User:RJGray/Sandboxcantor

Cantor's first set theory article Georg Cantor axiom of limitation of size GA_Review

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As far as visceral reactions go: I learned years ago that you write a "paper" and, if it is printed, it becomes an "article". This explains why I prefer the term "article." For me, "article" is used only when a paper is deemed worthy of being published. In fact, I had hoped that the author guidelines would prove that my viewpoint was true. However, the way some research journals place the terms "paper" and "article" indicates that they view the terms as synonymous. For example, Crelle's Journal, which published Cantor's 1874 article/paper, starts a sentence with: "Each paper" and ends it with: "the article". The American Mathematical Society talks about "Where to send files for accepted papers" and later states "Track an accepted article". The Israel Journal of Mathematics has a section on "Submission of articles" and immediately states "Papers submitted to the Israel Journal of Mathematics". The International Journal of Mathematics has "Papers must be submitted and later states: "Each article submitted ..."

Opinions on the Net are all over the place. Here's an example that seems to say the opposite of what you are saying. From Difference between research article and research paper:

What is the difference between Research Article and Research Paper?

• There is no difference as such between a research article and a research paper and both involve original research with findings.

• There is a trend to refer to term papers and academic papers written by students in colleges as research papers whereas articles submitted by scholars and scientists with their groundbreaking research are termed as research articles.

• Research articles are published in renowned scientific journals whereas papers written by students do not go to journals.

Personally, if this is a trend, I don't like it. It's taken me awhile to accept the terms "paper" and "article" as synonymous. I was helped by my daughter who is in grad school in soil science. When I asked her the difference, she immediately said they are equivalent.

Also, I was helped by Gregory H. Moore's Zermelo's Axiom of Choice: Its Origins, Development, and Influence (which is a Wikipedia reliable source). On the top of page 152, Moore states: "In order to grasp Zermelo's system and its relation to the Axiom of Choice, it will be useful to re-examine Russell's article of 1906." The last paragraph of that page begins: "During July 1907, unaware of Russell's paper [1906], Zermelo completed the article [1908a] containing his axiomatization, which in Russell's terminology was a theory of limitation of size." I've read this page before and never noticed the switch from paper to article in this sentence. It appears to me that Moore is taking advantage of paper and article being synonymous, and is avoiding repeating the word article (or paper). Moore seems to use the terms about equally; on page 158, he uses: papers, paper, article, paper, article, paper, article.

So I'm happy if you wish to use "paper" in your contributions, but changing other's use of the term "article" because of a personal visceral reaction doesn't seem to respect others visceral reactions.

Mathematical work

Set theory

Absolute infinite, well-ordering theorem, and paradoxes

In 1883, Cantor divided the infinite into the transfinite and the absolute. The transfinite is increasable in magnitude, while the absolute is unincreasable. For example, an ordinal α is transfinite because it can be increased to α + 1. On the other hand, all the ordinals form an absolutely infinite sequence, which cannot be increased because there are no larger ordinals to add to it. In 1883, Cantor also introduced the well-ordering principle "every set can be well-ordered" and stated that it is a "law of thought."

Cantor extended his work on the absolute infinite by using it in a proof. Around 1895, he began to regard his well-ordering principle as a theorem and attempted to prove it. In 1899, he sent Dedekind a proof of the equivalent aleph theorem: the cardinality of every infinite set is an aleph. First, he defined two types of multiplicities: consistent multiplicities (sets) and inconsistent multiplicities (absolutely infinite multiplicities). Next he assumed that the ordinals form a set, proved that this leads to a contradiction, and concluded that the ordinals form an inconsistent multiplicity. He used this inconsistent multiplicity to prove the aleph theorem. In 1932, Zermelo criticized the construction in Cantor's proof.

Cantor avoided paradoxes by recognizing that there are two types of multiplicities. So in his set theory, when it is assumed that the ordinals form a set, the resulting contradiction only implies that the ordinals form an inconsistent multiplicity. On the other hand, Bertrand Russell treated all collections as sets, which leads to paradoxes. In Russell's set theory, the ordinals form a set, so the resulting contradiction implies that the theory is inconsistent. From 1901 to 1903, Russell discovered three paradoxes implying that his set theory is inconsistent: the Burali-Forti paradox (which was just mentioned), Cantor's paradox, and Russell's paradox. Russell named paradoxes after both Cesare Burali-Forti and Cantor, even though neither of them believed that they had found paradoxes.

