peter Suber. I am unable to find a publication of Peter Suber that is explicitly about Skolem's paradox (list of publications). If Suber has indeed published something, then citing it would be worthwhile, since the article should at least mention contemporary work. But if Suber has not actually published about the paradox then we can't cite him. — Carl (CBM · talk) 14:00, 9 November 2008 (UTC)

I'm done editing the article for today, although I have asked other editors at the math project to look over the article and comment. — Carl (CBM · talk) 14:22, 9 November 2008 (UTC)

YOU DID NOT LOOK TO FAR DID YOU GO LOOK HERE [The Löwenheim-Skolem Theorem, http://www.earlham.edu/~peters/courses/logsys/low-skol.htm#amb3] there is ample argument supporting the fact that Skolems paradox is a paradox. And ample evidence about the unacceptablity of Skolems relativistic solution

WHAT YOU ARE ATTEMPTING TO DO IS HIDE ANY EVIDENCE THAT SHOWS SET THEORY IS INCONSISTENT DUE TO THE SKOLEM PARADOX —Preceding unsigned comment added by Xzungg (talk • contribs) 2008-11-09T14:38:27

Similarly, although Fraenkel said it was an antinomy in 1928, no contemporary texts support that viewpoint. So Fraenkel's thoughts are useful for historical study, but not useful for investigating the current status of the paradox. — Carl (CBM · talk) 14:11, 9 November 2008 (UTC)

So what if no contemporary texts support Fraenkel viewpoint you must present both accounts and stop distorting the facts just to save set theory from being invalidated —Preceding unsigned comment added by Xzungg (talk • contribs) 2008-11-09T14:38:27

A few replies:

• This web page by Peter Suber [1] is not a published paper, it is a handout for a lecture he gave in class. I have not found any published, peer-reviewed work by Suber on Skolem's paradox.
• The viewpoints of Skolem, Fraenkel, and von Neumann from the 1920s and 1930s are mainly of interest as historical data, rather than as contemporary viewpoints on the status of set theory. Many of these comments were made before Goedel's completeness and incompleteness theorems had clarified the way the first-order logic behaves.
• I do agree the article should present the viewpoints of Skolem, Fraenkel, and von Neumann. But it should do so in historical context, rather than inaccurately presenting these views as if they represent contemporary thought. This is no different than an article on medicine, which would not present theories from the 1800s on par with contemporary theories of disease. — Carl (CBM · talk) 14:48, 9 November 2008 (UTC)

You SAY "Many of these comments were made before Goedel's completeness and incompleteness theorems had clarified the way the first-order logic behaves." This is useless argument as the Australian philosopher Colin Leslie Dean has shown Godels incompleteness theorem to be invalid for a number of reasons: he uses in his proof the invalid axiom of reducility; he constructs impredicative statements-which text books on logic say are invalid; he has no idea of what truth is therefore his formal logic cannot be converted into English sentences and remains no more than meaningless symbols —Preceding unsigned comment added by Xzungg (talk • contribs) 12:48, 10 November 2008 (UTC)

I don't understand the issue with the Suber page. It seems to agree with CBM: "Most mathematicians agree that the Skolem paradox creates no contradiction." It largely deals with Skolem's paradox as a historical issue, not a current issue.

Personally, I find it hard to even understand why it was ever considered a paradox. It seems perfectly ordinary to me.

CRGreathouse (t | c) 02:58, 10 November 2008 (UTC)

You say suber notes ""Most mathematicians agree that the Skolem paradox creates no contradiction" but you left out the bit where he says "But that does not mean they agree on how to resolve it"

and further

Suber states "It is at least a paradox in the ancient sense: an astonishing and implausible result."

and

One reading of LST holds that it proves that the cardinality of the real numbers is the same as the cardinality of the rationals, namely, countable. (The two kinds of number could still differ in other ways, just as the naturals and rationals do despite their equal cardinality.) On this reading, the Skolem paradox would create a serious contradiction, for we have Cantor's proof, whose premises and reasoning are at least as strong as those for LST, that the set of reals has a greater cardinality than the set of rationals. and further

further

"The bad news is that the obvious alternatives are very ugly. The most common way to avoid the strongly paradoxical reading is to insist that the real numbers have some elusive, essential property not captured by system S. This view is usually associated with a Platonism that permits its proponents to say that the real numbers have certain properties independently of what we are able to say or prove about them. "

further regarding Skolems relatvist solution

"A widely held interpretation is that of Thoralf Skolem himself. He believed that LST showed a relativity in some of the fundamental concepts of set theory. Uncountable cardinalities in particular have no meaning apart from specific sets of axioms. A set may be uncountable within a certain formal system and countable when viewed from the standpoint of an unintended model of that system. This means that there simply are no sets whose cardinality is absolutely uncountable. For many, this view guts set theory, arithmetic, and analysis. It is also clearly incompatible with mathematical Platonism which holds that the real numbers exist, and are really uncountable, independently of what can be proved about them"

Dont now say Suber is not published so we can avoid taking note of him. If you cant refute his claims then it stands that the Skolem paradox is a paradox

I think the main reason it was considered paradoxical is that it was discovered in 1922, but Goedel's completeness theorem and the compactness theorem were not proved until 1929. And there was a lot of confusion about the difference between first-order logic and second-order logic for a long time (Henkin's paper on the semantics of higher-order logic was published in 1950!). I think Skolem's paradox is roughly comparable to the Banach-Tarski paradox, which doesn't seem paradoxical at all in the light of measure theory but does seem odd if stated in an informal manner. — Carl (CBM · talk) 03:21, 10 November 2008 (UTC)

You say And there was a lot of confusion about the difference between first-order logic and second-order logic for a long time. But as Suber points out "LST holds only in first-order theories. Higher-order logics are not afflicted with it"

No one here is arguing against Suber; his summary, when I looked at it a few months ago, seemed well done and even-handed. To be honest I'm less than convinced that you actually understand what he's saying. For example, did you understand that the line about "for many, this guts set theory, arithmetic, and analysis" is an argument against Skolem-style relativism and in favor of Platonism/realism? And that the fact that LST is limited to first-order theories means that the Skolem paradox is not paradoxical unless you expect that first-order logic should be able to capture all mathematical meaning, an expectation that has no warrant from the realist standpoint? --Trovatore (talk) 09:25, 10 November 2008 (UTC)

you say "and that the fact that LST is limited to first-order theories means that the Skolem paradox is not paradoxical...."

But as Suber notes the paradox holds in first order theories ie "LST holds only in first-order theories" and on one reading of LST there is a contradiction

"One reading of LST holds that it proves that the cardinality of the real numbers is the same as the cardinality of the rationals, namely, countable. (The two kinds of number could still differ in other ways, just as the naturals and rationals do despite their equal cardinality.) On this reading, the Skolem paradox would create a serious contradiction, for we have Cantor's proof, whose premises and reasoning are at least as strong as those for LST, that the set of reals has a greater cardinality than the set of rationals. "

All in all there is able evidence to state -in the main article -that the paradox holds in regard to certain readings, and that there is contemporary support for those mathematician -you insultingly call only of historical interest- who argued the Skolem paradox is a paradox what . To leave this out is to falsify the evidence —Preceding unsigned comment added by Xzungg (talk • contribs) 12:33, 10 November 2008 (UTC)

Actually, there isn't such evidence. That passage from Suber's lecture notes doesn't make particular sense to me - I don't know of any reading of the Skolem paradox which can be used to prove that the actual real numbers are a countable set, since countable models of set theory are not the standard model. Suber makes no effort to explain either how this supposed argument goes or where it can be found in the literature (and he doesn't have to, in a set of class notes).

This is one reason why we don't use unpublished class notes as references, because they often have poorly-worded passages that are OK for class notes but not suitable for scholarly work. — Carl (CBM · talk) 13:04, 10 November 2008 (UTC)

Hey your ignorance is your problem since when is your lack of knowledge a reason for excluding evidence Suber is clear "One reading of LST holds that it proves that the cardinality of the real numbers is the same as the cardinality of the rationals, namely, countable. (The two kinds of number could still differ in other ways, just as the naturals and rationals do despite their equal cardinality.) On this reading, the Skolem paradox would create a serious contradiction,"On this reading, the Skolem paradox would create a serious contradiction, for we have Cantor's proof, whose premises and reasoning are at least as strong as those for LST, that the set of reals has a greater cardinality than the set of rationals. Either prove him wrong or put these arguments in the main article —Preceding unsigned comment added by Xzungg (talk • contribs) 2008-11-10T13:47:18

Has the work by Suber been peer reviewed? If not then it cannot stand against the other peer reviewed work except on notability grounds. Is it notable? I don't think so from the evidence I've seen. Is he even possibly right? (and this is not a ground on which wikipedia includes things) I think his reasoning is muddled. Just consider for instance the sentence "Most mathematicians agree that the Skolem paradox creates no contradiction. But that does not mean they agree on how to resolve it". What exactly is 'it'? the lack of a paradox or Suber's perception of the existence of a paradox? Dmcq (talk) 14:06, 10 November 2008 (UTC)

You say What exactly is 'it'? Cant you understand English -it refers to the Skolem paradox Even B Bunch states " no one has any idea how to reconstruct axiomatic set theory so that this [Skolem] paradox does not occur" (B Bunch Mathematical fallacies and paradoxes, dover, 1982, p.167"

Re Dmcq: No, it isn't reviewed, it's just a class handout Suber made to go along with his lecture. And yes, his writing is muddled in several areas there, although to be fair he only wrote a class handout, not a research paper.

