wtf

Thanks for plussing my comments. Won't be responding. All the best.

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? 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. 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. I think we've stated our positions. 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. 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.

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? Of course it's a valid proof by contradiction. 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. 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. 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. 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. 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.

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. 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. 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. 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. 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. 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. 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. 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. 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.

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. 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. 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 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. 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. 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. 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. This is already clearly expressed in the two translations I've looked at. 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. 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. 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? Howler howler howler. Totally blows the proof out of the water. 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. 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. 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.

@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.

"Act in haste, repent at leisure." Ok, 1903. This will become relevant shortly. I will soon introduce evidence that as early as 1905, and no later than 1913, Brouwer introduced into mathematics the idea of "lawlike" sequences, that are defined by SEP as being generated by algorithms; and that it's entirely possible that Russell may have been familiar with these ideas that were "in the air" at the time. Or maybe he wasn't. But you can't accuse Russell, a famous philosopher, of using words imprecisely. You say, "But it translates, almost verbatim, what Cantor wrote." But it does NOT. It ADDS the concept of a "law" that determines the characters of the string, and that entirely invalidates Russell's claim to be explaining Cantor's proof. Cantor made no such mistake. I suspect Russell didn't really understand Cantor's argument. I honestly think that I've answered this point several times already, to you and to @studiot. Let me make this crystal clear. * I am perfectly satisfied with the correctness of the CDA and indeed Cantor's other two proofs of the uncountability of the reals, based on the modern treatments that I've seen. * I rarely get involved in "alternative" aka "skeptic" aka "crank" discussions of the subject. * I only mentioned to @phyti that he might benefit from looking at a contemporary account of the CDA. He responded by saying that he finds contemporary versions equally objectionable. That completed the conversation as far as I was concerned. * You replied to me, offering the Russell quote as a "better" explanation of the CDA. I pointed out that by requiring the strings to be generated by a "law," and then giving specific examples of algorithms, Russell underminded the CDA and missed the point entirely. And that although I could live with saying that Russell misspoke himself and that we should not be pedantic about the matter; your continued claim that Russell did not mean what he wrote, or even if he did, he can still prove the CDA, are wrong. * Finally, I am not actually familiar with Cantor's original paper on the subject, in German or in English translation, and I'm totally unable to offer any comment on it. * Therefore any attempts on your part to engage me in conversation regarding Cantor's original exposition is pointless. I have neither the knowledge nor the inclination to discuss the matter. If you say his original argument was informal, or imprecise, or lacking in some particular virtue, I will take your word for it and I won't bother to look any deeper. As I originally said to @phyti, the original paper introducing a revolutionary new idea is often more difficult and sometimes even error-prone, than the later, cleaned-up versions. I truly hope this is clear. Well the "almost" is the operative word. Russell specifies a law. A law is an algorithm, and was beginning to be recognized as such at the time of Russell's writing. Let me elaborate on this point. See Intuitionism in the philosophy of mathematics. A guy named Brouwer suggested that the only legitimate sequences were the ones given to us by laws or algorithms. From the SEP article: "A choice sequence is an infinite sequence of numbers (or finite objects) created by the free will. The sequence could be determined by a law or algorithm [my emphasis], such as the sequence consisting of only zeros, or of the prime numbers in increasing order, in which case we speak of a lawlike sequence, or it could not be subject to any law, in which case it is called lawless." As I say, Brouwer (according to the SEP article) published a philosophical article about his ideas as early as 1905, only two years after the Russell quote. And he published his first full-fledged paper on his mathematical ideas about intuitionism as early as 1913, only ten years after the Russell quote. I do not know if the British Russell and the Dutch Brouwer corresponded, directly or indirectly; or if perhaps these early ideas about intuitionism were "in the air" at the time, and known to Russell. It seems to me at least as likely that when Russell used the word "law," he did so deliberately, being a famous philosopher well up to speed on the intellectual trends of his time; as that he simply misspoke himself and did not mean to write what he clearly wrote. Here is Russell's quote as you initially wrote it to me: "... where the a’s are each an m or a w in some determinate manner. (For example, the first p terms of Ep might be m’s, the rest all w’s. Or an other law might be suggested, which insures that the E’s of our series are all different." You can see that the examples he give: "first p terms are m, the rest w ..." are obvious examples of algorithms. It's perfectly clear that he's thinking of algorithms. [Note: On rereading, I realize that he's saying he wants a SINGLE law to generate ALL the strings. This is truly hopeless. He can never prove the uncountability of the reals that way. This is even worse than I initially thought]. And as I believe you agreed earlier, the strings on the list need not even be different for the argument to go through. It seems clear to me that Russell did not actually grasp the subtleties of Cantor's argument, and made at least two mistakes. Indeed, even his claim that different laws must give rise to different strings is a mistake. Where is his proof that two distinct laws must necessarily give rise to distinct strings? For example the sequence of natural numbers given by the primes, and the sequence of numbers that satisfy Wilson's theorem, are the exact same sequence, even though at first glance the rules seem very different. https://en.wikipedia.org/wiki/Wilson's_theorem So I'd say Russell made three errors: One, requiring the symbol strings to be generated by laws; two, thinking the strings need to be distinct; and three, thinking that distinct laws always generate distinct strings. And even if these errors are (charitably) excused as not being material, they are at the very least confusing, because they are utterly irrelevant to the argument. In trying to "clarify" Cantor's argument, Russell has mangled it. Which is my entire point here. This is in reference to my understanding that you are starting with a countable list an adding one element at a time, in the hopes of (somehow?) proving the uncountability of the reals. I got this idea from your exposition about the L and L1 and L2 and E0 and E1 and the like. If I have misunderstood your argument, please state it more clearly. I also note that you did not respond to my challenge to prove your claim that you can prove the uncountability of the reals by starting with a list of all the computable reals, showing the list is incomplete, and thereby concluding that the reals are uncountable. That proves nothing of the kind! I have asked you for an explanation and you have as yet not supplied one. 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. You claim it's vague. i showed that the philosophical notion of a law being the same as an algorithm was "in the air" in the decade following Russell's quote, in the work of Brouwer. You say Russell didn't mean what he wrote. I say that a philosopher of Russell's stature would ALWAYS use words that meant exactly what he wanted to communicate, and that it's more probable that he meant what he wrote and simply made a technical error. We can never resolve this. I don't understand why you want to argue the point. I've stated several times that I'm perfectly willing to say, "Ok Russell made an oopsie, let's let it slide." But you don't even want to admit he made a mistake. You want to claim that by "law" he meant ... what, exactly? I can't imagine. SEP regards the terms law and algorithm as synonymous. I agree with that point. I just misunderstood your L, L1, L2, etc. argument, and still don't understand it, and I am requesting clarification. Where is your proof that you can restrict the CDA to start with the list of computable reals and end up proving the uncountability of the reals? You can't do that! At best you can prove only that there exists a noncomputable real number (or symbol string). 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; merely generate any arbitrary one in finitely many steps. That, it can clearly do. As another example, I noted earlier that the number pi is computable. Now clearly no algorithm can generate ALL of the digits in finitely many steps! But that is not what is required. Pi is computable because if you give me a natural number n, I can determine the n-th decimal digit of pi in finitely many steps. And I can do that for any n. So even though pi has infinitely many nonrepeating digits, each individual digit can be algorithmically determined in finitely many steps. That makes pi a computable real number.

