wtf Posted July 24, 2023 Posted July 24, 2023 (edited) @JeffJo I have reviewed an English translation of Cantor's 1891 CDA paper here: https://www.jamesrmeyer.com/infinite/cantors-original-1891-proof.php I find it impeccable. There is nothing wrong with it. I agree with its methodology and conclusion completely. I have no idea what your objection is. Also, I was never able to actually find the translation that you and @phyti are discussing. Perhaps there's a problem with the translation and not the original paper. As I say, the translation I linked is a perfect and beautifully simple proof. It shows that any arbitrary list of infinite symbols strings is necessarily missing an element; and that therefore there is no list of all such strings. That's the proof. Edited July 24, 2023 by wtf
JeffJo Posted July 24, 2023 Posted July 24, 2023 (edited) 10 hours ago, wtf said: @JeffJo I have reviewed an English translation of Cantor's 1891 CDA paper here: https://www.jamesrmeyer.com/infinite/cantors-original-1891-proof.php And my understanding is that the one at ... http://www.logicmuseum.com/cantor/diagarg.htm ... is closer to the original. This is because I have seen another translation that differed in only about five words. True, Meyer's has a more modern feel. But regardless, this is exactly what I have been summarizing the entire thread. And, there is one detail that I find important that Meyer leaves out. 10 hours ago, wtf said: I find it impeccable. There is nothing wrong with it. I agree with its methodology and conclusion completely. I have no idea what your objection is. Where have you gotten the impression that I object to the methodology, or conclusion? All I have done is properly use the name "CDA" for the only part that, well, diagonalizes anything. The part between where Meyer's paraphrasing says "This follows from the following proposition" and "From this proposition it follows immediately." And then I call what "follows immediately" the corollary. Do you find anything less than "impeccable" about those labels? That CDA is Meyer's "this proposition" and the corollary is what "follows immediately"? I use this separation, because only the part I call CDA uses diagonalization, and conclusions that "follow immediately" from another are usually called corollaries. Or did Turing change that, too? I've even explained this, and summarized those two parts for you, more than once. I do get the impression that you don't read my objections, and only react to the fact that I objected to something in what you said. What I did object to, was where you said the content of CDA - the "proposition" Meyer described - was "uncountability." That is not mentioned in the "proposition" that I call CDA. In fact, it requires a countable set. What I did object to, in your characterization of Russell, was saying he added a "howler." He did add something that wasn't necessary, but it is not the restriction you keep saying it is. He only required that the characters au,v should be found "in some determinate manner." He did not define what that meant. Specifically, Turing's clarifications about "laws" do not apply to it. Because Russell did not require a "law." And Turing was clarifying what "law" should mean. Not defining what it did mean in 1903. Russell did give one example of "determinate manner. " It includes "the first p terms of Ep might be m’s, the rest all w’s." We can tell that this is an example, not a requirement, by reading the words "for example." He did not say that a "law" must be used. He said "Or an other law might be suggested." That's still as an example. How do you get "requirement" from "might" and "suggested"? 10 hours ago, wtf said: [The Meyer version] shows that any arbitrary list of infinite symbols strings is necessarily missing an element. And I said the same thing in "CDA: Any countable subset of M, the set of all infinite-length binary strings, necessarily omits a string E0 that is in M." Mine strengthens two points: I emphasized that "countable" is a necessary property of the list. Yes, it should be implied by "list." I made it explicit that the list is a subset, not the entire set. This is the part that phtyi is missing, and that is taught incorrectly. This is where Meyer departs from Cantor. Cantor said the rows of the table are "elements of M." Not "all elements of M." Russell said "any denumerable collection of E’s", not merely a table of characters. Meyer merely said "assume that there are". And the reason I keep pointing out these details, that I admit are implied in Meyer, is because they are the details that people who doubt CDA don't see. 10 hours ago, wtf said: And that therefore there is no list of all such strings. That's the proof. And I said the same thing in "Corollary: M is uncountable." Edited July 24, 2023 by JeffJo
JeffJo Posted July 24, 2023 Posted July 24, 2023 13 hours ago, wtf said: Correct. He never did, because he knew better. I thought YOU were suggesting that, but now you say you are not. So what is the argument about L, L1, L2, E0, E1, etc.? If I misunderstood, please clarify the argument so I can understand it. Cantor uses the set of all infinite-length binary strings, in the form he called En. He calls this set M. He also says that the proof does not utilize the properties of irrational numbers, something his 1875 proof was criticized for. This is why I do not use numbers, especially if the arguments resort to properties (like yours do). Turing has defined "Law" in such a way that the subset of M that can be generated by Turing's Laws is countable. Call this set T. Russell did not refer to T; in fact, he could not, since he did not know Turing's definition. Nor did he know that attempting a definition might make the set countable. I'm saying that this is irrelevant. RUSSELL DID NOT REQUIRE "LAWS" BY ANY DEFINITION, LET ALONE TURING"S "LAWS." He suggested that a law might be used to generate strings in what he called "some determinate manner." All he required was "some determinate manner." Call this set D. He clearly did this so that, for string En, he could say whether an,n was an "m" or a "w", rather than either an "m" or a "w". But we can make a list of T and a apply diagonalization to it to get a new string. Russell clearly thinks this process is using "some determinate manner" for generating a string. So what Russell would think, if he knew of Turing's work, and I'm not bothering to learn LATEX so "<" means a subset, is T < D <= M All I'm trying to say is that Russel's "oopsie" is in not recognizing the possible difference between D and M. Not between T and D. And that if this ill-defined D is a strict subset of M, it is uncountable by the proof being discussed. I'm not saying that D is well defined by modern standards, or that Russell needed this step. I'm only saying that it doesn't invalidate the proof, and isn't a "howler."
phyti Posted July 25, 2023 Author Posted July 25, 2023 wtf; The natural numbers are computable. Given any natural number n, we have an algorithm ("keep adding 1 till you get where you're going") that reaches n in finitely many steps. The algorithm is not required to generate ALL the natural numbers at once; There is no problem with finite sets. I'm referring to algorithms for producing infinite sets.Pi will always be an approximation since it has no last position. The approximation works if it is within specifications. There is also a real world limit on what is achievable. If material can't be sized to less than say + or - an atom, specifying dimensions with more precision is useless. No one will ever see a complete (all possible) list of integers, since there is no largest, thus no last element. 'Complete' (beginning and ending), is verified by boundaries. 'Infinite' is literally without a boundary. Cantor was trying to do the impossible, assign a measure (cardinality) to a set that is immeasurable. The sequences of symbols must be unique to avoid counting duplicates which is cheating, when comparing how many. As for 'well ordered', 'determinate', I offer this. From Cantor on Set Theory: [para.2] A well-ordered set is a well-defined set in which the elements are bound to one another by a determinate given succession such that (i) there is a first element of the set; (ii) every single element (provided it is not the last in the succession) is followed by another determinate element; and (iii) for any desired finite or infinite set of elements there exists a determinate element which is their immediate successor in the succession (unless there is absolutely nothing in the succession following all of them). Two "well-ordered" sets are now said to be of the same Anzahl (with respect to their given successions) when a reciprocal one-to-one correlation of them is possible such that, if E anf F are any two elements of the one set, and E1 and F1 are the corresponding elements of the other, then the position of E and F in the succession of the first set always agrees with the position of E1 and F1 in the succession of the second set (i.e. when E precedes F in the succession of the first set, then E1 also precedes F1 in the succession of the second set). This correlation, if it is possible at all, is, as one easily sees, always completely determinate; and since in the widened number sequence there is always one and only one number alpha such that the numbers preceding it (from 1 on) on the natural succession have the same Anzahl, one must set the "Anzahl" of both these "well-ordered" sets directly to alpha, if alpha is an infinitely large number, and to the number alpha-1 which immediately precedes alpha, if alpha is a finite integer. ----------------- So what value does alpha represent, and why can't it accommodate a number increased by 1? The red phrase connects with Wittgenstein's quote, "Where the nonsense starts is with our habit of thinking of a large number as closer to infinity than a small one".