Re Xzungg: if you feel Suber's writing is clear, can you explain exactly what reading of the LST makes one think that the cardinality of the reals is the same as the cardinality of the rationals? This is what Suber is claiming, but it looks to me like he just chose his words poorly there. And, in any case, a set of unpublished class notes is not a reference that we can use on Wikipedia. The reason for that is that many people make poorly-worded statements in class notes, which would be caught by the peer review process if the paper were actually published. — Carl (CBM · talk) 14:18, 10 November 2008 (UTC)

You only say it is not a paradox based on Skolems relativistic solution but this is not accepted by many mathematician You only say it is not a paradox based on Skolems relativist solution but as Suber points out For many, this view guts set theory, arithmetic, and analysis. Either show us why it is not a paradox with out relying on Skolems solution or put Subers arguments in the main article. Even B Bunch states " no one has any idea how to reconstruct axiomatic set theory so that this [Skolem] paradox does not occur" (B Bunch Mathematical fallacies and paradoxes, dover, 1982, p.167"

No, Carl is not saying that. --Trovatore (talk) 05:52, 11 November 2008 (UTC)

You say "No, Carl is not saying that". Bunch says " no one has any idea how to reconstruct axiomatic set theory so that this [Skolem] paradox does not occur" (B Bunch Mathematical fallacies and paradoxes, dover, 1982, p.167" } and Suber says ""Most mathematicians agree that the Skolem paradox creates no contradiction But that does not mean they agree on how to resolve it [Skolem paradox] "Then why is it not a paradox and dont bring up Skolems unaccepted solution. IF YOU CANT DO THAT THEN THE PARADOX IS A PARADOX SO SAY SO IN THE MAIN ARTICLE —Preceding unsigned comment added by Xzungg (talk • contribs) 06:50, 11 November 2008 (UTC)

You are confusing two meanings of paradox

• statements leading to a contradiction
• statements leading to an apparent contradiction

When Suber says "Most mathematicians agree that the Skolem paradox creates no contradiction. But that does not mean they agree on how to resolve it". I read it as saying mathematicians agree there is no contradiction, however they have philosophical arguments about it. So the meaning of "it" is the Skolem paradox as an apparent rather than real contradiction. It is unfortunate that he seems to have got carried away skirting around the 'contradiction' word and confusing students. Talking about 'very very serious ambiguity' and suchlike without coming right out and saying he is talking about confusion and philosophy rather than a contradiction is ingenuous I feel. He has shown no contradiction. He has demonstrated no inconsistency in set theory.

Of course there are philosophical arguments about areas like this, just look at constructivism (mathematics) or Godel or more recently Gregory Chaitin. The Skolem paadox is not a paradox in the sense of Richard's paradox which is just wrong. It is a paradox in the sense like that the number of numbers is the same as the number of square numbers even though one seems much smaller than the other. That's another one that people argued over in the past. And frankly I don't think there is much argument about the Skolem paradox nowadays, it just seems common sense now people see how computers work and it is amazing that second order logic can go so much further.

As to the handout by Peter Suber on this, there is no evidence it is notable. As I said before wiki is not about the truth, it is about notability. If there was some contradiction shown I believe it would become notable fairly quickly. Nobody has any interest in suppressing anything like that, mathematics is not a religion, and a contradiction in set theory would provide loads of work for theses which I assure you is something everyone wants. Dmcq (talk) 10:56, 11 November 2008 (UTC)

you say "Most mathematicians agree that the Skolem paradox creates no contradiction. But that does not mean they agree on how to resolve it". I read it as saying mathematicians agree there is no contradiction, however they have philosophical arguments about it. So the meaning of "it" is the Skolem paradox as an apparent rather than real contradiction." I say rubbish Bunch says it is a paradox which no one knows how to solve. And Suber says it is a paradox-"Insofar as this is a paradox it is called Skolem's paradox. It is at least a paradox in the ancient sense: an astonishing and implausible result Suber dose mean a contradiction as he says in your disputed sentence ie "On this reading, the Skolem paradox would create a serious contradiction". So here we have two people saying it is a paradox

NOW

I ASKED YOU WHICH YOU HAVE AVOIDED ANSWERING Then why is it not a paradox and dont bring up Skolems unaccepted solution. IF YOU CANT DO THAT THEN THE PARADOX IS A PARADOX SO SAY SO IN THE MAIN Please answer or stand convicted of not having and answer and falsifing the record —Preceding unsigned comment added by Xzungg (talk • contribs) 11:10, 11 November 2008 (UTC)

To repeat you say it is not a paradox -thus agreeing with Subers observation ""Most mathematicians agree that the Skolem paradox creates no contradiction" So give us your solution. If you cant give a solution then Bunch is correct about no one knowing how to solve it and Suber is correct when he says "But that does not mean they [mathematician] agree on how to resolve it [Skolem paradox]" —Preceding unsigned comment added by Xzungg (talk • contribs) 11:25, 11 November 2008 (UTC)

You should not forget Wikipedia policies on using good secodary sources (review articles and such) to back such claims as what is or is not "commonly accepted" 62.142.58.210 (talk) 11:43, 16 February 2013 (UTC)

Lying that the Skolem paradox is not a paradox

Just give us a non Skolem solution to the paradox or stand convicted of falsifing the record

It is admmitted in this disccusion that "it's 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." As Suber pointed out "LST holds only in first-order theories " You have admitted that the paradox is a a consequence of formalizing set theory if first-order logic." but not second order . So say so in the main article

I have asked twice and no one will answer To repeat you say it is not a paradox -thus agreeing with Subers observation ""Most [ note not all] mathematicians agree that the Skolem paradox creates no contradiction" So give us your solution. If you cant give a solution then Bunch is correct about no one knowing how to solve it and Suber is correct when he says "But that does not mean they [mathematician] agree on how to resolve it [Skolem paradox]" IF YOU CANT DO THAT THEN THE PARADOX IS A PARADOX SO SAY SO IN THE MAIN Please answer or stand convicted of not having and answer and falsifing the record As I said give us a solution and not some reworking of Skolems relativist solution which as Suber notes "For many, this view guts set theory, arithmetic, and analysis" of which Abraham Fraenkel, John von Neumann and Skolem himself noted and

"I believed that it was so clear that axiomatization in terms of sets was not a satisfactory ultimate foundation of mathematics that mathematicians would, for the most part, not be very much concerned with it. But in recent times I have seen to my surprise that so many mathematicians think that these axioms of set theory provide the ideal foundation for mathematics; therefore it seemed to me that the time had come for a critique." – ([[Skolem]"The Bulletin of symbolic logic" Vol.6, no 2. June 2000, pp. 147 http://www.math.ucla.edu/~asl/bsl/0602/0602-001.ps.

Even John von Neumann noted that Skolems relativism was one more reason to upset set theory and destroy it

"At present we can do no more than note that we have one more reason here to entertain reservations about set theory and that for the time being no way of rehabilitating this theory is known." – ([[John von Neumann]"The Bulletin of symbolic logic" Vol.6, no 2. June 2000, pp. 148 http://www.math.ucla.edu/~asl/bsl/0602/0602-001.ps.

Abraham Fraenkel noted that Skolems relativism did not satisfactoraly 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

So just give a non Skolem solution to the paradox or stand convicted of falsifing the record —Preceding unsigned comment added by Xzungg (talk • contribs) 12:00, 11 November 2008 (UTC)

If there is no actual contradiction, as essentially all mathematical logicians agree, then there is no need for a "solution". How can someone solve a problem that doesn't exist?

You say "If there is no actual contradiction, as essentially all mathematical logicians agree, then there is no need for a "solution". How can someone solve a problem that doesn't exist?" What rubbish Skolem thought there was a problem that is why he did his unacceptable solution. John von Neumann Abraham Fraenkel saw there was a problem to. You cant not just say mathematicians say there is no paradox with out presenting their evidence As I have presented evidence from mathematician who say it is a paradox. Dont now say but contemporary mathematician sai it is no paradox with out giving us their evidence —Preceding unsigned comment added by Xzungg (talk • contribs) 01:56, 12 November 2008 (UTC)

It is true that there are philosophical issues that are illustrated by Skolem's result. But the mathematical community has found that the result is not actually a mathematical contradiction, it's 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.

You say "it's 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." As Suber pointed out "LST holds only in first-order theories " You have admitted that the paradox is a a consequence of formalizing set theory if first-order logic." but not second order —Preceding unsigned comment added by Xzungg (talk • contribs) 02:08, 12 November 2008 (UTC)

You seem to be focusing on comments that were made immediately after Skolem's paper. In doing so, you would ignore the enormous progress made in the decades after Skolem published in 1922. The article should focus on the contemporary understanding of the paradox, and include the historical viewpoints as history rather than modern theory. As I said, it's like an article on disease discussing the views of scientists before the germ theory was developed. — Carl (CBM · talk) 14:05, 11 November 2008 (UTC)

I fully agree with CBM. Please read up on notability and find citations showing Suber's notes satisfy that fundamental wiki criterion. If it does then it can be included whatever people think about it. That really is all it takes. For anyone here to verify that what is written in those notes means anything would be original research and wiki frowns on that, it would have to be done by a source that can be cited. You don't have to convince anyone it is true, only that it is notable and that you can produce citations. Dmcq (talk) 14:17, 11 November 2008 (UTC)

My concern is not so much with "notability", but with undue weight. The Wikipedia policy is to present all significant contemporary views, but not to emphasize views of a small minority as if they were commonly held. In this particular situation, every mathematical logic text I have ever seen from the last 50 years says that Skolem's paradox is not actually a contradiction in set theory.

However I am becoming more convinced of the need to add a section on "Philosophical interpretations", which I could source to published contemporary texts. Although mathematicians are no longer particularly interested in the paradox, it is a useful example for philosophers. — Carl (CBM · talk) 14:31, 11 November 2008 (UTC)

Separating the maths and the philosophy sounds like a very good idea. I vote for it. In fact a section devoted to the maths would be good! There's hardly any math at the moment, it is mostly mixed history and philosophy. Dmcq (talk) 18:24, 11 November 2008 (UTC)

So you'd like us to add another section that explains the mathematical resolution of the paradox in more detail? — Carl (CBM · talk) 22:10, 11 November 2008 (UTC)

Actually I've had another read and come to the conclusion the details included are the right ones and the wiki links are good on the mathematical side. I guess you're talking more about extending the bit about Putnam's argument to discussions in general like this. I have no idea how he'd think it discredits realism and it looks from how Xzungg goes on that others must be having ideas about philosophical implications too. Dmcq (talk) 00:20, 12 November 2008 (UTC)

It is admmitted in this disccusion that "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." As Suber pointed out "LST holds only in first-order theories " You have admitted that the paradox is a a consequence of formalizing set theory if first-order logic. SO SAY SO IN THE MAIN ARTICLE

Another way of telling the paradox, using natural numbers

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)

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)

The fact is mathematicians cant prove that the Skolem paradox is not a paradox

As any reader to this page will see I have asked people to prove that the Skolem paradox is not a paradox without recourse to the non accepted Skolem solution. All they say is mathematicians say it is not a paradox but they dont show us the proof of this. It seems people here just want us to accept the word of mathematician with out the presentation of any proof to support their claim. THE FACT IS THEY HAVE NO PROOF THAT THE SKOLEM PARADOX IS NOT A PARADOX.