In 1908, Zermelo published his axiom system for set theory. Zermelo had two motivations for developing the axiom system: eliminating the paradoxes and securing his proof of the well-ordering theorem. He proved this theorem in 1904, but his proof was criticized for a variety of reasons. His response to the criticism included a new proof and his axiom system. His axioms supported his new proof, and his axiom of separation eliminated the known paradoxes by restricting the formation of sets.

In 1923, John von Neumann developed an axiom system that eliminates the paradoxes by using an approach similar to Cantor's. He introduced classes that are too large to be sets—that is, classes that can be put into one-to-one correspondence with the class of all sets. These classes, now known as proper classes, bring Cantor's inconsistent multiplicities into an axiomatic system. Von Neumann defined a set as a class that is a member of some class and adopted the axiom: A class is not a set if and only if there is a one-to-one correspondence between it and the class of all sets. This axiom eliminates the paradoxes by not allowing these large classes to be sets. He also used his axiom to prove the well-ordering theorem: Like Cantor, von Neumann assumed that the ordinals form a set. The resulting contradiction implies that the class of all ordinals is not a set. Then his axiom provides a one-to-one correspondence between this class and the class of all sets. This correspondence well-orders the class of all sets, which implies the well-ordering theorem. In 1930, Zermelo defined models of set theory that satisfy von Neumann's axiom.

• Moore, Gregory H. (1982), Zermelo's Axiom of Choice: Its Origins, Development & Influence, Springer, ISBN 978-1-4613-9480-8.
• Moore, Gregory H. (1988), "The Roots of Russell's Paradox", Russell, 8: 46–56.
• Moore, Gregory H.; Garciadiego, Alejandro (1981), "Burali-Forti's Paradox: A Reappraisal of Its Origins", Historia Mathematica, 8: 319–350, doi:10.1016/0315-0860(81)90070-7.
• Purkert, Walter (1989), "Cantor's Views on the Foundations of Mathematics", in Rowe, David E.; McCleary, John (eds.) (eds.), The History of Modern Mathematics, Volume 1, Academic Press, pp. 49–65, ISBN 0-12-599662-4 {{citation}}: |editor-first2= has generic name (help).
• Zermelo, Ernst (1908), "Untersuchungen über die Grundlagen der Mengenlehre I", Mathematische Annalen, 65 (2): 261–281, doi:10.1007/bf01449999.
• Zermelo, Ernst (1930), "Über Grenzzahlen und Mengenbereiche: neue Untersuchungen über die Grundlagen der Mengenlehre" (PDF), Fundamenta Mathematicae, 16: 29–47.
• van Heijenoort, Jean (1967), From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, ISBN 978-0-674-32449-7.

OLD REFS:

Around 1895, Cantor decided that his well-ordering principle is a theorem and attempted to prove it. In an 1897 letter to Hilbert and an 1899 letter to Dedekind, Cantor gave his proof, which used his absolute infinity. He defined two types of multiplicities: consistent multiplicities (sets) and inconsistent multiplicities (absolutely infinite multiplicities). He proved that assumption that the ordinals form a set leads to a contradiction, and concluded that they form an inconsistent multiplicity. Cantor used this inconsistent multiplicity to prove the well-ordering theorem. He argued by contradiction: if a set cannot be well-ordered, then "one easily sees" that the inconsistent multiplicity of all ordinals can be mapped one-to-one into it. This implies that the set must be an inconsistent multiplicity, which is a contradiction. Zermelo later criticized Cantor's proof.

From 1901 to 1903, Bertrand Russell found contradictions in set theory by treating all of Cantor's multiplicities as sets. Russell discovered three paradoxes: the paradox of the largest cardinal (which was later named Cantor's paradox), Russell's paradox, and the paradox of the largest ordinal (which Russell named the Burali-Forti paradox). Neither Cantor nor Burali-Forti believed that they had found paradoxes.