wtf Posted July 25, 2023 Posted July 25, 2023 (edited) On 7/24/2023 at 4:55 AM, JeffJo said: And my understanding is that the one at ... http://www.logicmuseum.com/cantor/diagarg.htm ... is closer to the original. Ok thanks for the link. Now I reiterate my point that Cantor's original proof even in this translation is fine. I note the following phraseology: "If E1, E2, …, Ev, … is any simply infinite [einfach unendliche] series of elements of the manifold M, then there always exists an element E0 of M, which cannot be connected with any element Ev." Cantor uses the phrase "simply infinite series," by which I think you and I would agree that he mean what we would call a "list"; that is, a countably infinite well-ordered set having the order type of the naturals 0, 1, 2, 3, ... From this I draw two conclusions: 1) In the translation you gave me, I still think Cantor has made an impeccable, there's that word again, argument; and 2) I reiterate the counterproductive nature of using 19th century academic papers in translation, rather than contemporary versions of the argument. The problem is that we end up arguing over how to interpret archaic phrases that today have much more precise technical meanings; and we end up not discussing the actual mathematical argument in question. If we wanted to complain, we could note that when Cantor says manifold, he means what we call a set; and that a manifold is something else entirely these days. And even that what Cantor meant by a set was an entirely different animal than what we think of as a set today! Cantor had not seen Russell's paradox nor ZFC let alone modern set theory. So why are we studying Cantor's original paper at all, when we really want to discuss the ARGUMENT ITSELF? Is this thread about the history? Or the math? Maybe that is the question. To the extent that this thread is about trying to interpret translations of 140 year old papers rather than work from modern sources, we end up arguing about the "original intent" of the authors, rather than the mathematics itself. On 7/24/2023 at 4:55 AM, JeffJo said: Where have you gotten the impression that I object to the methodology, or conclusion? Because you've been bringing it to my attention repeatedly. You keep trying to tell me that you have objections to some particular translation of Cantor's 1891 paper, and I say ok, I haven't seen the paper but if you say so, ok. And then next post you complain again that I am not sufficiently engaging with your point. I conceded your point two posts ago. But now that you motivated me to look at translations of the original argument, I really don't see the problem. You want to rephrase the argument so that it's more clear to yourself, and that's fine. But the original arguments in both translations look fine to me too. I just don't place the vital importance on this issue than you do. You want to emphasize that the list in question is countable; and Cantor says it's a "simply infinite series." And to my mind, those are the same thing. That's what Cantor meant. It seems to be important to you that Cantor didn't use the word countable. I can't do anything about that. It's important to you, it's not important to me. I'm fine leaving it at that. I think "simply infinite series" is perfectly clear, if you map it through the 19th century academic German prose filter. I just don't see the issue. And I think that the entire issue could be avoided if you would just start with a more modern exposition, instead of trying to parse the original words of Cantor. So I can't help seeing this whole thing as a tempest in a teapot. Russell's teapot, as it were. On 7/24/2023 at 4:55 AM, JeffJo said: All I have done is properly use the name "CDA" for the only part that, well, diagonalizes anything. The part between where Meyer's paraphrasing says "This follows from the following proposition" and "From this proposition it follows immediately." And then I call what "follows immediately" the corollary. Ok. This is what I understand you to be saying. Step 1: Any arbitrary countable set (or "list," or "simply infinite series") of infinitely long symbol strings is necessarily missing an element, and is therefore not the entire collection of infinitely long symbol strings. Step 2: Therefore the entire collection of infinitely long symbol strings is not countable. Now I am casually saying "That's the argument," and you are saying, "No, Step 1 is the argument and Step 2 is the corollary." Have I got that right? If so, I concede the point. It's utterly trivial. I totally don't get why you think this matters to anything. BUT if you do, then I accept that, and I am perfectly happy to totally concede the point. Is that ok? Or have I once again misunderstood you? If so, I don't really want to know On 7/24/2023 at 4:55 AM, JeffJo said: I've even explained this, and summarized those two parts for you, more than once. I do get the impression that you don't read my objections, and only react to the fact that I objected to something in what you said. Let me see if I can put it in context. It's not that I don't read your objections, it's that I think the point you are making is entirely trivial, and I have a very low interest in engaging with it. So every time I tell you that I'm not particularly interested in what Cantor originally wrote, since to discuss the CDA it's better to work from modern sources; you then object that I'm not engaging with your point. But I am engaging with your point. I'm telling you that I haven't got a high interest in engaging with it. To study the CDA study modern treatments. That's my advice. AND, to the extent that I have now looked at two different translations of the 1891 paper, I personally find them perfectly satisfactory from a mathematical point of view. And you don't. That's ok. Let's agree to disagree. I find it an unbelievably minor an unimportant point. But I confess I am missing the point of your concerns. And in the end you may have to live with that. On 7/24/2023 at 4:55 AM, JeffJo said: Do you find anything less than "impeccable" about those labels? That CDA is Meyer's "this proposition" and the corollary is what "follows immediately"? I use this separation, because only the part I call CDA uses diagonalization, and conclusions that "follow immediately" from another are usually called corollaries. Or did Turing change that, too? Ok my Step 1 is the argument and Step 2 is the corollary. If this point is important to you, I concede it utterly. What was that Turing remark about? Did I say something wrong? There's been 130 years of progress in mathematics, logic, and computer science since 1891. That's why it's so futile to try to have a mathematical conversation based on a paper that is now of immense historical value, but can be confusing or misleading from our modern viewpoint. On 7/24/2023 at 4:55 AM, JeffJo said: What I did object to, in your characterization of Russell, was saying he added a "howler." I reiterate that from a modern viewpoint, what Russell wrote was a howler. And even viewed from the context of his own time, I have presented evidence that a fellow European mathematical philosopher, Brouwer, was thinking about "lawlike" and "lawless" sequences in published work as early as 1905 and no later than 1913; and that's possible that Russell knew about Brouwer's work. It's possible he didn't, too. What Russell mean by law, we can never know. I'm perfectly willing to say that it was a harmless inaccuracy in light of what we know today. You want to argue with my calling it even a harmless inaccuracy. You want me to deny that Russell, a professional philosopher, was so careless with his words. I call it a howler, you have a different opinion. Is that ok with you? It's ok with me. I've explained my reasoning several times already so I'll refrain from repeating myself. If you don't like the word howler, just say to yourself, "I hate the word howler." Then let it go. Please. On 7/24/2023 at 4:55 AM, JeffJo said: He did add something that wasn't necessary, but it is not the restriction you keep saying it is. In the light of how we view laws today, as algorithms, it's a fatal flaw in Russell's version of Cantor's argument. In the light of 1903, who the heck knows what Russell meant. Brouwer certainly knew the difference between a sequence that was lawlike versus one that was lawless. And he was a near-contemporary of Russell living on the same continent, if you