THUS This most of this article is a pack of lies and an attempt to keep from the publick a fact that the Skolem paradox is a real paradox —Preceding unsigned comment added by Xzungg (talk • contribs) 14:49, 11 November 2008 (UTC)

Why do you say that Skolem's solution (that is, that countability is not an absolute property) is not accepted? In fact, it is accepted by all contemporary mathematical logic texts. Here is a direct quote from Foundations of Set Theory, 1973, by Abraham Adolf Fraenkel, Yehoshua Bar-Hillel, Azriel Levy, and Dirk van Dalen, p. 303:

"It is easy to see that Skolem's paradox is no paradox at all. When we say that a is uncountable in A we mean that there is no one-one function f in A which maps a on ω but this does not deny the existence of a one-one function mapping a on ω which does not belong to A." (emphasis in the original)

Yes, that is the same Abraham Fraenkel mentioned in this article. I copied the quote from google books; I'm planning to take this book from the library this afternoon because it has other commentary I can use to improve the article. — Carl (CBM · talk) 15:01, 11 November 2008 (UTC)

You quote Abraham Fraenkel so do I Abraham Fraenkel is on record for saying it is an antinomy and that Skloems solution is not accepted- it is interesting that you want bring this point up in the main article

"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

you say "Why do you say that Skolem's solution (that is, that countability is not an absolute property) is not accepted? In fact, it is accepted by all contemporary mathematical logic texts.I say it is not accepted by all mathematicians AS SUBER POINTS OUT IE "For many, this view guts set theory, arithmetic, and analysis " And Skolem Abraham Fraenkel and John von Neumann said it destroyed set theory

Skolem himself noted and

"I believed that it was so clear that axiomatization in terms of sets was not a satisfactory ultimate foundation of mathematics that mathematicians would, for the most part, not be very much concerned with it. But in recent times I have seen to my surprise that so many mathematicians think that these axioms of set theory provide the ideal foundation for mathematics; therefore it seemed to me that the time had come for a critique." – ([[Skolem]"The Bulletin of symbolic logic" Vol.6, no 2. June 2000, pp. 147 http://www.math.ucla.edu/~asl/bsl/0602/0602-001.ps.

Even John von Neumann noted that Skolems relativism was one more reason to upset set theory and destroy it

"At present we can do no more than note that we have one more reason here to entertain reservations about set theory and that for the time being no way of rehabilitating this theory is known." – ([[John von Neumann]"The Bulletin of symbolic logic" Vol.6, no 2. June 2000, pp. 148 http://www.math.ucla.edu/~asl/bsl/0602/0602-001.ps.

Abraham Fraenkel noted that Skolems relativism did not satisfactoraly 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

Here is another quote, from van Heijenoort's introduction to Skolem's 1922 paper. This is van Heijenoort speaking:

"For Skolem the discrepancy between an intuitive set-theoretic notion and its formal counterpart leads to the “relativity” of set-theoretic notions. ... The existence of such a “relativity” is sometimes referred to as the Lowenheim-Skolem paradox. But, of course, it is not a paradox in the sense of antinomy; it is a novel and unexpected feature of formal systems." (van Heijenoort 1967, p. 290)

— Carl (CBM · talk) 16:03, 11 November 2008 (UTC)

van Heijenoort opinion is in contradiction with Bunch Suber and Skolem himself who all say it is a paradox

Hmm, is that really Skolem's take on it? That I'm not sure about. I thought Skolem's view was that there is no such thing as "real" uncountability, but only uncountability relative to a model. Xzungg is right that this latter view is seriously problematic for analysis. What he's wrong about is that LST implies any such thing. --Trovatore (talk) 18:29, 11 November 2008 (UTC)

Mathematician only say its not a paradox based on Skolems relativist solution but this solution asAbraham Fraenkel and John von Neumann Suber points out "For many, this view guts set theory, arithmetic, and analysis"

You say "If there is no actual contradiction, as essentially all mathematical logicians agree, then there is no need for a "solution". How can someone solve a problem that doesn't exist?" What rubbish Skolem thought there was a problem that is why he did his unacceptable solution. John von Neumann Abraham Fraenkel saw there was a problem to. You cant not just say mathematicians say there is no paradox with out presenting their evidence As I have presented evidence from mathematician who say it is a paradox. Dont now say but contemporary mathematician say it is no paradox with out giving us their evidence

No, you're wrong about that. The resolution is not that there is only uncountability relative to a model and no real uncountability. The resolution is rather that there is uncountability relative to a model, and there is real uncountability, and they are not necessarily the same. This is not the fault of set theory. It's a limitation on the expressive power of first-order logic.

By the way, kindly do not change the emphasis in others' comments. --Trovatore (talk) 01:55, 12 November 2008 (UTC)

GIVE US THE SOLUTION TO THE SKOLEM PARADOX THE ONE MATHEMATICIAN BASE THEIR CLIAM ON THAT IT IS NOT A PARADOX You say "If there is no actual contradiction, as essentially all mathematical logicians agree, then there is no need for a "solution". How can someone solve a problem that doesn't exist?" What rubbish Skolem thought there was a problem that is why he did his unacceptable solution. John von Neumann Abraham Fraenkel saw there was a problem to. You cant not just say mathematicians say there is no paradox with out presenting their evidence As I have presented evidence from mathematician who say it is a paradox. Dont now say but contemporary mathematician saY it is no paradox with out giving us their evidence IE SOLUTION THEY BASE THAT CLAIM ON

It is admitted in this disccusion that "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." As Suber pointed out "LST holds only in first-order theories " You have admitted that the paradox is a a consequence of formalizing set theory if first-order logic. SO SAY SO IN THE MAIN ARTICLE

You're still missing the point. It is not that formalizing set theory into first-order logic creates a paradox -- that makes it sound as though first-order logic is too strong, whereas what is actually going on here is that first-order logic is too weak to capture all of the intended meaning of the notion of "uncountable". The only way to get a paradox here (in the sense of something contrary to intuition) is if you have an intuition that first-order logic ought to be able to capture all of that semantic content. But that intuition is just simply wrong, and the resolution is to get a better intuition, after which there is no more paradox. --Trovatore (talk) 02:52, 12 November 2008 (UTC)

You say "It is not that formalizing set theory into first-order logic creates a paradox -- that makes it sound as though first-order logic is too strong, whereas what is actually going on here is that first-order logic is too weak to capture all of the intended meaning of the notion of "uncountable"." Well that is just to bad as you have admitted the Skolem paradox is a consequence of first order logic -to bad if that is due to first order logic being weak. You state "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.". SO SAY SO IN THE MAIN ARTICLE —Preceding unsigned comment added by Xzungg (talk • contribs) 06:20, 12 November 2008 (UTC)

It is admitted that the Skolem paradox is a paradox if first order logic is used

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)

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)

You say "You can see that no mathematician considers it a paradox. At least you can't quote one published after 1935" You obviously dont read this disccussion I quote Bunch Bunch says " no one has any idea how to reconstruct axiomatic set theory so that this [Skolem] paradox does not occur" (B Bunch Mathematical fallacies and paradoxes, dover, 1982, p.167" } i HAVE EVEN QUOTED YOUR OWN EDITOR WHO ADMITS IT IS A PARADOX IF USEING FIRST ORDER LOGIC ie'[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.".

You deleted my entry in the main article I even quoted your own editor who AMITTED THE SKOLEM PARADOX IS A PARADOX IE '[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.". READERS NOTE THAT THE EDITORS OF THIS ARTICLE ARE TRYING TO FALSIFy THE FACTS —Preceding unsigned comment added by Xzungg (talk • contribs) 09:58, 12 November 2008 (UTC)

Readers note 2 things - they cant proove it is not a paradox, they even admit it is a paradox if first order logic is used 1) mathematicians cant give a proof that the Skolem paradox is not a paradox. When they say it is not a paradox they are basing that on an unaccepted proof by Skolem-Who himself admitted destroyed set theory

Skolem states "I believed that it was so clear that axiomatization in terms of sets was not a satisfactory ultimate foundation of mathematics that mathematicians would, for the most part, not be very much concerned with it. But in recent times I have seen to my surprise that so many mathematicians think that these axioms of set theory provide the ideal foundation for mathematics; therefore it seemed to me that the time had come for a critique." – ([[Skolem]"The Bulletin of symbolic logic" Vol.6, no 2. June 2000, pp. 147 http://www.math.ucla.edu/~asl/bsl/0602/0602-001.ps.

Even John von Neumann noted that Skolems relativism was one more reason to upset set theory and destroy it

"At present we can do no more than note that we have one more reason here to entertain reservations about set theory and that for the time being no way of rehabilitating this theory is known." – ([[John von Neumann]"The Bulletin of symbolic logic" Vol.6, no 2. June 2000, pp. 148 http://www.math.ucla.edu/~asl/bsl/0602/0602-001.ps.

Abraham Fraenkel noted that Skolems relativism did not satisfactoraly 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

2) they even admit it is a paradox if first order logic is used. From their own editors mouth '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." —Preceding unsigned comment added by Xzungg (talk • contribs) 10:05, 12 November 2008 (UTC)

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)

You say "I reverted your latest edits Xzungg, it was not Arthur Rubin. I did it because they were blatant point of view pushing." I quoted one of your own editors who admitted the Skolem paradox is a paradox if first order logic is used-are you saying he is "point of view pushing"

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.