For TALK:

I have rewritten the "Paradoxes of set theory" subsection. I thank the editor who referenced Hallett's book in the original subsection. This book was a great starting point. I also thank the editors who referenced Moore's articles in the Burali-Forti paradox and Cantor's paradox articles. These articles have been very helpful. What follows is an explanation of my rewrite and how I went about it. My explanation is a bit long because I came across many interesting facts during my research for the rewrite. In fact, I'm considering using some of this material in a rewrite of the Wikipedia article "Absolute infinite," which currently has a maintenance template in it. I'll start with Moore's statement about the paradoxes and the reactions to them:

… later opinions have been influenced so strongly by the traumatic view of the paradoxes which Russell set forth in The Principles of Mathematics [1903]. One should observe, first of all, that Cantor exhibited no alarm over the state of set theory in his letter [to Hilbert]—in sharp contrast to Gottlob Frege's dismay upon learning in 1902 of Russell's paradox. What Cantor remarked was merely that certain multitudes are, in effect, too large to be considered as unities (or sets) and so are termed absolutely infinite. Significantly, he retained such absolutely infinite, or inconsistent, multitudes and even employed them in the proof of the Aleph Theorem that he sent to Dedekind. Thus Cantor did not treat these apparent difficulties as paradoxes or contradictions, but as tools with which to fashion new mathematical discoveries. (Moore, Gregory H. (1982), Zermelo's Axiom of Choice: Its Origins, Development & Influence, p. 53.)

So it appears that caution is needed when reading accounts of the paradoxes of set theory. Because of this, I have been cautious and tried to make sure that every sentence in this subsection states a fact and not an opinion. I've used Hallett's book, and Moore's book and articles because they reference primary sources, such as letters. Also, Moore (starting in 1981) seems to have done the most detailed analyses of the paradoxes. For those interested in the paradoxes, I recommend his articles, which are available online.

I've rewritten this subsection so that it covers Cantor's ideas and two mathematical solutions to the paradoxes: (1) Zermelo's 1908 axiomatic solution that restricting the formation of sets, and (2) von Neumann's axiomatic solution using proper classes. There is a third solution, Russell's theory of types, but that theory is less relevant to this article and is more complex to explain. I included the well-ordering theorem because it motivated Cantor to extend his use of the absolute infinite, because it was proved unsuccessfully by Cantor and successfully by von Neumann with the use of an inconsistent multiplicity or a proper class, and because it was one of Zermelo's motivations for developing his axiom system.

Cantor not only knew about the contradictions that occur by assuming certain multiplicities are sets, but he also considered the problem of proving consistency. In an 1899 letter to Dedekind, "… Cantor declared that one could not even demonstrate the consistency of every finite set. Such consistency was 'a simple indemonstrable truth,' which he termed the Axiom of Arithmetic [Cantor 1932, 447–448]. In a similar fashion he regarded the consistency of each aleph as an indemonstrable truth, which he named the Axiom of Extended Transfinite Arithmetic." (Moore 1982, p. 54.) With the work of Gödel, we know that we can't prove the consistency of finite set theory (within finite set theory) if it supports elementary number theory.

Cantor is clearer in his letters than in his articles. Concerning his 1883 definition of a set: In a 1907 letter to Grace Chisholm Young, Cantor stated that when he wrote his 1883 Grundlagen, he saw clearly that the ordinals form an inconsistent multiplicity rather than a set. He also pointed out in remark (1) of the endnotes: "I said explicitly that I designate as "sets" only those multiplicities that can be conceived as unities, i. e. objects, …." (Moore and Garciadiego 1981, p. 342.) However, in the Grundlagen, he did not explicitly state that conceiving all the ordinals as a unity leads to a contradiction.

Concerning his 1895 definition: By a "set" we are to understand any collection into a whole M of definite and separate objects m of our intuition or our thought. (Cantor 1955, p. 85. I've used "set" rather than the old term "aggregate.") It has been claimed that Cantor's definition leads to "naive set theory." However, in an 1897 letter to Hilbert, it is clear that this Cantor did not intend his definition to be interpreted in this way:

I say of a set that it can be regarded as comprehensible … if it is possible (as is the case with finite sets) to conceive of all its elements as a totality without implying a contradiction. … For that reason I also defined the term "set" at the very beginning of the first part of my paper [this is his 1895 paper whose "set" definition is given above] … as a collection (meaning either finite or transfinite). But a collection is only possible if it is possible to unite it." (Purkert, Walter (1989), "Cantor's Views on the Foundations of Mathematics", in Rowe, David E.; McCleary, John (eds.) (eds.), The History of Modern Mathematics, Volume 1, Academic Press, p. 61 {{citation}}: |editor-first2= has generic name (help).)