consider England part of the Continent So I've made a point, I've given some evidence. I rest my case. Using "law" is a howler, it totally blows up the argument. That's just how I see it. I'm ok that you see it differently. On 7/24/2023 at 4:55 AM, JeffJo said: And I said the same thing in "CDA: Any countable subset of M, the set of all infinite-length binary strings, necessarily omits a string E0 that is in M." Mine strengthens two points: This is already clearly expressed in the two translations I've looked at. On 7/24/2023 at 4:55 AM, JeffJo said: And I said the same thing in "Corollary: M is uncountable." Step 1, Step 2. I'm fine with that. Argument/corollary. You know in math books this happens all the time, one book's definition is another's theorem and vice versa. It doesn't matter to the underlying ideas. On 7/24/2023 at 7:13 AM, JeffJo said: Cantor uses the set of all infinite-length binary strings, in the form he called En. He calls this set M. He also says that the proof does not utilize the properties of irrational numbers, something his 1875 proof was criticized for. This is why I do not use numbers, especially if the arguments resort to properties (like yours do). That's fine. But if the symbols are "0" and "1" instead of e and w, and we interpret the strings of 0's and 1's as binary representations of reals (between in the unit interval, ie all strings have an implied binary point to the left) then we've mapped symbols to real numbers. So it makes absolutely no difference. But why the interest in Cantor's original formulation? As you point out, he was trying to answer objections that nobody has anymore. Why not talk about CDA from the modern viewpoint? It's so much clearer, we don't get into these semantic confusions. On 7/24/2023 at 7:13 AM, JeffJo said: I'm saying that this is irrelevant. RUSSELL DID NOT REQUIRE "LAWS" BY ANY DEFINITION, LET ALONE TURING"S "LAWS." I get that this is your opinion. Mine is that Russell committed an egregious howler. We have different opinions. Did I mention Russell committed a howler? On 7/24/2023 at 7:13 AM, JeffJo said: I'm only saying that it doesn't invalidate the proof, and isn't a "howler." Howler howler howler. Totally blows the proof out of the water. 6 hours ago, phyti said: Pi will always be an approximation since it has no last position. The approximation works if it is within specifications. There is also a real world limit on what is achievable. If material can't be sized to less than say + or - an atom, specifying dimensions with more precision is useless. Pi is exact. It even has a finite representation via the Leibniz series https://en.wikipedia.org/wiki/Leibniz_formula_for_π pi/4 = 1 - 1/3 + 1/5 - 1/7 + ... I'm having trouble getting the closed-form formula to render today, but you can find it on the Wiki page I linked. It exactly expresses pi in 14 symbols. 6 hours ago, phyti said: As for 'well ordered', 'determinate', I offer this. From Cantor on Set Theory: Why are you trying to understand a somewhat difficult subject like well ordering by studying archaic texts? May I suggest Wikipedia? https://en.wikipedia.org/wiki/Well-order A set is well-ordered if it's a totally ordered set in which every nonempty subset has a least element. 6 hours ago, phyti said: if alpha is an infinitely large number, and to the number alpha-1 which immediately precedes alpha, if alpha is a finite integer. ----------------- So what value does alpha represent, and why can't it accommodate a number increased by 1? You asked me this a couple of weeks ago. What Cantor means is a transfinite ordinal. https://en.wikipedia.org/wiki/Transfinite_number The transfinite ordinals are an interesting subject, and we could talk about them, but not in this thread. And in any event, please learn math from contemporary sources. It's incredibly difficult to try to understand the transfinite ordinals from Cantor's original writings. Edited July 25, 2023 by wtf
JeffJo Posted July 26, 2023 Posted July 26, 2023 18 hours ago, wtf said: I reiterate my point that Cantor's original proof even in this translation is fine. I never said it was anything other than "fine." The point is that what is usually taught as CDA is this: Assume that a list of the entire set is possible. Prove that any list is missing an element E0. So that wasn't the entire set, as assumed. (Revisit this below) This contradiction disproves the assumption. This is invalid as a proof by contradiction, since the "assumption" that the entire set was in the list is not used to derive the supposed contradiction. So it isn't a contradiction, but it is a valid conclusion if you ignore the claim that it was assumed. But the actual proof is: For any countable subset that exists. Prove that this countable subset is missing an element E0. It follows that the entire set is uncountable, since E0 would have to both be in, and not be in, the set. And the reason this is important, is that most objections to the proof revolve around the point I asked you revisit. The "missing" string can be put into the list, so it isn't a contradiction. These doubters can't be reasoned with as long as they misinterpret the proof, and phyti is an example. He won't let go of that point. The reason I keep stressing that the "content" of CDA is not uncountability, is that one way to characterize this misinterpretation is that CDA actually depends on the argument itself being about a countable subset of M. Which I told you many times; you were also holding on to a misinterpretation of the argument, and only changed it when you read the logicmuseum translation. You were distracted from it by your objection to the Russell translation, missing the points that (A) Russell never required a Law but (B) translated every other part faithfully, including the removal of words like "manifold" that you object to. Which was my point in quoting it. 19 hours ago, wtf said: If the symbols are "0" and "1" instead of e and w, and we interpret the strings of 0's and 1's as binary representations of reals (between in the unit interval, ie all strings have an implied binary point to the left) then we've mapped symbols to real numbers. So it makes absolutely no difference. The difference is: We need some gymnastics for that mapping to become a bijection, since in the trivial mapping "100000..." and "011111..." both map to "1/2." Many of the consequences that one could be tempted to put forward will deal with actual properties of numbers, which Cantor explicitly said have no part in the proof. And he is right. There is a better analogy. Let the 0's and 1's be "false" and "true." Each infinite string of 0s and 1s then defines a unique subset of the natural numbers. This way, CDA is an example, for infinite sets, of Cantor's Theorem. That the power set of any set A has greater cardinality that set A itself. The Theorem is trivial for finite sets, but it as important to show that it is true for an infinite set.
phyti Posted July 27, 2023 Author Posted July 27, 2023 ; Cantor shows an example of 3 E-sequences. E1=(m, m, m, m, ...) E2=(w, w, w, w, ...) E3=(m, w, m, w, ...) ... Cantor then assigns a (row, column) coordinate system (u,v) to the array of E-sequences. The character at (1, 1) is an element of E1 The character at (2, 2) is an element of E2 The character at (3, 3) is an element of E3 ... Cantor then defines E0=(b1, b2, b3, ..) where bn is the negation of the same coordinates already assigned to the array of E-sequences. The trivial conclusion, the diagonal coordinates (v,v) cannot be both m and w simultaneously. His contradiction, that a thing cannot be in two mutually exclusive states simultaneously is true. A thing can't be in two different places at the same time. The contradiction is his own creation, by manipulation of a sequence that excludes (prevents) the appearance of others. It’s not an error or inability that causes any sequence to be missing. --------------------------- Since the array of E-sequences is random, there are at least as many arrays/configurations as sequences. For the example there are 6, (123, 231, 312, ...). Applying the cda method to each would mean more sequences are missing than there are existing! There is also the vertical argument, where a random column V is selected, and each character is swapped for its negation. This forms V0 which differs from every E where they intersect. In this case there are as many V0's as sequences, resulting in no array!