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)

2008-11-12

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)

You ask "can you explain briefly what contradiction in set theory you are actually worried about" I say we have a contradiction, because Skolem proved that unaccountable sets cannot be uniquely defined in set theory but the fact is they can be. In other words Skolem proved every countable axiomatisation of set theory in first-order logic, if consistent, has a model that is countable [A],but it can be proved the existence of sets that are not countable [~A]. THUS WE HAVE a contradiction A~A To paraphrase the main entry Löwenheim (1915) and Skolem (1920, 1923) proved the Löwenheim-Skolem theorem. The downward form of this theorem shows that if a countable first-order axiomatisation is satisfied by any infinite structure [A], then the same axioms are satisfied by some countable structure [~A]. THUS A CONTRADICTION A~A —Preceding unsigned comment added by Xzungg (talk • contribs) 15:23, 12 November 2008 (UTC)

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)

I have pointed out time and time again Skolems solution is not accepted as it destroys set theory . You said "Skolem went on to explain why there was no contradiction." Without telling the readers that his solution is not accepted as it guts sets theory

Skolem states "I believed that it was so clear that axiomatization in terms of sets was not a satisfactory ultimate foundation of mathematics that mathematicians would, for the most part, not be very much concerned with it. But in recent times I have seen to my surprise that so many mathematicians think that these axioms of set theory provide the ideal foundation for mathematics; therefore it seemed to me that the time had come for a critique." – ([[Skolem]"The Bulletin of symbolic logic" Vol.6, no 2. June 2000, pp. 147 http://www.math.ucla.edu/~asl/bsl/0602/0602-001.ps.

Even John von Neumann noted that Skolems relativism was one more reason to upset set theory and destroy it

"At present we can do no more than note that we have one more reason here to entertain reservations about set theory and that for the time being no way of rehabilitating this theory is known." – ([[John von Neumann]"The Bulletin of symbolic logic" Vol.6, no 2. June 2000, pp. 148 http://www.math.ucla.edu/~asl/bsl/0602/0602-001.ps.

Abraham Fraenkel noted that Skolems relativism did not satisfactoraly 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 —Preceding unsigned comment added by Xzungg (talk • contribs) 02:19, 13 November 2008 (UTC)

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)

"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)

@Renassault:: 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)

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)

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)

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)

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)

You say ":There are numerous references in the article that verify that the following solution is indeed accepted" and site Skolem as one-YOU CONVIENTIENTLY LEFT OUT THAT HE EVEN SKOLEM SAID HIS SOLUTION DESTROYED SET THEORY

I have found a reference that says most mathematicians at the time of Skolems solution agreed it was unacceptable

Faenkels states

"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

Even John von Neumann noted that Skolems relativism was one more reason to upset set theory and destroy it

"At present we can do no more than note that we have one more reason here to entertain reservations about set theory and that for the time being no way of rehabilitating this theory is known." – ([[John von Neumann]"The Bulletin of symbolic logic" Vol.6, no 2. June 2000, pp. 148 http://www.math.ucla.edu/~asl/bsl/0602/0602-001.ps

And Skolem himself

"I believed that it was so clear that axiomatization in terms of sets was not a satisfactory ultimate foundation of mathematics that mathematicians would, for the most part, not be very much concerned with it. But in recent times I have seen to my surprise that so many mathematicians think that these axioms of set theory provide the ideal foundation for mathematics; therefore it seemed to me that the time had come for a critique." – ([[Skolem]"The Bulletin of symbolic logic" Vol.6, no 2. June 2000, pp. 147 http://www.math.ucla.edu/~asl/bsl/0602/0602-001.ps.

So I have found FOUR references that say Skolems solution is unacceptable because it destroyed set theory. Following Van Dalen comments if you looked you would find more who said it was unacceptable because it destroyed set theory. If we take Bunch and Subers comments then I have found SIX references that say the Skolem solution is not accepted Suber states

For many, this view guts set theory, arithmetic, and analysis

And B Bunch makes it clear Skolems solution is not accepted

" no one has any idea[ MEANING SKOLEM TO] how to reconstruct axiomatic set theory so that this [Skolem] paradox does not occur" (B Bunch Mathematical fallacies and paradoxes, dover, 1982, p.167"

—Preceding unsigned comment added by Xzungg (talk • contribs)

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)

"real paradox"

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)

Rubbish your own editors has said it is a paradox if first order logic is used

"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.

And Buch has called it a paradox

" no one has any idea how to reconstruct axiomatic set theory so that this [Skolem] paradox does not occur" (B Bunch Mathematical fallacies and paradoxes, dover, 1982, p.167"

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)

Also, please avoid using terms like "rubbish" when responding to other editors. — Carl (CBM · talk) 16:30, 12 November 2008 (UTC)

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)

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)

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)

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)

When IS countability absolute?

I suppose this is a little off-topic, but is there a theorem of the form

For any model M of set theory, there is a submodel N of bounded cardinality such that:

1. Every M-countable element of N is N-countable, or, more generally,
2. Every M-countable M-subset of N is in N?

— Arthur Rubin (talk) 15:42, 12 November 2008 (UTC)

Sure. For example N could just equal M. Do you want N to be an element of M or something? Maybe you want to require M to be transitive? Not sure exactly what you're getting at.

One way of putting it might be to dispense with M and just ask what condition you can put on a transitive ε-model N to guarantee that for every set x in N, if x is really countable, then N satisfies that x is countable. For that it's sufficient for the true $\aleph _{1}$ to be an element of N, or that N contain all the reals. Whoops--no it isn't Ah, but here's what is true: If N contains all the reals, then if A is a set of reals that is really countable, and A is in N, then N satisfies that A is countable. --Trovatore (talk) 18:57, 12 November 2008 (UTC)

You missed "bounded cardinality", such as $\beth _{1}$ or $\beth _{2}$ (assuming the meta-theory includes the axiom of choice, anyway.) And I think I wanted to say elementary submodel and transitive submodel. It's a version of the downward Löwenheim-Skolem theorem with countability preserved. — Arthur Rubin (talk) 19:28, 12 November 2008 (UTC)

You want N to have that cardinality in the sense of M, or in reality? --Trovatore (talk) 20:07, 12 November 2008 (UTC)

Reality — whatever that means. — Arthur Rubin (talk) 01:21, 13 November 2008 (UTC)

Skolem paradox means set theory is inconsistent as well as maths being in contradiction

It was asked "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"

There is both a 1)contradiction in mathematics and 2) set theory being inconsistent

In the main article it is stated

Skolem (1922) pointed out the seeming contradiction between the Löwenheim-Skolem theorem on the one hand, which implies that there is a countable model of Zermelo's axioms, and Cantor's theorem on the other hand, which states that uncountable sets exist, and which is provable from Zermelo's axioms. "So far as I know," Skolem writes, "no one has called attention to this paradoxical state of affairs. By virtue of the axioms we can prove the existence of higher cardinalities... How can it be, then, that the entire domain B [a countable model of Zermelo's axioms] can already be enumerated by means of finite positive integers?" (Skolem 1922, p. 295, translation by Bauer-Mengelberg)

Thus we have 1)Cantors theorem states that uncountable sets exist 2)set theory can prove that uncountable sets exist 3) The downward Löwenheim-Skolem theorem. proves that there is a countable model of set theory. Thus there is a contradiction between The downward Löwenheim-Skolem theorem and Cantors theory . Thus A) maths is in contradiction ie two proofs prove contradictory results.

Now it is stated in the main article

Skolem's paradox is the mathematical fact that every countable axiomatisation of set theory in first-order logic, if consistent, has a model that is countable, even if it is possible to prove, from those same axioms, the existence of sets that are not countable.

Skolems proof shows 1)that by using the axioms of set theory we can prove that set theory has a model which is countable But 2) useing the same axioms of set theory it is possible to prove that there are uncountable sets. Thus the same axioms prove contradictory things. Thus B) set theory is inconsistent

Now dont say there is 1) no contradiction in maths and 2) set theory is not inconsistent because Skolems solution has proved that there is no paradox. Because as I have shown Skolem and many others have said Skolems solution destroys set theory. It is admitted in the article and by an editor that it is a paradox if first order logic is used on set theory

Skolem's result applies only to what is now called first-order logic,

You have one big dillemma if you accept Skolems solution then 1) set theory is destroyed or 2) it is a paradox and maths and set theory are inconsistent

—Preceding unsigned comment added by Xzungg (talk • contribs)

To show there is a contradiction in maths what one needs to do is shown for example in Russel's paradox in the 'Formal derivation' section. There a contraction is derived. The formal derivation of a contradiction would then have to be peer reviewed, or it might just become notable by being reported in newspapers around the world as is very possible if it happened, and then it would be included as a wikipedia entry and loads of aspiring PhD's round the world would have another thesis subject to have a go at. It is not enough to just say there is a contradiction. G. H. Hardy used to send postcards saying he had solved the Riemann hypothesis before going on a dangerous journey. Dmcq (talk) 09:19, 13 November 2008 (UTC)

You say to show a contradiction in maths one needs to so do a formal derivation. I say you have a very limited view of the matter. 1) It is shown that there is a contradiction in maths ie The downward Löwenheim-Skolem contradicts Cantors theorem- this is ample evidence for there being a contradiction in maths. And 2) it has been shown that from the same axioms of set theory you can get contradictory results-this is ample proof that set theory is inconsistent. In both cases there are proofs given ie 1) The downward Löwenheim-Skolem theorem has been proven and as such it is seen that it contradicts Cantors theorem 2) The downward Löwenheim-Skolem theorem using the axioms of set theorem proves there are countable sets BUT the same axioms can be used to prove there are uncountable sets. Here we have two proofs useing the same axioms which contradict each other. Thus we have proofs that maths is contradictory and set theory inconsistent, not a formal derivations but proofs nevertheless. It should be noted that set theory being inconsistent is a view but forward also by the Australian philosopher Colin Leslie Dean Godels incompleteness theorem to be invalid —Preceding unsigned comment added by Xzungg (talk • contribs)

Xzungg: there is no actual contradiction in Skolem's paradox. It only appears to be a contradiction because the same word "uncountable" is used to mean different things during the argument. Even Skolem, in 1922, realized this and explained that there is no actual contradiction. If there were a way of actually proving that ZF set theory was inconsistent, this would be a great interest to mathematicians, but at the moment no proof of its inconsistency is known. This discussion is starting to go in circles, so I am going to work on other things. If you make any new points here, I will respond, but if you simply repeat the same arguments, I may not respond to them. — Carl (CBM · talk) 14:00, 13 November 2008 (UTC)