I did not mention "limitation of size" because Cantor viewed the difference between the transfinite and the absolute infinite originally in terms of increasable/unincreasable and later in terms of consistent/inconsistent. Hallett says that the limitation of size hypothesis (all contradictory collections are too big) is a "spiritual descendant of Cantor's way of thinking represented in his 1899 correspondence." (Hallett 1986, p. 176.) Hallett is interested in the development of ideas and is looking for possible ancestors. However, this subsection deals in history and Cantor did not take the step of formulating "limitation of size." Hallett goes on to say: "But in published form LSH [limitation of size hypothesis] and its use as a starting point for building a contradiction-free set theory stems from Russell and Jourdain."

I found Zermelo's handling of the paradoxes interesting. In his 1908 set theory article, he pointed out that his axioms exclude the known paradoxes. Then, in his 1930 article about models of set theory, he gave a new explanation of the paradoxes. He postulated that there exists an unbounded sequence of strongly inaccessible cardinals κ and built an unbounded sequence of models V_κ that satisfy von Neumann's axiom. Using this approach of successively building models, he explained why the paradoxes are only apparent "contradictions". In the following quotation, "ultrafinite non- or super-set" is Zermelo's way of saying "proper class."

Scientific reactionaries and anti-mathematicians have so eagerly and lovingly appealed to the 'ultrafinite antinomies' in their struggle against set theory. But these are only apparent 'contradictions', and depend solely on confusing set theory itself, which is not categorically determined by its axioms, with individual models representing it. What appears as a 'ultrafinite non- or super-set' in one model is, in the succeeding model, a perfectly good, valid set with a cardinal number and an ordinal type, and is itself a foundation stone for the construction of a new domain [model]. (Ewald, William B. (1996), From Immanuel Kant to David Hilbert: A Source Book in the Foundations of Mathematics, Volume 2, p. 1233.)

To get a feeling for Cantor's absolute infinite, imagine being inside one of Zermelo's models V_κ where κ is a strongly inaccessible cardinal. There is no bound on the ordinals of the model. Also, you cannot increase the class of all ordinals since there are no larger ordinals to add to it. Therefore, the class of ordinals is unincreasable, which is a defining feature of Cantor's absolute. Looking at the model from the outside, this class of ordinals is the set of ordinals < κ. Of course, κ is not in the model.

I learned a lot by doing research for this subsection. I hope that readers will finding my rewrite informative and interesting. SIGN!!

Paradoxes of set theory

Discussions of set-theoretic paradoxes began to appear around the end of the nineteenth century. Some of these implied fundamental problems with Cantor's set theory program. In an 1897 paper on an unrelated topic, Cesare Burali-Forti set out the first such paradox, the Burali-Forti paradox: the ordinal number of the set of all ordinals must be an ordinal and this leads to a contradiction. Cantor discovered this paradox in 1895, and described it in an 1896 letter to Hilbert. Criticism mounted to the point where Cantor launched counter-arguments in 1903, intended to defend the basic tenets of his set theory.

In 1899, Cantor discovered his eponymous paradox: what is the cardinal number of the set of all sets? Clearly it must be the greatest possible cardinal. Yet for any set A, the cardinal number of the power set of A is strictly larger than the cardinal number of A (this fact is now known as Cantor's theorem). This paradox, together with Burali-Forti paradox, led Cantor to formulate a concept called limitation of size, according to which the collection of all ordinals, or of all sets, was an "inconsistent multiplicity" that was "too large" to be a set. Such collections later became known as proper classes.