wtf Posted July 28, 2023 Posted July 28, 2023 (edited) On 7/26/2023 at 11:58 AM, JeffJo said: I never said it was anything other than "fine." This was in reference to the original Cantor 1891 proof. I confess I've been seeing you point me to this proof (in translation), then complain about the proof. But if you've been pointing me to the proof but complaining about some OTHER version of the proof, then I have misunderstood you. On 7/26/2023 at 11:58 AM, JeffJo said: The point is that what is usually taught as CDA is this: Assume that a list of the entire set is possible. Prove that any list is missing an element E0. So that wasn't the entire set, as assumed. (Revisit this below) This contradiction disproves the assumption. This is invalid as a proof by contradiction, since the "assumption" that the entire set was in the list is not used to derive the supposed contradiction. So it isn't a contradiction, but it is a valid conclusion if you ignore the claim that it was assumed. This is a proof by contradiction, which was not Cantor's original version. And I agree with you that many students and people one sees online are confused by the proof by contradiction. I disagree that it's invalid. It's perfectly valid. You assume that you have a complete list, then it turns out you don't, so there is no complete list. I see no problem with that. But I do agree with you that the direct proof, starting from an arbitrary list, is more clear. Note that in this context "list" implies countability by definition. A list is a countable set with the well-order type of the usual order on the natural numbers. On 7/26/2023 at 11:58 AM, JeffJo said: But the actual proof is: For any countable subset that exists. Prove that this countable subset is missing an element E0. It follows that the entire set is uncountable, since E0 would have to both be in, and not be in, the set. You call that the actual proof. I call it the direct form, as opposed to the proof by contradiction. I regard both proofs as equally valid. You haven't convinced me that the proof by contradiction is not actually a proof. Of course the direct proof is exactly the form of the argument that Cantor has given us. He used the phrase, "simply infinite series," by which I understand him to mean what we would today call a list, or a countably infinite set with the order type of the usual order on the natural numbers. On 7/26/2023 at 11:58 AM, JeffJo said: And the reason this is important, is that most objections to the proof revolve around the point I asked you revisit. The "missing" string can be put into the list, so it isn't a contradiction. These doubters can't be reasoned with as long as they misinterpret the proof, and phyti is an example. He won't let go of that point. Well, you could never have evidence for that. By "most objections" do you mean the kind of confusion one sees on Reddit, or on various math forums like this one? Or objections made by the legion of Cantor cranks online? Either way, it really doesn't mean anything. Many people are confused about many things online. Mathematicians aren't confused about the CDA in either form. I haven't read @phyti's posts in detail nor followed most of this thread, so I can't comment. As I've mentioned, I've seen his arguments on other forums going back several years, and it's possible that I've engaged with his argument previously. Or perhaps not, I can't really say. Either way, I can't comment on his ideas as expressed in this thread. On 7/26/2023 at 11:58 AM, JeffJo said: The reason I keep stressing that the "content" of CDA is not uncountability, is that one way to characterize this misinterpretation is that CDA actually depends on the argument itself being about a countable subset of M. Well, it really doesn't. If we assume that M, the collection of all possible strings, may be put into a list, we find that the antidiagonal is not on the list, and therefore M was not the entire collection of all possible strings after all. QED in my opinion, but I gather you don't agree. On 7/26/2023 at 11:58 AM, JeffJo said: Which I told you many times; you were also holding on to a misinterpretation of the argument, and only changed it when you read the logicmuseum translation. I have never had any misinterpretations of the CDA. Never. You deeply misconstrue my words and my meaning if you believe that. I changed nothing. I originally thought you were complaining about Cantor's CDA formulation. When I finally read it, I saw that it expressed the exact same argument that you call the only correct version. Leaving me more puzzled than ever about your point. I get that you think that only the direct proof, starting from a list, or a "simply infinite series," in the 19th century German formulation, is the correct and valid version. But that's the version Cantor gave! So what is the issue. I do agree that there is much confusion online regarding the CDA. But most of the confused people are cranks, so I don't know what value you find in complaining about them in the abstract. And, finally, the proof by contradiction version is perfectly valid, so I disagree with you about that. On 7/26/2023 at 11:58 AM, JeffJo said: You were distracted from it by your objection to the Russell translation, missing the points that (A) Russell never required a Law but (B) translated every other part faithfully, including the removal of words like "manifold" that you object to. Which was my point in quoting it. I was not distracted at all. I was initially totally uninterested in discussing the CDA, which vexed you mightily. When I finally did look at the translations, I was surprised to find that they were perfectly unobjectionable, being proofs starting from the countability assumption, ie a list, or a "simply infinite series." Let me defer talking about Russell till I get to the end of my replies. I have another strong piece of evidence that Russell knew exactly what he was saying, that he did intend to use the word law in the sense of an algorithm, and that in so doing, he was flat out wrong. I'll get to that later. On 7/26/2023 at 11:58 AM, JeffJo said: The difference is: We need some gymnastics for that mapping to become a bijection, since in the trivial mapping "100000..." and "011111..." both map to "1/2." Many of the consequences that one could be tempted to put forward will deal with actual properties of numbers, which Cantor explicitly said have no part in the proof. And he is right. There is a better analogy. I'll take your point about the dual representation problem. If we do the argument with bitstrings instead of real numbers that objection goes away. 0 and 1 are no different than e and w; and the fact that you put so much energy into insisting that there is a difference gives me pause. I imagine Cantor took pains to say it was "not about properties of numbers," because his original 1874 proof was based on the topological properties of the real number line, specifically the nested interval property. In 1874 that was a very subtle business. These days it's standard fare for undergrad math majors in their first real analysis class, so it's no longer controversial. Once again, I see confusion being introduced by referring to Cantor's 19th century work. Things that were confusing then are not confusing today. On 7/26/2023 at 11:58 AM, JeffJo said: Let the 0's and 1's be "false" and "true." Each infinite string of 0s and 1s then defines a unique subset of the natural numbers. This way, CDA is an example, for infinite sets, of Cantor's Theorem. That the power set of any set A has greater cardinality that set A itself. The Theorem is trivial for finite sets, but it as important to show that it is true for an infinite set. Ok. The CDA is a special case of Cantor's theorem. Agreed. To sum up: I think we agree that the direct proof is preferable to the contradiction proof. We agree that many people online are confused about the CDA, but many people online are confused about a great many things. Throwing rocks at the great unwashed masses of the Internet seems futile. Now to Russell. Russell is one of the authors (along with Whitehead) of the famous Principia Mathematica. In this volume, the authors endeavored mightily to show that mathematics could be derived directly from logic. https://en.wikipedia.org/wiki/Principia_Mathematica As we know, Gödel totally destroyed that hope, and showed that there must always be mathematical truths that are beyond the power of any axiomatic system to prove. Russell and Whitehead's book came out between 1910 and 1913, but it's fair to assume that Russell must have even in 1903 believed that math can be derived from logic. So in my opinion, his