You say "Even Skolem, in 1922, realized this and explained that there is no actual contradiction." How more times do I have to say this Even Skolem said his solution to the paradox destroyed set theory So you are left with a dillemma if you accept Skolems solution then 1) set theory is destroyed or 2) it is a paradox and maths and set theory are inconsistent —Preceding unsigned comment added by Xzungg (talk • contribs)

It sounds like Xzungg and this 'Australian philosopher Colin Leslie Dean' have a great deal in common and ought to get together :) Lawyer type arguments just aren't going to cut much ice as far as mathematicians are concerned. You might be interested that Godel was just about persuaded not to go on about logical problems in the US constitution when applying for citizenship. Dmcq (talk) 14:10, 13 November 2008 (UTC)

I went hunting for this, because I've read it too -- it's a funny story -- in Dawson's biography of Goedel but I couldn't find it. Anyway, in the process I bumped into a discussion of Goedel's assertions re "impredicativity" and intuitionism (cf page 156-157). Some philosophers and mathematicians -- Goedel and Kleene and apparently Brouwer among them -- believe(d) that all the true antinomies, i.e. the truly mathematically paradoxical outcomes, are the result of one or more impredicative statements (e.g. axioms, premises or definitions) somewhere in an argument. Kleene discusses this at length in his 1952 text. For example, if the set can be an element of itself then we have a problem. If Skolem's "conundrum" is a true paradox then it will contain, somewhere, an impredicative premise or statement. So as the lead-in paragraph implies, if this "conundrum" is a paradox in this classical sense then it will contain an impredicative premise or definition somewhere. If not, then not. As best as I can determine: Impredicative <--> paradox. Bill Wvbailey (talk) 20:15, 14 November 2008 (UTC)

Theres a reference to a short bit about the story in wiki's Godel article. It is quite easy to make a maths theory that has contradictions without using "impredicativity"! On the other hand some limited self reference needn't cause problems, it would be difficult to talk about anything being infinite otherwise. So it isn't an if and only if business. Dmcq (talk) 20:56, 14 November 2008 (UTC)

Am truly curious (not sniping, am actually curious), what would be an example of a contradiction in a modern "well-formed" theory (one that mathematicians actually use and isn't bizarrely hypothetical to make a point) that is not impredicative? Kleene et. al. (as does wiki) give the l.u.b. example as an example of impredicativity, one that probably bothers the intuitionists just because of this. Kleene for one refuses to let this one go (as late as the 10th impression of his book 1991, cf page 43; he emended the book in 1971 cf page VI.) He also discusses "Skolem's paradox" in detail in his chapter 75 pp. 421-432; I just found this, haven't read it yet. It is in context of "Axiom systems" and "Decidability". Bill Wvbailey (talk) 22:22, 14 November 2008 (UTC)

I don't think you're going to find a known example of a contradiction in a modern theory that mathematicians actually use. Impredicative or otherwise. --Trovatore (talk) 23:55, 14 November 2008 (UTC)

You'd have to cast your net a bit wider round maths for a theory which looked good like naive set theory but was later shown to allow a contradiction, I can't think of one offhand. It's quite possible there's a few to be found. It would be quite easy for instance for someone to take a bunch of assumptions like the ones Arrow's impossibility theorem proves can't work together and assume them as a basis for further work before finding the whole basis was flawed. Dmcq (talk) 00:21, 15 November 2008 (UTC)

The Kleene-Rosser paradox showing that Church's original formulation of lambda calculus circa 1930 was inconsistent comes to mind. Of course that is not a set theory and I don't know if 1930 counts as modern. I feel like I'm half-remembering something else like that too, but am not sure. 207.241.239.70 (talk) 02:44, 15 November 2008 (UTC)

What is written in the article agrees with Kleene 1952:426-7. Kleene calls it Skolem's "paradox" [sic]. But he raises the spectre of the "set-theoretic paradoxes . . . [to avoid them] we must accept the set-theoretic concepts, in particular that of non-enumerability, as being relative, so that a set which is non-enumerable in a given axiomatization may become enumerable in another, and no absolute non-enumerability exists. This relativation of set theory was proposed by Skolem (1922-3, 1929, 1929-30). ¶ The Lowenheim theorem, since it leads to Skolem's "paradox", can be regarded as the first of the modern incompleteness theorems. For further discussion, see Skolem 1938." (p. 427). Kleene then goes on to extend the results to "Axiomatic arithmetic" with his proof that "no finite or effectively enumerable infinite set of elementary axioms can characterize the natural number sequence 0, 1, 2, . . ., a, a', . . ." (p. 429). Thus his Postulate Group B ("Postulates for number theory") are "incomplete as a characterization of the natural number sequence" (p. 429). He ends up invoking Goedel's incompleteness theorem. Kleene's discussion leads me to think that this Skolem's "paradox" (and its extensions) is pretty important -- at least from a historical context. Perhaps someone more adept than me (this is tough stuff) could extend the article with regards to these further developments. Bill Wvbailey (talk) 16:30, 15 November 2008 (UTC)

I have to say I'm fairly shocked that Kleene could write such a thing as late as 1952. Not for the choice of the word paradox, which is fine ("paradox" = "apparent contradiction"). But the wording about a set being countable or not countable in an "axiomatization", rather than in a model, is a straightforward category error (axiomatizations don't have sets in them; rather, they prove or don't prove things about sets, and you can't in general identify a set with a name for it in some axiomatic system). And the leap to the claim that "no absolute non-enumerability exists" is simply unwarranted—while it's licit for Kleene to have believed that, it ought to have been clear to him that it didn't follow from the proposition at issue. Of course I don't see the whole context here.

I agree that this is evidence for a historical take on it that should perhaps be treated in the article, but it would be better to put it into a context that includes the later consensus, which is rather different from what Kleene appears to have thought. I expect to have a chance to talk to a couple of Kleene's former students in January — maybe they can enlighten me as to what he may have been thinking here. --Trovatore (talk) 23:16, 15 November 2008 (UTC)

I should clarify that Kleene puts quote-marks around the world 'paradox', i.e. he refers to it as " . . . Skolem's "paradox" . . .". But he also does the same thing to the notion/explanation of "non-enumerable inside the theory while enumerable outside the theory": "Although there is this "explanation", the "paradox" still confronts us with the following alternative. Either we must maintain that the concepts of an arbitrary subset of a given set, and of a non-enumerable set, are a priori concepts which elude characterization by any finite or enumerbly infinite system of elementary axioms; or else (if we stick to what can be explicitly characterized by elementary axioms, as we may well wish to in consequence of the set-theoremtic paradoxes §11) we must accept the set-theoretic concepts, in particular that of non-enumeability as being relative ...[the rest of this quote is the one above]" (p. 426). With regards to "models" he doesn't seem too impressed. Earlier in this same section (§75) he states "Prior to Hilbert's proof theory or metamathematics, proofs of consistency of an axiomatic system or theory were by exhibiting a model for the theory (§14)." (p. 422). When we go to §14, we discover that this is his section §14 Formalism. In it he states "Consistency proofs by the method of a model are relative. The theory for which a model is set up is consistent, if that from which the model is taken is consistent. ¶ Only when the latter is unimpeachable does the model give us an absolute proof of consistency. . .. ¶ For proving absolutely the consistency of classical number theory, of analysis, and of set theory (suitably axiomatized), the method of a model offers no hope. No mathematical source is apparent for a model which would not merely take us back to one of the theories previously reduced by the method of a model to these. ¶ The impossibility of drawing upon the perceptual or physical world for a model is argued in Hilbert and Bernays 1934 pp. 15 - 17. They illustrate it by considering Zeno's first paradox . . .." (p. 54). "It is a deep philosophical question what the 'truth' or objectivity is which pertains to this theoretical world construction going far beyond the given. This is closely connected with the question, what motiviates us to take as basis the particular axiom system chosen. For this consistency is a necessary but not sufficient argument." (p. 58) [This notty question also came up while I was doing research on the Brouwer-Hilbert controversy article.] These are not the first places that Kleene leaves us hanging. As I mentioned above, he is not too happy with the impredicative l.u.b. . . . "The same argument can be used to uphold the impredicative defintions in the paradoxes" (p. 43). Once again Kleene brings us back to the (true) paradoxes (he takes them very seriously cf p. 36) and the deep arguments behind metamathematics, Formalism and Intuitionism. I dunno, I don't have enough background etc to make much sense of this, excepting that Kleene seems rather radical w.r.t. the (true) paradoxes (and I find his wording ambiguous about his own take on the Skolem "paradox"). Perhaps he settled some of this later in his career as new theories etc unfolded -- but he was making minor emendations to his text as late as 1971 (cf note page VI of 10th reprint.) It would be interesting to know if/how his philosophy etc changed over the years. Bill Wvbailey (talk) 19:57, 16 November 2008 (UTC)

You say —Preceding unsigned comment added by Xzungg (talk • contribs)

I had a look on Google to see what 'You say' is supposed to mean. Too unspecific unfortunately. Amusingly though lots of the top hit were 'You say I only hear what I want to". Perhaps Google is more intelligent than I give it credit for? :) Dmcq (talk) 13:32, 14 November 2008 (UTC)

Internal vs. external countability

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)

Removal of irrelevant forum-like discussions

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)

Joshtrimble's reformulation

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)

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)

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)

I am sorry to have lifted the text from from the Stanford Encyclopedia of Philosophy, word for word. I gave it a cite link :) I am just unsure of the rules here and am jumping right into editing wiki's and learning as I go. Next time, I will reword it to make it more accessable to a general audience. Sorry again and thank you. Joshtrimble (talk) 14:39, 22 January 2009 (UTC)

first-order sentence asserting non-countability

It may be helpful to include such a sentence in the page. Tkuvho (talk) 19:53, 5 February 2011 (UTC)

Countable

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)

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)

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)

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)

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)

Error on HOL

The article has the phrase "if set theory is formalized using higher-order logic with full semantics then it does not have any countable models." Unfortunately there is no formalization of the full semantics. See also:

"HOL with standard semantics is more expressive than first-order logic. For example, HOL admits categorical axiomatizations of the natural numbers, and of the real numbers, which are impossible with first-order logic. However, by a result of Gödel, HOL with standard semantics does not admit an effective, sound, and complete proof calculus." https://wiki.riteme.site/wiki/Higher-order_logic

So I guess the phrase should be replaced by something more appropriate, since usually "formalized" means an effective, sound and complete proof calculus. Formally the full semantic can only be gradually approached, see for example this article http://plato.stanford.edu/entries/logic-higher-order/

Jan Burse (talk) 03:47, 13 July 2015 (UTC)

That's a lot stronger sense of the word "formalized" than I think I use, or than I think is standard. AFAIK to "formalize" something just means to make it formal; to put it into symbols instead of using natural language. Do you have a reference for this sense of the word? Does anyone else think this language is confusing? --Trovatore (talk) 04:00, 13 July 2015 (UTC)

I guess the author wanted to point out that there are higher order logics (not necessarely with an effective, sound and complete proof calculus) which have Löwenheim numbers which are bigger than ℵ0. Namely the article on HOL says: "For example, the Löwenheim number of second-order logic is already larger than the first measurable cardinal, if such a cardinal exists." https://wiki.riteme.site/wiki/Higher-order_logic

I was looking around recently, whether there are such HOLs without full semantics, i.e. whether the Löwenheim number can also be bigger than ℵ0 if we don't require full semantics, and possibly if there are some HOLs around which have even an effective and sound (not necessarely complete for the full semantics) proof calculus. But I did not yet complete my research. I have some hints that such logics might exist. Jan Burse (talk) 12:53, 13 July 2015 (UTC)

Sounds like fun, but I don't see what it has to do with the article's use of the word "formalize". --Trovatore (talk) 18:59, 13 July 2015 (UTC)

For a reference see for example http://www.amazon.com/Mathematical-Logic-Edition-Undergraduate-Mathematics/dp/0387942580 In this basic text book incompletness of second order logic is shown via a different route than Gödel. Namely it uses some result by Trahtenbrot. The application to second order logic is not found in the English wikipedia, but is mentioned in the German wikipedia, giving proof to cultural bias in wikipedia even in precise science such as mathematical logic, probably reflecting the different centers of intellectual gravity around the world.

The heading of the chapter of the book is "Limitations of the Formal Method", so I guess yes if you equate "Formalization" with "The Formal Method", i.e. finding an effective, sound and complete proof calculus, then yes, the sentence in the article needs some revision. But maybe we are dealing here with a case of ambiguity, between using logic to model something, i.e. when someone is just happy to have symbols instead of natural language, and when somebody is using logic to formalize something, which needs a little bit more than just symbols.

Jan Burse (talk) 21:05, 20 July 2015 (UTC)

I would personally view the completeness theorem as a property of some formal systems and not others, rather than an intrinsic property needed to consider something to be a "formal system". I view a formalization as consisting of a formal language (syntax), a deductive system, and a semantics (collection of interpretations). The deductive system may or may not be effective (e.g. in proof theory one may often consider systems of arithmetic that have all true $\Pi^0_1$ formulas as axioms); the semantics may or may not consist of "all possible" interpretations (compare full semantics and Henkin semantics for second-order logic), and the completeness theorem may or may not hold.

Nevertheless, I do think there is a more classical usage of "formal system" which requires effectiveness, and which might be more interested in completeness - this would have been the viewpoint of traditional foundations of mathematics in the early and mid 20th century. The sentence in the article could be edited to say "If set theory is studied using higher-order logic using full semantics, ...", and that would convey the same point. — Carl (CBM · talk) 23:38, 20 July 2015 (UTC)

GA Review

This review is transcluded from Talk:Skolem's paradox/GA1. The edit link for this section can be used to add comments to the review.

Nominator: Pagliaccious (talk · contribs) 04:15, 22 August 2024 (UTC)

Reviewer: David Eppstein (talk · contribs) 23:23, 26 August 2024 (UTC)

First read-through: meaning and completeness

First sentence: should "first-order model of set theory" maybe be "model of first-order set theory"? And how can a model prove anything? It is mathematicians who prove things, not models. Or by extension you could maybe say that an axiomatization proves something (its theorems) but it is still not the model that is the subject of "prove".

Somewhere in the article, and maybe also in the lead, it should be clarified what a model is. I am familiar with them but I think we should not expect all readers to be. I think the punch line of the paradox should also be made more explicit: because an uncountable set exists, such a set is an element of the model, despite the model being (externally) countable.

In the lead "Formally, Skolem's paradox is that every countable axiomatization of set theory in first-order logic, if it is consistent, has a model that is countable": isn't this really just the Löwenheim–Skolem theorem, not the paradox itself? What meaning does "Formally" add? And in the next sentence, about proving existence of uncountable sets, maybe Cantor should be briefly credited?

I suspect that the last line of the lead 'More recently, the paper "Models and Reality"...' is intended to be a summary of the last paragraph of the "Later opinions" section, on Putnam and reactions to Putnam, but this is unclear because that paragraph never mentions "Models and Reality" by name.

You have the dates of Cantor's uncountability theory and the Löwenheim–Skolem theorem, but not of Skolem's observation that they are (philosophically at least) contradictory. Can it be dated?

Doesn't the second paragraph of "The result and its implications" lead to a different paradox? The paradox as stated earlier is "this model is countable but yet it (its domain) contains an element that (in the theory of the model) is uncountable". But the second paragraph doesn't talk about an element that is uncountable in the theory of the model; it talks about a set that is not in the model (the set of all subsets of the model). And I don't see how the part about "meaning we cannot put each set of the model B in relation with some natural number" follows. It somehow seems to be assuming that each subset of the model is a set of the model; why? This paragraph seems to be a mixed-up combination of the Skolem paradox and a different paradox: if B is a countable model, then there exists a set of all subsets of B (with the cardinality of the continuum) but we know that "the set of all sets" cannot actually exist. (The resolution to this paradox being that this set of all subsets of B is not an element of B so it does not belong to the model.)

In the third paragraph of the same section "Skolem resolved the paradox by concluding that the existence of such a set cannot be proven in a countable model". What do you mean by proving something in a model? I think the resolution is that, if U is an element of B that models an uncountable set, and C is an element of B that models a countable set, then there does not exist an element phi of B that models a bijection from U to C. It's not merely that there is no proof element in B that phi models such a bijection, but that this bijection is not part of the model at all.

How does the "formally" of the fourth paragraph introduce a different level of formality than the "formally" of the second paragraph? Especially as both paragraphs appear to consist of informal prose rather than formalized logical deductions? In any case I think this fourth paragraph much more closely describes the paradox and its resolution than the second. One quibble: "There are two special elements of M; they are": maybe instead of "they are", more accurate would be "they model"?

In the fifth paragraph, we again have this confusing issue of whether a model contains an element that models a bijection, or whether the bijection can be proven to exist. Why not "relative to one model, no enumerating function puts some set into correspondence with the natural numbers, but relative to another model, this correspondence may exist"? What does provability have to do with it, except for the side point that if an object is actually proven to exist (in the theory) then it must exist (in the model that models the theory)?

In the "Reception" section, how exactly does "Skolem's result" (by which I imagine is intended the paradox, not the Lowenheim–Skolem theorem) prove that first-order set theory cannot be categorical? I mean, it cannot be categorical, but how does that follow from the paradox?

The van Dalen & Ebbinghaus reference at the end of the Reception paper discusses a 1937 paper of Zermelo which, van Dalen & Ebbinghaus state, was intended to refute the Skolem paradox. Why is this work not discussed in more detail in this section? The sentence "It is now known that Skolem's paradox is unique to first-order logic; if set theory is studied using higher-order logic with full semantics, then it does not have any countable models, due to the semantics being used." appears to be referenced to van Dalen & Ebbinghaus p145 but that page says nothing about higher-order logic not having countable models; can this be sourced properly?

A minor formatting note: the |30em in the reflist of the References section is no longer needed (reflists are put into columns by default), but Template:Reflist suggests using 20em instead for articles with shortened footnotes, as used here. When I view this article on my laptop I get only one tall column of references, so a smaller number like 20em would make it more likely that a compact two-column format could be used.

I considered the possibility of discussing the proofs of both Lowenheim-Skolem and Cantor's uncountability theorem, but ultimately decided against requesting that. I think it would be too duplicative of material that belongs better on the articles on those theorems. The only important part here is merely that both are proofs in the first-order theory and therefore must be true of any model of the theory.

The illustrations are more decorative than informative, but they are properly licensed and have informative and relevant captions; I think they're ok, and it's hard to imagine what else might be used as an illustration for this article.

I think that covers WP:GACR criteria 1b, 3, and 6. Criteria 4 and 5 are unlikely to be problematic, but I still need to do another read-through (another day soon, most likely) for low-level copyediting (1a) and for cross-checking the references against the material they reference (2; touched on briefly above but not thoroughly checked).

—David Eppstein (talk) 06:59, 27 August 2024 (UTC)

Hello David Eppstein. Thank you for taking the time to write such a thorough review. I think that I've addressed all of your comments:

• Throughout the article, I've done my best to remove "prove" or "proven" where possible, replacing it with "satisfied" whenever necessary, as you described at several points.
• I've clarified what a model is in the lead. If you think that the article would be improved by a longer explanation, I would gladly move it to the "Background" section and expand it.
• I've removed all the "Formally" introductory phrases, mentioned Cantor in the lead, and fixed the mention of Putnam
• Added date of Skolem's paper
• Removed the second paragraph of the Result section
• Concerning the "first-order set theory cannot be categorical" claim: this article is a bit of a forgotten battleground, as you might tell from the talk page. This claim is an artifact from this era of the page, which I missed when sourcing and cleaning up the existing prose. It seems to only be mentioned on this course webpage, but nothing published. This source was a topic of much discussion on the talk page. I'm happy to remove it from the article.
• I've sourced the "Skolem's paradox is unique to first-order logic" statement properly. The van Dalen & Ebbinghaus only sourced the following sentence, on Zermelo's 1937 paper.
• Concerning this 1937 paper of Zermelo: in fact, it is not so much a paper as a handwritten rough draft. It appears in English on pages 155-156 of the van Dalen and Ebbinghaus paper. From what I understand, this is Zermelo's last attempt at an attack on "finitism" by means of attacking the paradox, and it was left unfinished, unpublished, and uncirculated until the van Dalen and Ebbinghaus paper. It's interesting within the context of the history of the paradox, but I don't know if it's very important to a broader understanding of the paradox's history, as it was left unpublished. I've added a short sentence clarifying that the refutation was unfinished.
• Fixed the ref formatting
• For the images, although you said that they seemed to be fine, I've decided to remove all but Skolem's portrait. I agree that they're only decorative, and Zermelo's and Putnam's pictures were a bit irrelevant.