One common view among mathematicians is that these paradoxes, together with Russell's paradox, demonstrate that it is not possible to take a "naive", or non-axiomatic, approach to set theory without risking contradiction, and it is certain that they were among the motivations for Zermelo and others to produce axiomatizations of set theory. Others note, however, that the paradoxes do not obtain in an informal view motivated by the iterative hierarchy, which can be seen as explaining the idea of limitation of size. Some also question whether the Fregean formulation of naive set theory (which was the system directly refuted by the Russell paradox) is really a faithful interpretation of the Cantorian conception.

I took out a footnote that I used in an earlier version:

Next he assumed that the ordinals form a set, proved that this leads to a contradiction, and concluded that the ordinals form an inconsistent multiplicity.

He proved that the assumption that the ordinals or the cardinals form a set leads to a contradiction, and stated that this contradiction proves that they form inconsistent multiplicities.

In a 1907 letter to Grace Chisholm Young, Cantor stated that he "saw this [contradiction] clearly" when he wrote his 1883 Grundlagen. In an 1897 paper on an unrelated topic, Cesare Burali-Forti set out the first such paradox, the Burali-Forti paradox: the ordinal number of the set of all ordinals must be an ordinal and this leads to a contradiction. Cantor discovered this paradox in 1895, and described it in an 1896 letter to Hilbert. Criticism mounted to the point where Cantor launched counter-arguments in 1903, intended to defend the basic tenets of his set theory.

In 1899, Cantor discovered his eponymous paradox: what is the cardinal number of the set of all sets? Clearly it must be the greatest possible cardinal. Yet for any set A, the cardinal number of the power set of A is strictly larger than the cardinal number of A (this fact is now known as Cantor's theorem). This paradox, together with Burali-Forti paradox, led Cantor to formulate a concept called limitation of size, according to which the collection of all ordinals, or of all sets, was an "inconsistent multiplicity" that was "too large" to be a set. Such collections later became known as proper classes.

One common view among mathematicians is that these paradoxes, together with Russell's paradox, demonstrate that it is not possible to take a "naive", or non-axiomatic, approach to set theory without risking contradiction, and it is certain that they were among the motivations for Zermelo and others to produce axiomatizations of set theory. Others note, however, that the paradoxes do not obtain in an informal view motivated by the iterative hierarchy, which can be seen as explaining the idea of limitation of size. Some also question whether the Fregean formulation of naive set theory (which was the system directly refuted by the Russell paradox) is really a faithful interpretation of the Cantorian conception.

Gödel's L and the axiom of limitation of size

The proof of Theorem 2, | V_κ | = κ, for the models V_κ where κ is a strongly inaccessible cardinal proof does not generate an explicit function from κ to V_κ. This is because the α+1 case depends on 2^λ being well-ordered so that 2^λ < κ makes sense. Since the axiom of choice only specifies the existence of a well-ordering, this proof does not generate an explicit function from κ to V_κ.

In his 1940 monograph on the relative consistency of the axiom of choice and the generalized continuum hypothesis, Gödel starts with a model V of NBG without the axiom of global choice. He uses tranfinite recursion to define a function F(α) that builds one set for each ordinal. He then defines the class L of constructible sets as the domain of F(α). He then proves that L is a model of NBG and that F(α) builds the same sets in L as it does in V. Since F(α) builds all the sets in L, this implies that F(α) is a one-to-one correspondence from the class of ordinals to the class of all sets.

http://math.bu.edu/people/aki/9.pdf Ulam

http://math.bu.edu/people/aki/10.pdf Zermelo

http://math.bu.edu/people/aki/11.pdf Levy

http://math.bu.edu/people/aki/12.pdf Gödel (short article)

In his 1940 monograph, based on 1938 lectures, Gödel formulated L via a transfinite recursion that generated L set by set. His incompleteness proof had featured “Gödel numbering”, the encoding of formulas by natural numbers, and his L recursion was a veritable Gödel numbering with ordinals, one that relies on their extent as given beforehand to generate a universe of sets. This approach may have obfuscated the satisfaction aspects of the construction, but on the other hand it did make more evident other aspects: Since there is a direct, definable well-ordering of L, choice functions abound in L, and AC holds there. Also, L was seen to have the important property of absoluteness through the simple operations involved in Gödel’s recursion, one consequence of which is that for any inner model M, the construction of L in the sense of M again leads to the same class L. Decades later many inner models based on first-order definability would be investigated for which absoluteness considerations would be pivotal, and Gödel had formulated the canonical inner model, rather analogous to the algebraic numbers for fields of characteristic zero.