use of the word "law," along with his giving an example of what we would obviously today call an algorithm (p e's followed by all w's etc.) was actually a sly pitch for his hobby horse belief. Why else would he have inserted it, since even you agree it's irrelevant to the argument. If you say Russell didn't mean what he said, or used the word casually without thinking, you insult the great man. You are saying that Russell, one of the most famous of the twentieth century philosophers, used words carelessly, and did not intend to mean what he wrote. My position, on the other hand, is based on seeing Russell as someone who used words and mathematical symbols with incredible precision, and always wrote exactly what he meant to convey. He would have been in contact with other prominent European mathematical philosophers. He would have been aware of the question of whether everything in math was the logical consequence of "laws," or whether perhaps that was not the case. Russell believed passionately that everything in math was deducible from laws. Hilbert had that dream too. Gödel and Turing utterly destroyed that hope for all time. So in short, you think Russell was a dummy and didn't know what he was saying. I think Russell was very smart, very precise, very careful to write exactly what he meant, and aware of the intellectual questions of the day. When he said law, he was subtly nudging the reader to agree with his early logicism, in a context where that remark was entirely irrelevant to the discussion at hand. See https://en.wikipedia.org/wiki/Logicism for background on the idea that math can be reduced to logic. And one more point. You say that Russell only used the word "law" to make sure that the letters in the strings were "determinate." You agree with me that with the modern definition of functions, no law is necessary. A function that inputs a natural number and outputs the result of a random coin flip has a determinate result at each natural number; even though such a sequence of flips can never be generated by any law. It's my contention that Russell knew exactly what a mathematical function was; and if he didn't, he should have. Again, your argument rests on Russell being a dummy; mine rests on his being smart, extremely well-informed on matters of mathematical philosophy as they stood in 1903. and a precise user of words. When he said law, it's because he believed everything in math was reducible to laws. In Principia Mathematica he wrote 2000 pages in support of that belief. It's unreasonable to assume he used the word law without knowing exactly what he meant. Edited July 28, 2023 by wtf
JeffJo Posted August 2, 2023 Posted August 2, 2023 (edited) On 7/28/2023 at 12:10 AM, wtf said: This was in reference to the original Cantor 1891 proof. I confess I've been seeing you point me to this proof (in translation), then complain about the proof. But if you've been pointing me to the proof but complaining about some OTHER version of the proof, then I have misunderstood you. And this is why I think you haven't tried to see what I'm saying, since I never "complained about" that proof. There were complaints associated with how others view it and you would have seen that it you even tried. On 7/28/2023 at 12:10 AM, wtf said: This is a proof by contradiction, which was not Cantor's original version. ... You assume that you have a complete list, then it turns out you don't, so there is no complete list. I see no problem with that. Incorrect, it isn't a valid proof by contradiction. Correct, it was not Cantor's version. Incorrect, Cantor's corollary is a proof by contradiction. It isn't a proof-by contradiction because it never uses the part of the assumption that says every element of M is in the list. What if I assume that the moon is made of individually-wrapped packets of bleu and green cheese, in a ratio that is equal to the square root of 2. Then I show that this leads to a contradition. Did I prove that the moon is not made of green cheese? Or anything about cheese? No, because I didn't use any of that part of the claimed assumption. Simply saying "I assume this" is meaningless if it isn't used in the derivation. It literally can be struck from the so-called proof without changing anything except the claim that there is a contradiction. But it does form the basis of a proof. You could argue that contraposition implies B->not(A), so A and B can never be true at the same time. Or, as Cantor did, that a secondary proof-by-contradiction follows from it. But what people learn is invalid. On 7/28/2023 at 12:10 AM, wtf said: I'll take your point about the dual representation problem. If we do the argument with bitstrings instead of real numbers that objection goes away. 0 and 1 are no different than e and w; and the fact that you put so much energy into insisting that there is a difference gives me pause. And again, I never said that the two different sets of characters makes a difference. I have repeatedly said that the set using real numbers is different. "10000..." and "011111..." are different bit strings, but both are 1/2. The "properties of numbers" don't apply to the bitstrings. And the numbers do not represent the special case of Cantor's Theorem. On 7/28/2023 at 12:10 AM, wtf said: Let me defer talking about Russell till I get to the end of my replies. I have another strong piece of evidence that Russell knew exactly what he was saying, that he did intend to use the word law in the sense of an algorithm, and that in so doing, he was flat out wrong. And yet he still DID NOT REQUIRE THAT LAW, ALGORTITHM, OR ANYTHING LIKE IT. He only required "some determinant" method, and said it could be a law or algorithm. I agree that this is a fine point, but you are relying on that very fine point in your claim. Edited August 2, 2023 by JeffJo
wtf Posted August 3, 2023 Posted August 3, 2023 (edited) On 8/1/2023 at 6:14 PM, JeffJo said: And this is why I think you haven't tried to see what I'm saying, since I never "complained about" that proof. There were complaints associated with how others view it and you would have seen that it you even tried. Ah, others. But I have already asked you in a previous post who these others are. Are they professional mathematicians? Earnest students? Online professional cranks? Or perhaps serious finitists, ultrafinitists, intuitionists, or constructivists, all of whom make serious, intellectually coherent objections to the standard view of mathematical infinity. I already asked you that question earlier, falsifying your claim that I "would have seen that if I tried." I did try. I asked you who these "others" are. You chose not to answer, but rather double down on your disappointment that I have not chosen to join you in your condemnation of these "others." Can you give me their names and addresses so that I can send them some epistolary opprobrium? Have you got their phone numbers so that I can leave them accusatory voicemails? "Yo mama so fat" jokes? And most importantly of all ... Are these "others" in the room with us right now? On 8/1/2023 at 6:14 PM, JeffJo said: Incorrect, it isn't a valid proof by contradiction. Correct, it was not Cantor's version. Incorrect, Cantor's corollary is a proof by contradiction. It isn't a proof-by contradiction because it never uses the part of the assumption that says every element of M is in the list. Of course it's a valid proof by contradiction. On 8/1/2023 at 6:14 PM, JeffJo said: What if I assume that the moon is made of individually-wrapped packets of bleu and green cheese, in a ratio that is equal to the square root of 2. Then I show that this leads to a contradition. Did I prove that the moon is not made of green cheese? Yes. Yes yes yes yes yes. You have either proven that the moon is not made of green cheese, OR that your deduction system is already inconsistent. We usually make the implicit assumption that our deduction system does not lead to contradictions; and therefore that that if we introduce a new assumption that does lead to a contradiction, it's the new assumption that's false. Example: If 2 + 2 = 5, then I am the Pope. This is a valid statement, because it's of the form "False implies False." I see that perhaps you are unclear on material implication, and you should review this information before proceeding further in your erroneous reasoning. Now (stipulating, for purposes of this discussion, that I am not actually the Pope. After all we're all anonymous here and maybe I am the Pope. But for what follows, I stipulate that I'm not the Pope) ... so now, how can we turn this around to prove that 2 + 2 is NOT 5? Well, if 2 + 2 = 5, then I am