Kind regards, Pagliaccious (talk) 14:45, 28 August 2024 (UTC)

Second reading: Copyediting and some potential additional sources

The previous edits look good; here are some new, mostly less significant, concerns.

The article isn't very consistent about whether the first time a mathematician is named (rather than merely mentioning something else named after them), or later, it should be just the last name or the full name: "Thoralf Skolem", "Cantor proved" (but then later "Georg Cantor"), "Ernst Zermelo", "Hilary Putnam", "Löwenheim", "Abraham Fraenkel", "John von Neumann", "Stephen Cole Kleene", "Geoffrey Hunter", "Reuben Goodstein", "Hao Wang", "Brouwer", "Carl Posy", "Cohen's method", "Kanamori", "Hilary Putnam", "Timothy Bays", "Tim Button".

Background: "When Zermelo proposed his axioms for set theory": link Zermelo set theory (these are the 1908 axioms). But later in the section you have "Zermelo's axioms of set theory" linked to Zermelo–Fraenkel set theory; this is confusing, because it's not clear whether the 1908 axioms or ZF is the intended meaning. If you called it "the axioms of Zermelo–Fraenkel set theory" here, it would be clearer, and would not make it look like the paradox applies only to an obsolete variant of the theory. The same confusion between two different axiom systems persists into the first paragraph of the next section, which in any case does not need to link them again.

"There is only a countably infinite amount of ordered pairs" reads awkwardly, maybe better "There are only countably many ordered pairs" or maybe something like "The set of ordered pairs is countable"

"Zermelo at first considered": do we need to link Zermelo again here?

"Hilary Putnam considered it": here "it" can only mean the Skolem paradox, but as there is no preceding noun in the same paragraph and the last noun from the previous paragraph is "formal systems", the referent is grammatically unclear.

"the most recent": Is this time-specific wording necessary? They are both over a decade old, so not particularly recent, and I don't think it would be helpful to demand sourcing for the claim that these really are the most recent significant discussions.

Speaking of which, is there anything worthy of mention in the following recent publications?

• Penchev, "Skolem's Paradox and Quantum Information. Relativity of Completeness according to Gödel", Philosophical Alternatives Journal
• Hosseini and Kimiagari, "Higher-Order Skolem's Paradoxes and the Practice of Mathematics: a Note", Disputatio
• Hanna, "A Neo-Organicist Approach to the Löwenheim-Skolem Theorem and “Skolem’s Paradox”", Science for Humans
• Shapiro, The limits of logic: higher-order logic and the Löwenheim-Skolem theorem (a book-length work which calls the paradox "the other main theme of this volume")

I think the following ones really may be worthy of mention:

• Roman Suszko, "Canonic axiomatic systems", Studia philosophica (Poznán), 1951
• Pogonowski, "On the axiom of canonicity", Log. Log. Philos., 2023
• Riccardo Bruni, Review of Pogonowski, MR4562923
• Jan Kalicki, Review of Suszko, J. Symbolic Logic, doi:10.2307/2267712

I have only seen the Kalicki and Bruni reviews, not the original papers. The reviewers both write that Suszko developed an axiomatic theory of sets in which the paradox is obtained without going through the Lowenheim-Skolem theorem. From Bruni's review, the Pogonowski paper clarifies Suszko's and puts it into historical context.

—David Eppstein (talk) 01:42, 31 August 2024 (UTC)

Hello David Eppstein. I believe that I've made all of your suggested changes:

• The full name of mathematicians is given at their first mention and in all wikilinks, and their last name is given at all subsequent mentions.
• I've done my best to clear up the difference in context between Zermelo's axioms, ZFC, and standard models of first-order set theory in general. The paradox holds for all "standard" (to use a nebulous term from Bays and Eklund) models of first-order set theory, probably most notably ZFC; however, Skolem's 1922 paper is written with specifically Zermelo's 1908 axioms in mind, so I chose to leave "Zermelo's axioms" in the paragraph beginning with "In 1922, Skolem pointed out..." The second part of Skolem's paper is his proposal of the axiom of replacement, independent of Fraenkel's own 1922 proposal. After this historical description, I've added a statement clarifying that the result holds for ZFC and other "standard" models.
• I've implemented the rest of your copyediting suggestions.

Thank you for listing these much newer publications. I'll take a look at them and see if I can't find more to include in the article. Kind regards, Pagliaccious (talk) 15:04, 31 August 2024 (UTC)

I've been looking at the first set of publications this afternoon. The Hanna and Penchev papers seem to be short appeals to the paradox in fields outside of logic/mathematics, so I'll include a short sentence on these uses at the end of the "Later opinions" section. The short paper by Hosseini and Kimiagari is about a higher-order class of "paradox" which they associate with Skolem's paradox by calling it an "extension" of his paradox, but I don't know if it's notable enough to mention. Shapiro's book is a collection of papers, many of which are already included as references in the article, but many of which I think ought to be referenced.

As for the "Axiom of Canonicity," Pogonowski's paper seems very interesting, and I'd certainly like to include it in the article, but I cannot find Suszko's work online or in any library catalog. I'll have everything from these publications finished by tomorrow. Pagliaccious (talk) 20:54, 31 August 2024 (UTC)

Third reading: source check (mostly)

I finally took the time to look through the talk page and...wow. Not the sort of thing I would have expected controversy over. Fortunately long done.

Last line "the Skolem's idea": either "Skolem's idea" or "the Skolem idea"

Minor reference formatting details (beyond the scope of the GA requirements): consider adding |language=de to the German-language references. I would not generally include publishers on journal publications but it's mostly harmless. On the other hand, the book source From Frege to Gödel: A Source Book in Mathematical Logic is missing its publisher (Harvard University Press). And we're not very consistent about which journals get their names linked, nor on whether the titles of chapters and journal articles are sentence case or title case. In Hunter 1971, "Macmillan" is misspelled. Kleene 1967 is again missing a publisher. Moore 1980 is unlinked (doi:10.1080/01445348008837006) and its journal is uncapitalized. Klenk 1976 and Resnik 1966 could use the |jstor= parameter rather than merely a link (it would have made it more obvious to me that I had access to the link). Kanemori 2012 is missing page numbers.

I'm not entirely convinced that Penchev 2020 counts as a peer-reviewed and reliable source. The "Epistemology eJournal" that it's in [2] is basically just a topic subclassification of the SSRN preprint server without peer-review [3]. On the other hand, Penchev appears to be an associate professor in the Bulgarian Academy of Science [4] so maybe he counts as a "established subject-matter expert" under WP:EXPERTSPS? In any case if this is kept then the name of the publisher should be split from the name of the sort-of-journal.

Footnote [1], the only footnote in the lead (a direct quote) is appropriate and valid for both its uses.

The first paragraph in "Background" implies that Cantor's theorem is his 1874 proof of the existence of an uncountable set. But checking footnote [2] I learn that the 1874 paper proved only that the reals are uncountable, by a different method than diagonalization, much weaker from how our article on Cantor's theorem states the theorem (that every set has a larger power set).

The claim that Cantor's is "One of the earliest results in set theory" appears unsourced.

Bayes 2007 is linked with |url=, putting the link on the book title. It should be linked with |article-url=, putting the link on the chapter title. Long ago |url= used to be smart and guessed what to link, usually correctly, but the citation template maintainers in their infinite wisdom decided to require different url parameter names for different link locations. This reference might have been formatted before that change, but this issue still bites me all the time. The same thing is also happening for Hanna 2024. There's another issue on footnote [3], that the link pagination does not match the book pagination, I assume because the pagination changed across different editions of the book. So in reading the source from its link I had to do some arithmetic to find the correct page for this note. This footnote doesn't go into much detail about what it is used for (the definition of countability) but this is such a basic and easy to source notion that I think it's ok.

Footnote [4]: I don't see Cantor's theorem on page 203 of Zermelo 1967. That page has a related theorem that a set cannot be its own powerset, and a comment that this avoids Russell's paradox. I ran out of limited-preview pages to check whether maybe Cantor's theorem itself is on a subsequent page.

The claim that Lowenheim's was "the first proof of what Skolem would prove more generally" appears unsourced (sources [5] and [6] are only to their two papers, not to any subsequent source that clarifies the priority). Reference [7] should surely be Bays 2007, not Bays 2000 (which has no pages with this number), and re-used as a named reference from reference [3] rather than given as a separate footnote. The only part of the rest of the paragraph (sourced to [7]) that I didn't see in the source is that this is "the downward form" of LS.

Reference [8]: now I see where the odd "proved by a model" formulation was coming from. Anyway, ok for its use here.

"holds for any standard model of first-order set theory, such as ZFC": the part before the comma is in footnote [9]. The specific application to ZFC is not (the source uses ZF as an example) but can be covered by repeating footnote [3].

Reference [10] led me to read footnote 3 of Bays 2007, which I think is not a correct formulation of being a singleton. Not a big deal, just an illustration of how hard it is to get these formulations correct. Page 4 of the preprint version does raise an issue we don't mention: when we say that we every element \hat m of the model is countable (even the one that models an uncountable set) do we mean that the actual elements of \hat m are countable or that the elements of the model that model elements of \hat m are countable? In a countable model both are true but they could be different in general. The (now third) formalization of our "The result and its implications" picks the second kind of countability, of the elements of the model that model elements of \hat m, and that makes sense because who cares what the elements of a model actually are (they could be natural numbers or ur-elements for all we care), but maybe we should mention this distinction? Anyway, this footnote is good for this content.