http://math.bu.edu/people/aki/13.pdf Gödel

http://math.bu.edu/people/aki/14.pdf Cohen

http://math.bu.edu/people/aki/15.pdf Italian article

http://math.bu.edu/people/aki/16.pdf Set theory from Cantor to Cohen

http://math.bu.edu/people/aki/18.pdf Suslin's problem

http://math.bu.edu/people/aki/19.pdf Kunen

http://math.bu.edu/people/aki/20.pdf In praise of Replacement

http://math.bu.edu/people/aki/21.pdf Large cardinals with forcing

http://math.bu.edu/people/aki/22.pdf The Mathematical Infinite as a Matter of Method

http://math.bu.edu/people/aki/23.pdf Mathematical Knowledge: Motley and Complexity of Proof

Extra from Zermelo's models and the axiom of limitation of size

The proof of the axiom of global choice in V_κ is more direct than von Neumann's proof. First note that κ (being a von Neumann cardinal) is a well-ordered class of cardinality κ. Since Theorem 2 states that V_κ has cardinality κ, there is a one-to-one correspondence between κ and V_κ. This correspondence produces a well-ordering of V_κ, which implies the axiom of global choice.

Miscellaneous

Shorter examples of wrapping problems in Internet Explorer

In the following sentence, Internet Explorer wraps the subscript between the "g" and "(": x_g(n).

In the following sentence, Internet Explorer wraps the subscript between the "g" and "(": g(n).

I think that it's ridiculous that in such simple expressions, an editor needs to use a nowrap template to avoid the wrapping. Wikipedia should take care of it for us.

I fixed the subsection "Example of Cantor's construction" so that, with one small exception, it wraps the way I want it to in Chrome. Internet Explorer makes many simple wrapping errors such as the ones above.

I can fix it by using the nowrap template on "x_g(n)", but this shouldn't Problems with "&nbsp;" in Internet Explorer and with "nowrap" in Chrome

Thank you David for telling me that I may be seeing a bug. I've done some more experimenting and found a sentence that I can get to wrap properly in either Chrome and Internet Explorer, but not both.

The following sentence which uses "&nbsp;" wraps properly in Chrome, but not in Internet Explorer. For example, by shrinking your window, the 3 occurrences of "g(" can wrap before the "(".

A one-to-one correspondence between T and R is given by the function: g(t) = t if t ∈ T₀, g(t_{2n – 1}) = t_n, and g(t_2n) = a_n.

The next sentence which uses "nowrap" wraps properly in Internet Explorer, but not in Chrome. The 1st occurrence works for Chrome, but Chrome can wrap within the subscripts in the next 2 occurrences.

A one-to-one correspondence between T and R is given by the function: g(t) = t if t ∈ T₀, g(t_{2n – 1}) = t_n, and g(t_2n) = a_n.

So I've just fixed the 1st occurrence so that at least it works in both browsers. I find these examples very interesting. I am specifying the same no-wrap regions in two different ways. Since I'm getting two different behaviors in Chrome or Internet Explorer, it seems unclear whether the bug is in a browser or in Wikipedia code. However, since I can get the proper no-wrapping behavior by using different text the problem can be fixed in Wikipedia. Actually, I could do it myself with we had a browser template so I could write: {{browser | Chrome | … }} {{browser | Internet Explorer | … }} {{browser | default | … }} . Of course, it would be preferable for this to be done by the Wikipedia people who maintain "&nbsp;" and "nowrap". Can my examples be communicated to them? Thanks, SIGN

In the following sentence, by shrinking your window, in the first use of nowrap, you can get the subscript 1 in "a_n1" or one of the other subscripts to wrap to the next line. In the second use of nowrap, you can get the comma in "n_ν," to wrap to the next line. Is there anywhere we can submit a bug report on nowrap? I switched over to "&nbsp;".

In his letter introducing the concept of countability, Cantor stated without proof that the set of positive rational numbers is countable, as are sets of the form (a_{n1, n2, …, nν}) where n₁, n₂, …, n_ν, and ν are positive integers.