the Pope. But I am not the Pope. Therefore, 2 + 2 is not 5. Note that there is no semantic connection or correspondence between the basic mathematical fact of 2 + 2, and the leadership of the Catholic church. There does not need to be any such semantic correspondence. So yes, in your example, we HAVE INDEED shown that the moon is not made of green cheese, IF it happens to be the case that assuming the moon is made of green cheese leads to some other contradiction, AND with the implicit assumption that our deduction system does not already contain contradictions. It is imperative for you to understand this. If 2 + 2 = 5 then I am the Pope. When you understand that, you will be enlightened. Now go in peace and sin no more. You can cos as much as you like. And don't tan for too long. On 8/1/2023 at 6:14 PM, JeffJo said: Or anything about cheese? No, because I didn't use any of that part of the claimed assumption. Simply saying "I assume this" is meaningless if it isn't used in the derivation. It literally can be struck from the so-called proof without changing anything except the claim that there is a contradiction. I have already explained this. If you don't understand material implication it's imperative that you do. If you assume that your deduction system is free of contradictions and you make a new assumption that does lead to a contradiction, you may conclude that the new assumption is false. The new assumption need not have any semantic or meaningful relation to the proposition in question. On 8/1/2023 at 6:14 PM, JeffJo said: But it does form the basis of a proof. You could argue that contraposition implies B->not(A), so A and B can never be true at the same time. Or, as Cantor did, that a secondary proof-by-contradiction follows from it. But what people learn is invalid. Who are these "people?" You are extremely unhappy that I'm not joining you in vitriolic condemnation of "people" whom you will not identify by name or characteristic or membership in some subversive organization. Did you miss a post or two ago when I asked you to tell me who these people are that you are angry with? Are they professors or students or cranks? You want me to be angry at someone, I ask again: Who are these people you want me to be angry at? You won't say. You only claim that I'm not angry enough at them, whoever they are. Can't you see how silly all this is? I have no idea who you are upset at. Now of course as I've already mentioned, I've seen plenty of cranks online over the years. But I don't even know if those are the people you're angry at. You seem to be unhappy with anyone who thinks that the contradiction form of Cantor's argument is valid. But I'm one who thinks the contradiction form is valid. So you are not only angry with me, but you are angry that I am not angry at me. If I told you that I am angry, furious, cross, vexed, irritated, indignant, aggrieved, piqued, irked, and displeased at MYSELF for having the effrontery and sheer gall to believe that the contradiction form of Cantor's argument is valid, would that make you happy? Ok well I have my own address, I'll dash myself off an angry missive forthwith. On 8/1/2023 at 6:14 PM, JeffJo said: And again, I never said that the two different sets of characters makes a difference. I have repeatedly said that the set using real numbers is different. "10000..." and "011111..." are different bit strings, but both are 1/2. The "properties of numbers" don't apply to the bitstrings. And the numbers do not represent the special case of Cantor's Theorem. You keep doing this. I answered this objection in my PREVIOUS POST. I said ok, forget the binary interpretation, just do the argument for bitstrings. I even explained WHY Cantor was sensitive to the "properties of numbers" objection. And you turn around and make the exact same complaint as if I haven't already just answered it. On 8/1/2023 at 6:14 PM, JeffJo said: And yet he still DID NOT REQUIRE THAT LAW, ALGORTITHM, OR ANYTHING LIKE IT. He only required "some determinant" method, and said it could be a law or algorithm. I agree that this is a fine point, but you are relying on that very fine point in your claim. Russell wrote a 2000 page book claiming that all of mathematics could be reduced to the laws of logic. And he was a professional philosopher, in the business of the precise usage of words. It's unreasonable to believe that when he said he wanted to use a law so that the strings were determinate, he had no idea what he's saying. It's more sensible to assume he knew exactly what he was writing, and that he was making a subtle pitch for his philosophy of logicism. He actually gave a perfect example of an algorithm: 'p' instances of e followed by all the rest w's, for each positive integer 'p'. It's perfectly clear what he meant. He said he wanted a "law" so that the strings would be "determinate," and then he gave an example of an algorithm. His intended meaning could not be more clear; especially in light of his strong belief in logicism. It's unreasonable to imagine that he didn't mean to express his belief in the core thesis of what would become his 2000 page book. But never mind all that. To sum this up: 1: Understand material implication, so that you will understand proof by contradiction; and 2: Tell me exactly who these "others" are so I know who you want me to be angry with. Name and shame. If you want me to be angry with Fred Bloggs of Duluth, say so. Maybe I know the guy and I'm like, "Yeah I hate Fred Bloggs, he cheated me at cribbage, he still owes me a buck fifty." Maybe I'm already angry with some of these "others," if you'd only tell me who they are. Edited August 3, 2023 by wtf
JeffJo Posted August 3, 2023 Posted August 3, 2023 38 minutes ago, wtf said: Ah, others. But I have already asked you in a previous post who these others are. As I said before, anybody who doubts the result. Yes, they are cranks. Yes, I want to make this point clear to show why such cranks are wrong. 39 minutes ago, wtf said: Of course it's a valid proof by contradiction. I'm talking about the version that gets taught, the one that starts "Assume you have a list of all of these strings (or real numbers in [0,1])" and ends with "This contradicts the assumption that all of these strings is in the list." Take the word "all" out of it, and the steps of the proof all work the same except that no contradiction is found. This doesn't prove, yet, that it is impossible to have a complete list. But that proof isn't far. No, it really isn't a valid proof-by-contradiction this way. But as Cantor did it, without that word, the corollary is a valid proof by contradiction. It is imperative for you to understand this. 43 minutes ago, wtf said: [Russell] actually gave a perfect example of an algorithm: 'p' instances of e followed by all the rest w's, for each positive integer 'p'. It's perfectly clear what he meant. He said he wanted a "law" so that the strings would be "determinate," and then he gave an example of an algorithm. No, he said that an algorithm like that was an example of something that is "determinate." So was a Law. He did not require "determinate" to be either of those things.
JeffJo Posted August 4, 2023 Posted August 4, 2023 Proof by contradiction: Show that A-->C for some statements A and C. Show that A-->not(C) for the same statements. If A is true, then our mathematics is inconsistent. We don't want that, so we say that A is not true in this mathematics. Partial proof-by-contradiction. Mention that statement B could be true. Show that A-->C for some statements A and C. Show that A-->not(C) for the same statements. We can say that A is not true, by contradiction. We can't say anything about B. Simply saying that it could be true does not mean we have assumed it for the purposes of proof. We need to use it in the derivation of the contradiction. Not even that B can't be true in conjunction with A being true, since that is a null statement. No statement is true in conjunction with A being true. This is my "green cheese" example, where A="there is a rational number that, when squared, equals 2" and B="the moon is made of some mix of cheese." Not a proof by contradiction: Mention that statement B could be true. Show that A-->not(B). This is not a proof by contradiction, yet. This is not a contradiction, since need to use it in the derivation. It should be easy to extent this to "A and B cannot be true together." One way is that A-->not(B) means B-->not(A), so there can be no situation where both are true. This is a direct proof. Another is a secondary proof-by-contradiction, like Cantor did. The contradiction is not "B¬(B)" as is taught for CDA.