Reference [11]: Ok for the description of countability relative to a model, but not for the claim "Skolem used the term "relative"", which could I guess be sourced directly to Skolem. "He described this as the "most important" result" could also use a footnote to Skolem.

[12] Kunen: [page number needed]. [13] Enderton p152: states and discusses the Skolem paradox but does not appear to source its formulation in contemporary set theory as "countability is not an absolute property". So this claim still needs a better source.

Quote to Fraenkel, footnote [14]: our article says Fraenkel's Introduction to set theory (1928), with no page number. The source says Einleitung in die Mengenlehre (3rd ed., 1928, p. 333). Maybe we could at least clarify that this wording is a translation of Fraenkel, and give the page number for Fraenkel?

Von Neumann, [15][16]: Ok

"Zermelo ... spoke against it starting in 1929": contradicted by the source [15], which on that page only says that he gave talks about his foundational views beginning in 1929 (without mentioning any specific position on Skolem's paradox) and then on p.151 states "It is not quite clear whether Zermelo really knew Skolem's precise treat- ment in [10] or whether he was really aware of its scope when he gave his second-order definition, because he does not mention Skolem, whereas later, Skolem is the crystallization point of his criticism." and goes on to quote Skolem as suggesting that Zermelo may have been unaware of this work. It is only on [15]/p153 that one finds a direct response by Zermelo to Skolem, cited to a reference uncertainly dated to 1931.

"argued against the finitary metamathematics" [17]: cited to p.519 of Kanemori but I think this extends across pp. 519–520.

Half-paragraph beginning "Zermelo argued that his axioms should instead be studied in second-order logic", cited to van Dalen p.151": I think that this statement about 2nd-order logic can be sourced to that citation, but I do not see the remaining claims (Skolem does not apply to 2nd-order logic; Zermelo published in 1930; proved categoricity results; led to discovery of cumulative hierarchy; formalized infinitary logic) on that page of that source.

I'm not seeing "now a standard technique for constructing countable models" in footnote [21]; where is it?

"if set theory is studied using higher-order logic with full semantics, then it does not have any countable models": source [9] says only that Lowenheim-Skolem does not hold. Which I guess implies that there are no countable models but it would be better to have an explicit statement for that.

[26]: the relevant quote extends over pp. 304–305.

[28]: I did not have Google Books access to check this page of this source. But I am unsure about the phrasing "absolute countability was first championed by L. E. J. Brouwer from the vantage of mathematical intuitionism": does this mean that Brouwer was the first to connect absolute countability to intuitionism? Or does it mean that Brouwer was the first champion of absolute countability overall, despite [27]'s claims that this position was taken by Skolem himself?

"Both the Skolemites and Brouwer oppose mathematical Platonism": source [29] appears to only discuss the Skolemites and Platonists.

"Posy denies the idea that Brouwer's position was a reaction to any set-theoretic paradox": Posy speaks more specifically of "the set theoretic paradoxes discovered at the turn of the twentieth century", which suggests he was thinking of other paradoxes than this one, formulated somewhat later.

I don't see any discussion of forcing, let alone it being an extension of Skolem's paradox, on [31] Kanemori 2012 pp. 47-48 [5]. His discussion of forcing begins on p.51 and the analogy to Skolem is on p.53.

I think that's the last big read-through; once these are addressed it should pass GA. —David Eppstein (talk) 01:53, 2 September 2024 (UTC)

Hello David Eppstein. I've made all of the changes you suggested, except for looking at the Bays 2007 countability distinction, fixing my "Zermelo ... spoke against it starting in 1929" confusion, and finding a reference for the "if set theory is studied using higher-order logic with full semantics, then it does not have any countable models" claim. I'll hopefully have that sorted in a few days. As for the rest:

• I've included JSTOR and doi links where able
• I've made all titles title case. I had previously been following whatever appeared at the top of the work.
• I used the Penchev paper to back up the claim that Skolem's paradox is used outside of math/logic because I was unable to source an English translation of the other Penchev paper you suggested I take a look at, "Skolem's Paradox and Quantum Information. Relativity of Completeness according to Gödel". I could only find a Bulgarian version. If I'm able to translate it, I can replace the old Penchev reference with this one, but in the mean time I've separated the journal and publisher names like you suggested.
• For the Cantor result, I added a ref from Kanamori ("Set theory was born on that December 1873 day when Cantor established that the collection of real numbers is uncountable", 1873 being the year Cantor wrote to Dedekind describing his yet-unpublished result). To connect this result to the more general "Cantor's theorem" article I wrote a short clarifying sentence.
• You're correct that Cantor's theorem is missing from page 203 of Zermelo 1967; it is on page 200: van Heijenoort writes in the preface that "[Zermelo] then proves theorems about sets. The development goes as far as Cantor's theorem".
• For the Löwenheim–Skolem theorem, I've changed a footnote instead to reference van Heijenoort's prefaces to the articles by the two authors, which explains that Skolem "simplified" and "generalized" the Löwenheim–Skolem theorem which Löwenheim first showed.
• I've added a ref describing the "downward" theorem (Nourani, page 160: "The part of the theorem asserting that a structure has elementary substructures of all smaller infinite cardinalities is known as the downward Löwenheim–Skolem Theorem.")
• I've sourced the Skolem quotes (most important result, the use of "relative")
• I've added the Kunen page notes. Kunen describes the notion of non-absoluteness with respect to countability; you're quite right that Enderton does not describe this. I've added a second reference to replace Enderton which clearly describes it. This claim and the Enderton is another bit which I missed from an older version of the article. In fact, the claim of non-absoluteness predates the Nourani 2014 reference, so I'm a little wary of circular referencing, but the Kunen reference backs this up nonetheless.
• The van Dalen p 151 source should actually be 152. I was referencing footnote 12. This only backs up the "discovery of cumulative hierarchy" and "formalized infinitary logic" claims. For the two preceding sentences and their claims (Zermelo argued that his axioms require 2nd-order logic; Zermelo published in 1930; proved categoricity results) I've added further sources and removed the "categoricity" statement.
• Fixed the "Henkin's proof" reference. The claim "now a standard technique" is due to Baldwin, page 5, and the "for constructing countable models of a consistent first-order theory" claim is more or less explicit in a new Hodges reference.
• I slightly rephrased the statement on Brouwer and paradoxes. For the claim that Brouwer was opposed to Platonism, this is included in that Posy 1974 reference on the same page. I then added a new statement about Skolem's intuitionism. I don't believe that the source is online, so here is the quote: "I mention, only to put aside, the interesting issue of whether Skolem's intuitionism was a product of Brouwer’s influence or had some other source, perhaps the ideas of the school within which he was trained. Since, however, Skolem remarked in (1929a, 217) that the ideas of his 1923 paper were developed ‘independently of Brouwer and without knowing his writings’ I am inclined to discount the first source."

Please let me know if there are any issues with these changes.

I have a few questions about your suggestions. For the quote to Fraenkel, you suggested that I clarify that this is a translation of Fraenkel. I assume that I ought to do this for all of the block quotes, since the Skolem and von Neumann quotes are also originally German. However, since I'm not entirely sure whether this is a translation by van Dalen and Ebbinghaus, I can't write "(translated from the original German by van Dalen and Ebbinghaus)". Do you have any ideas for how to present this more elegantly? For now I've used footnotes, and I'm working on sourcing page numbers in the German originals. On a more minor note, what do you mean by "linking journal names"? And a question about footnotes, since I've never written an article of this sort before: for references to "primary" literature, specifically quoting short phrases of authors ("most important" result, etc), should I include a full quote as a footnote?

Kind regards, Pagliaccious (talk) 03:12, 4 September 2024 (UTC)

Ok, I think that's enough that I can pass this for GA now and trust you to carry on making the remaining changes as you see fit.

Re your questions: If they are translations copied from somewhere then we definitely need to say who translated them somewhere, at least in the footnotes. If they were just translated by some past Wikipedia editor then I think it's adequate to cite only the original German, as long as the citation makes clear that it is not English (for instance by using the language parameter). I don't think we need to provide extended context for the quotes in the footnotes.

As for linking journal names, I meant using something like | journal = [[The Journal of Symbolic Logic]] in the citation template (if you use the source editor), so that readers would be able to click on the link and find out some context about the journal. It's far from necessary, and should only be done when we actually have an article about that journal. But we currently only have a single link like that (to Crelle's journal on Cantor 1874) and it looks lonely.

One more really minor thing, while you're still cleaning up other stuff: Hodges 1985 is listed as being published by "CUP Archive". I don't know why web sources use that name, but this is really Cambridge University Press. —David Eppstein (talk) 06:45, 4 September 2024 (UTC)

Standard model of first-order set theory

Mostly jotting this down so I don't forget it. I don't have an immediate good solution and don't have time to address it now.

Article sez:

Though Skolem gave his result with respect to Zermelo's axioms, it holds for any standard model of first-order set theory, such as ZFC.

The problem with this wording is that you can read it as saying ZFC is a model, rather than a theory, which is already a fairly common misconception.

Not sure of a good rewording right now. --Trovatore (talk) 21:01, 5 September 2024 (UTC)

Maybe "for any standard first-order theory of sets, such as ZFC"? —David Eppstein (talk) 21:28, 5 September 2024 (UTC)

I think that works. I didn't have time to read the surrounding text and work out whether "model" was being used in its proper sense in some way we want to preserve. --Trovatore (talk) 22:07, 5 September 2024 (UTC)

You're quite right that it should be simply "standard first-order theory". Using "model" is misleading here. The source cited says "Skolem for the first time proves the Lowenheim-Skolem theorem downwards for standard first-order logic" and later "The Skolem paradox runs as follows. There is a theorem of ZF—and of all other standard set theories—which says that there is at least one nondenumerable set; more specifically, the set of real numbers, commonly called R, is such a set. At the same time, the Lowenheim-Skolem theorem downwards entails that ZF has a denumerable model. Hence, in a denumerable model it is true that there is a nondenumerable set. Paradox." Thanks for pointing this out. Kind regards, Pagliaccious (talk) 03:06, 6 September 2024 (UTC)