In the following sentence, nowrap is necessary because sfrac can cause wrap after "(":

The function can be quite general—for example, a_{n1, n2, n3, n4, n5} = (⁠n₁/n₂⁠)^⁠1/n₃⁠ + tan(⁠n₄/n₅⁠).

In his letter introducing the concept of countability, Cantor stated without proof that the set of positive rational numbers is countable, as are sets of the form (a_{n1, n2, …, nν}) where n₁, n₂, …, n_ν, and ν are positive integers.

Your GA nomination of Curve-shortening flow (From David Eppstein's talk page: interesting because of "In the news" or "Did you know")

The article Curve-shortening flow you nominated as a good article has passed ; see Talk:Curve-shortening flow for comments about the article. Well done! If the article has not already been on the main page as an "In the news" or "Did you know" item, you can nominate it to appear in Did you know. Message delivered by Legobot, on behalf of Mark viking -- Mark viking (talk) 19:21, 23 April 2016 (UTC)

In the 1890's, in letters to Hilbert, Cantor started calling

I'd like to address the two comments: "rm interpolated sentence which is … unlikely to be true, if there is a genuine disagreement" and "I agree: this sentence being removed seems to express a strange opinion that is contradicted by the well-cited sentences before it."

The analysis of Cantor's proof that shows it to be constructive does contradict the opinions of the mathematicians who regard the proof as non-constructive, but it does not contradict those who regard the proof as constructive. Personally, I know how confusing this can be. Years ago, I read some math history books and popular math books, and learned that Cantor's proof is non-constructive. Later, I came across some books that pointed out that Cantor's methods are constructive. I found the disagreement between these books confusing because a proof cannot be both constructive and non-constructive. So I read Cantor's article and found that his proof is constructive. At first, I couldn't understand how some books could be in error about his proof, but I've found some reasons why this has happened. (If you're interested, I can discuss this on this Talk page.)

So the math literature is confusing, but I've used it to point out an advantage that Wikipedia has over some books. People who know I contribute to Wikipedia have asked me how accurate Wikipedia articles are. I tell them that with so many people reading and correcting them, Wikipedia articles can be more accurate than some books (and the articles provide references to check, which books don't always have). This Wikipedia article is an example of this.

Thanks again to Carl and William. Hopefully, the new sentence handles your concerns. I'm trying to guide readers, who may find the disagreement between mathematicians confusing, to the method that mathematics provides to determine whether Cantor's proof is constructive—namely, the analysis of his article to see if it constructs transcendentals. Hopefully, these readers will then be interested enough to continue reading through the analysis sections.

(For the future if the discussion continues) I do know that all the books and articles I've referenced are reliable sources by Wikipedia standards, but this is a case where some sources are more reliable on this subject than others. (Warning: what comes next contains both facts and personal opinions.) My opinion is that Abraham Fraenkel, a set theorist who carefully studied Cantor's original articles and wrote an excellent biography of Cantor, is definitely more reliable than Birkhoff and MacLane, and Spivak who were busy writing excellent textbooks and probably did not have the time to check Cantor's original article. In fact, I have evidence that Birkhoff and MacLane, and Spivak took their information about "Cantor's proof" from Hardy and Wright's book. In the case of Spivak's Calculus, he gives the same exposition that Hardy and Wright do: Liouville's construction of a transcendental and the non-constructive proof that he attributes to Cantor (pp. 368-370). He states: "Cantor … showed, without exhibiting a single transcendental number, that most numbers are transcendental." In his bibliography (p. 515), he states: "Few books have won so enthusiastic an audience as … An Introduction to the Theory of Numbers (third edition), by G. H. Hardy and E. M. Wright; …." Birkhoff and MacLane also reference Hardy and Wright's book, and I remember they also contrast what they call "Cantor's proof" with Liouville's proof. (I don't have immediate access to their book so I can't give you the page numbers.)