wtf Posted August 5, 2023 Posted August 5, 2023 (edited) On 8/3/2023 at 4:46 PM, JeffJo said: As I said before, anybody who doubts the result. Yes, they are cranks. Yes, I want to make this point clear to show why such cranks are wrong. Ok. And you are upset with me for not being angry at those people? Is that what this is about? I think we're near the end here, at least I am. But still. You have said multiple times that you're frustrated that I'm not sharing your displeasure with these unnamed people who "doubt the result." What exactly would you like me to do or say? On 8/3/2023 at 4:46 PM, JeffJo said: I'm talking about the version that gets taught, the one that starts "Assume you have a list of all of these strings (or real numbers in [0,1])" and ends with "This contradicts the assumption that all of these strings is in the list." And who are these bad teachers? I get that you don't like that version of the proof. You won't even link a single example. If if you do, I still won't agree with you, as we've been over this many times. I think we might have to agree to disagree. Nothing new's been said for a while. On 8/3/2023 at 4:46 PM, JeffJo said: Take the word "all" out of it, and the steps of the proof all work the same except that no contradiction is found. This doesn't prove, yet, that it is impossible to have a complete list. But that proof isn't far. I agree that it's a cleaner proof, but even with "all" it's perfectly fine. I understand you don't agree. I can live with that. On 8/3/2023 at 4:46 PM, JeffJo said: No, it really isn't a valid proof-by-contradiction this way. But as Cantor did it, without that word, the corollary is a valid proof by contradiction. It is imperative for you to understand this. I think we've stated our positions. On 8/3/2023 at 4:46 PM, JeffJo said: No, he said that an algorithm like that was an example of something that is "determinate." So was a Law. He did not require "determinate" to be either of those things. It's not possible to know what Russell was thinking, but he wrote what he wrote and it's reasonable to take him at his word, especially in light of his logicism. 2 hours ago, JeffJo said: as is taught Was this for me? I didn't look at it in detail. I did note that once again, you complained about these teachers but didn't name them or link to their offending expositions. You're flailing at strawmen. But you know, if you assume you have a list of all the strings then it turns out you don't, that's a perfectly good proof by contradiction. The best I can do is agree to disagree with you on this point. I think I'm done here, I have nothing to add. I've enjoyed our chat. I'm out of steam. Edited August 5, 2023 by wtf
phyti Posted August 5, 2023 Author Posted August 5, 2023 My disagreement is with Cantor's diagonal argument, his method of proving his transfinite/infinite set M has a greater cardinality than N the natural/counting integers. It doesn't depend on any class of numbers. It is based on Cantor's own writings. Enclosed is a revision of the analysis. Cantor's illusion.pdf
JeffJo Posted August 6, 2023 Posted August 6, 2023 (edited) On 8/4/2023 at 8:28 PM, wtf said: Ok. And you are upset with me for not being angry at those people? No, I tried to correct you when you made false statements about details you quite clearly had not read completely. And you continue to try to make it look like I said something wrong. On 8/4/2023 at 8:28 PM, wtf said: And who are these bad teachers? Google "Cantor's Diagonal Argument." If the first line of what they say is the proof contains the phrase "list of all <whatever>," that is a bad teacher. Hint: at least 99% do. On 8/4/2023 at 8:28 PM, wtf said: I agree that it's a cleaner proof, but even with "all" it's perfectly fine. No, it is not. And it is easy to demonstrate, but you have to try. You do not. To see it, look for any line in the proof that depends on the presence of that word, except the one that says "the assumption leads to a statement that contradicts 'all'." This isn't deep mathematics. On 8/4/2023 at 8:28 PM, wtf said: It's not possible to know what Russell was thinking,.. Exactly. So why do you keep making arguments that are based on what you think he was thinking? What he knew, or could have known, or you think he should have known is irrelevant. Only what the proof requires. On 8/4/2023 at 8:28 PM, wtf said: ... but he wrote what he wrote and it's reasonable to take him at his word. And his word was "determinate." An example is the algorithm, but no word said it must be one. Another example is a law, but again no word said it had to be one. Again, this isn't deep mathematics. On 8/4/2023 at 8:28 PM, wtf said: I didn't look at it in detail. As you keep saying. Yet you keep objecting as if you had. ++++++ On 8/5/2023 at 11:16 AM, phyti said: My disagreement is with Cantor's diagonal argument, his method of proving his transfinite/infinite set M has a greater cardinality than N the natural/counting integers. Your disagreement is wrong, because it relies on a misinterpretation of the proof. "This analysis shows Cantor's diagonal argument published in 1891 focused on the ordered set of characters within a sequence while neglecting the orientation of the sequence, resulting in his conclusion of a missing sequence. It did not "focus on" any particular "ordered set of characters." It defined the set M of all possible, infinite-length binary strings. It said that for any ordering (not just a specific one as you use) of a subset of M (not the entire set M, as you use), that this subset necessarily omits a constructable element E0 that is in M. It doe snot utilize, or depend on, orientatton in any way. Your disagreement is wrong. Edited August 6, 2023 by JeffJo
wtf Posted August 6, 2023 Posted August 6, 2023 5 hours ago, JeffJo said: ++++++ Thanks for plussing my comments. Won't be responding. All the best.
JeffJo Posted August 7, 2023 Posted August 7, 2023 (edited) 14 hours ago, wtf said: Thanks for plussing my comments. Won't be responding. All the best. That's a separator, but I think you know that. This forum does not seem to allow consecutive posts. And you don't respond to comments, you lecture on what you think must be wrong, even after admitting that you don't bother to understand what the points are. And those points are: Many people think that the outline of Cantor's second uncountability proof goes something like this: Assume that the entirety of some infinite set T (this is based on Wikipedia's naming conventions, since they name one that Cantor didn't) can be put into a bijection with the set of all natural numbers (starting with 1) N. That is, it can be listed t1, t2, t3, ... . Use any such bijection to construct a new element t0 that (A) must be in T but (B) is not in the list used in this construction. Since this contradicts the assumption that every element of T is in the list, proof-by-contradiction applies and the assumption is false. THIS IS AN INCORRECT APPLICATION OF PROOF-BY-CONTRADICTION. You have to use all parts of the assumption to derive the contradiction. This outline does not use the part of the assumption that the entirety of T is in the list. It does, however, prove directly that the entirety is not in the list. This is close to, but not quite, what Cantor was trying to prove in the end. Nearly every "crank" objection to CDA uses this flaw in some way. The crank in question in this thread, phyti, starts by using a different bijection than what was used in the construction. Then the missing element can be found in another "orientation" of a that bijection. A more common, but less sophisticated, objection is that t0 can always be added to the bijection in the manner of Hilbert's Hotel. There are others, like https://hesperusisbosphorus.wordpress.com/2016/07/31/what-is-wrong-with-cantors-diagonal-argument/ . But given your claimed familiarity, I didn't think it necessary to list them. Call me silly, but I believe that if we could get more self-proclaimed experts to acknowledge this flaw in how CDA is understood, we might get fewer cranks. But these experts have to get over getting toes stepped on first. The correct outline is this: Assume that S is any infinite subset of T that can be put into such a bijection. (Cantor's lone, but minor, flaw is that he should have given an example to prove that it is possible. That way he is working with bijections that are known to exist, not supposing they might.) Construct t0 the same way. It is in T but isn't in S. If S=T, we would have the contradiction that t0 both is, and is not, in T. This is the contradiction that Cantor used, and it is different than the one that cranks think they have debunked. Specifically, it is about t0, not T, Bertrand Russell updated the terminology, in English not German, from Cantor's 1890 paper. This is something you asked to see. It says almost the exact same things, in the exact same way, with two exceptions: Russell corrected the omission of an example. For that example, he suggested one specific algorithm. He also suggested there could be "laws" defining more examples. BUT THIS WAS ONLY AN EXAMPLE FOR THE FIRST PART OF THE PROOF. Russell never suggested an algorithm or law could define the entire set. Only that it could define a listable subset. Russell never REQUIRED such an algorithm or law to define the subset S, let alone T. He said they could define S. Russell's conclusion that T can't be listed - that is, the part I call the corollary - does not suppose a definition, an algorithm, or a law. Let alone require one. Your claim that Russell committed a "howler" is based on the same misunderstanding of the difference between the two proof outlines I described. The same misunderstanding that cranks use. And that you assert does not exist, but have not defended that assertion. I do not believe that you have even considered how they are different. I feel I am justified in thinking that, based on how you responded with equal force to things you later said you didn't look at. And running away from the issue seems to be proof of this. Edited August 7, 2023 by JeffJo
wtf Posted August 7, 2023 Posted August 7, 2023 (edited) 2 hours ago, JeffJo said: Call me silly I've refrained from making any personal remarks, preferring to concentrate on the math and on the mathematical philosophy of the early twentieth century, in particular Brouwer's intuitionism and Russell's logicism. But if you insist ... Edited August 7, 2023 by wtf
phyti Posted August 10, 2023 Author Posted August 10, 2023 Jeff; It did not "focus on" any particular "ordered set of characters." It defined the set M of all possible, infinite-length binary strings. It said that for any ordering (not just a specific one as you use) of a subset of M (not the entire set M, as you use), that this subset necessarily omits a constructable element E0 that is in M. It does not utilize, or depend on, orientatton in any way. The A-list is the conventional form for a list, one sequence per line or row. Cantor defined a different form D using the diagonal elements, one character from each sequence, from the A-list which becomes the unconventional form, the B-list. The blue characters in the A-list do not represent a sequence in that list, and he does not alter the A-list. He then forms E0 the negation of D, overlays it on the A-list as a horizontal sequence, and states it would conflict with D, if D was literally in the A-list. He sees what he wants to see to prove his point, that the A-list is missing E0. He ignores the fact; if E0 is missing, so is D, they exist in pairs as shown in the B-list. In the conventional A-list all sequences are randomly formed and entered randomly. Since they are all parallel, there is no interference of one with another, a property of random lists. Yet when Cantor mixes the diagonal with the horizontal sequences, it imposes restrictions on the locations, a form of order where there should be no order! His goal, Cantor states M has a different cardinality than N the integers. "I maintain now that such a manifold [Mannigfaltigkeit] M does not have the power of the series 1, 2, 3, …, v, …." Cantor would be wasting his time demonstrating a subset M1 has fewer elements than the set M, when it does so by definition. The 'ordered set of characters' is the order they have within a sequence. Ignoring the orientation, they appear the same, but they differ, and that is the problem. If you think E0 cannot exist as u83 in the A-list, explain why. Cantor fooled himself first then many others. It's the illusionist's form of misdirection, 'look over there (while I do something over here)'.
JeffJo Posted August 29, 2023 Posted August 29, 2023 On 8/10/2023 at 1:41 PM, phyti said: The A-list is the conventional form for a list, one sequence per line or row. Mathematics does no recognize a "conventional for for a list." What you describe is just a convenient graphical representation of bijection. It has no significance. On 8/10/2023 at 1:41 PM, phyti said: Cantor defined a different form D using the diagonal elements, Cantor defined a string in the exact same way every other string he uses is defined. Each is a function mapping N to {'m','w'}. You are twisting it to make statements that are not true. There is no "A-list" or "B-list" except in your misrepresentations. Cantor uses another function from N to his set M, of all such strings. There is no division. On 8/10/2023 at 1:41 PM, phyti said: He ignores the fact; if E0 is missing, so is D, they exist in pairs as shown in the B-list. He ignores nothing, because "If E0 is missing, so is D" is not true. 000000000000... 100000000000... 110000000000... etc. D is the first element here. On 8/10/2023 at 1:41 PM, phyti said: Cantor would be wasting his time demonstrating a subset M1 has fewer elements than the set M, when it does so by definition. Cantor never limits the set he uses to being a proper subset. The only restriction on what you call M1is that it is countable. You have proven nothing. You have thrown words at Cantor's proof that have nothing to do with it.
phyti Posted November 23, 2023 Author Posted November 23, 2023 ; Let's assume Cantor is correct about the missing E0. There are many possible random lists with different orderings of sequences. Eg. swapping the 1st k sequences with k randomly selected from other parts of the list. Thus, the number of lists > the number of sequences. Cantor forms a diagonal d and its negation E0, beginning with sequence 1. of a random list L1. Rename E0 as x1. Cantor forms a diagonal d and its negation E0, beginning with sequence 1. of a different list L2. Rename E0 as x2. Cantor forms a diagonal d and its negation E0, beginning with sequence 1. of a different list L3. Rename E0 as x3. Continuing to apply Cantor's method, there is a missing sequence x for every list. Thus the number of missing sequences > the number of sequences!
KJW Posted November 24, 2023 Posted November 24, 2023 (edited) On 7/11/2023 at 3:29 AM, phyti said: The beginning of an infinite random list, with one infinite sequence per line, and line 1 blank. A random sequence can be placed anywhere in the list except at the end, since an infinite list has no end, but we have access to its beginning s1 s2 010010... s3 101011... s4 111000... s5 000111... s6 011011... s7 111111... sx 110000... Using Cantor's diag. method, sx is not in the list. Put sx in line 1, now it is. Except that when you place sx in line 1, you've created a new list, requiring a new application of Cantor's diagonal method, generating a new sequence that is not in the new list (nor in the old list). Given any list of sequences of binary digits, Cantor's diagonal method generates from that list a sequence of binary digits that: differs from the first sequence of that list in the first position, differs from the second sequence of that list in the second position, differs from the third sequence of that list in the third position, ... differs from the n-th sequence of that list in the n-th position, differs from the (n+1)-th sequence of that list in the (n+1)-th position, ... etc and therefore differs from every sequence of that list (that is, not in that list). Edited November 24, 2023 by KJW
phyti Posted November 24, 2023 Author Posted November 24, 2023 KJW; Let's assume Cantor is correct about the missing E0. The purpose is to explore the results of his conclusion. There are many possible random lists with different orderings of sequences. Consider forming a finite list of 5 different sequences (abbrev. s). There are 5 s for line 1, 4 s for line 2, ..., 2 s for line4, and 1s for line 5. There are 5!=120 different lists. The content of each list is constant, with only a different order. I.e. there are more lists than sequences. If Cantor applies his diagonal method to each list, the number of missing sequences > the number of sequences! That qualifies as a contradiction.
KJW Posted November 24, 2023 Posted November 24, 2023 29 minutes ago, phyti said: If Cantor applies his diagonal method to each list, the number of missing sequences > the number of sequences! So? 29 minutes ago, phyti said: That qualifies as a contradiction. I don't think so. In fact, that the number of missing sequences is greater than the number of sequences is kind of what Cantor is proving. Although Cantor's diagonal method is only generating a single sequence missing from a list, because the number of sequences in a list is aleph-0, and the number of possible sequences is aleph-1, then the number of sequences missing from a list is also aleph-1. Cantor is proving that aleph-1 is greater than aleph-0. 1
phyti Posted December 9, 2023 Author Posted December 9, 2023 If we copy Cantor's method, we can expect to get the same results. That does not prove his concept of transfinite sets is correct, only consistent. The new analysis begins with an early objection regarding the geometric form of Cantor's list. Does the progression of finite lists lead to transfinite lists? Cantor's Illusion.pdfCantor's Illusion.pdf
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