I have also figured out where E. T. Bell and G. H. Hardy probably took their information about "Cantor's proof" from. It seems to me that the mathematicians who say that Cantor's proof is non-constructive are talking about something outside of their area of expertise. Modern set theorists, mathematical logicians, and recursion theorists all seem to think that Cantor's proof is constructive (perhaps because the diagonal argument is used constructively in the first of Gödel's incompleteness theorems and in the Halting problem). There are mathematicians outside these areas who consider Cantor's proof to be constructive. For example, the Wikipedia article quotes Irving Kaplansky, an algebraist. However, after the remarks I quote, he says: "(I owe these remarks to R. M. Robinson.) Robinson was a mathematical logician.

For Wikipedia article (Look at Kristen letter below for changes -- don't forget to ref old Monthly article!)

Cantor's article also contains a proof of the existence of transcendental numbers. As early as 1930, mathematicians have disagreed on whether Cantor's proof is constructive or non-constructive. Books as recent as 2014 and 2015 indicate that this disagreement has not been resolved. A careful study of Cantor's article will determine whether or not his proof is constructive. Cantor's correspondence with Richard Dedekind shows the development of his ideas and reveals that he had a choice between two proofs, one that uses the uncountability of the real numbers and one that does not.

Hi Kristen,

Good news! I just found out that the title of the Wikipedia article I wrote will not be changed because of a lack of consensus.

Good point about enumeration being packed with meaning. I was planning to link the word "enumeration" and found out that the Enumeration article states: "An enumeration is a complete, ordered listing of all the items in a collection. The term is commonly used in mathematics and theoretical computer science (as well as applied computer science) to refer to a listing of all of the elements of a set." So this indicates that it shouldn't be used. The correct mathematical word to use here is "sequence" but as you noted this can be confusing to the reader since we have a sequence of sequences, which is probably why "enumeration" was used. Your suggestion of "list" may be the best alternative even though it's not a term used in mathematics.

Georg Cantor's first set theory article was published in 1874 and contains the first theorems of transfinite set theory, which studies infinite sets and their properties. One of these theorems is "Cantor's revolutionary discovery" that the set of all real numbers is uncountably, rather than countably, infinite. This theorem is proved using Cantor's first uncountability proof, which differs from the more familiar proof using his diagonal argument. The title of the article, "On a Property of the Collection of All Real Algebraic Numbers," refers to its first theorem: the set of real algebraic numbers is countable.

On a Property of the Collection of All Real Algebraic Numbers is Georg Cantor's first set theory article. It was published in 1874 in Crelle's Journal, and it contains the first theorems of transfinite set theory, which studies infinite sets and their properties. One of these theorems is "Cantor's revolutionary discovery" that the set of all real numbers is uncountably, rather than countably, infinite. This theorem is proved using Cantor's first uncountability proof, which differs from the more familiar proof using his diagonal argument. The title of the article refers to its first theorem: the set of real algebraic numbers is countable.

On a Property of the Collection of All Real Algebraic Numbers is Georg Cantor's first set theory article. It was published in 1874 in Crelle's Journal, and it contains "Cantor's revolutionary discovery" that the set of all real numbers is uncountably, rather than countably, infinite.. This theorem is proved using Cantor's first uncountability proof, which differs from the more familiar proof using his diagonal argument. The title of the article, "On a Property of the Collection of All Real Algebraic Numbers," refers to its first theorem: the set of real algebraic numbers is countable.

Cantor's article also contains a proof of the existence of transcendental numbers. As early as 1930, mathematicians have disagreed on whether this proof is constructive or non-constructive. Books as recent as 2014 and 2015 indicate that this disagreement has not been resolved. A careful study of Cantor's article will determine whether or not his proof is constructive. Cantor's correspondence with Richard Dedekind shows the development of his ideas and reveals that he had a choice between two proofs, one that uses the uncountability of the real numbers and one that does not.

Historians of mathematics have examined Cantor's article and the circumstances in which it was written. For example, they have discovered that Cantor was advised to leave out his uncountability theorem in the article he submitted; he added it during proofreading. They have traced this and other facts about the article to the influence of Karl Weierstrass and Leopold Kronecker. Historians have also studied Dedekind's contributions to the article, including his contributions to the theorem on the countability of the real algebraic numbers. In addition, they have looked at the article's legacy—namely, the impact of the uncountability theorem and the concept of countability on mathematics.