Ok here are my specific responses to your post. I agree that I did interpret "law" as effective procedures, computation, algorithm, etc., as was already being done even in Russell's time, though I'm not sure if you gave the date of his quote. But I can't imagine what ELSE he could have meant. A law is a finitely-expressed, deterministic procedure for obtaining an output from an input. It's an algorithm, computation, effective procedure, etc. Just because he didn't think of this at the time doesn't mean it wasn't a material error. Just as we don't regard the believers in the phlogiston theory of heat as having been right, simply because they were well-intentioned and couldn't see into the future. From what we know now, they were simply wrong. As was Russell. Yes. It's determined in the sense that each digit position has some particular value in it. And since it's the antidiagonal of a list of ALL the computable numbers, it can't be computable. We agree on this. You really lost me by referring to "Russell's intended restriction" and mine. I have no idea what specific ideas you are referring to. It would be good to clarify that. So here you are using "in a determinate manner" in the wrong way. The antidiag of the list of all computable numbers is determined, but not law-determined. It's not computable. That's because, as I noted, the enumeration itself is not computable. This is exactly where your argument is going wrong. Now the trouble is that you can NOT conclude that anything is uncountable. Having found that the antidiag of the list of all computable numbers is not itself on the list, for all you know, the real numbers are countable. You can just keep adding new antidiags one at a time to your list and it's always countable. You are adding one element at a time to a countable list. The list is still countable. In other words, Russell and you have given up trying to show that the reals are uncountable. All you are showing is that there exists a noncomputable number (or string if you prefer). You have not shown anything is uncountable at all. Where is your argument? You made a claim but not an argument. You keep throwing one more item at a time into a countable set, and the resulting set is still countable. It's perfectly clear that when he says there is a law, he means algorithm, computation, whatever. I'm perfectly willing to agree that he had a lapse in judgment and we can ignore what he wrote. But I can not agree that, taken literally, his error is not material. His error entirely destroys the argument he's trying to make. If he doesn't mean rule, algorithm, finitary mechanical procedure, etc., then what else CAN he mean? Russell certainly knew what a function was. A function deterministically produces an output for each input; but it NEED NOT do so according to a "law." It's pointless to keep arguing about this. Nothing at all to do with the axiom of choice. You do not need choice to refer to an infinite string of symbols. Russell would surely have known that if he thought about it at all. Infinite strings of symbols exist because each one is a function from the natural numbers to the set of symbols. (Input 47, output the symbol in slot 47. It's a function from the naturals to the set of symbols). No choice needed for that. Perhaps you are referring to constructivism, the philosophy of math that says nothing exists unless it can be specifically (algorithmically) constructed. https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_mathematics) Constructivism is related to intuitionism. These ideas started in the 1920s. I see no evidence that Russell was an intuitionist. He would perfectly well know that you don't need a "rule" or a "law" to define a function. Now again I agree with you that he misspoke himself and we should just ignore it and move on. But you cannot insist that we take him literally and then claim it doesn't matter. Russell should have and most probably did know better. Or perhaps (I'm hazy on the history) the modern, set-theoretic understanding of function as a set of ordered pairs was not yet well-known to Russell. In that case he's with the phlogiston-believers. Not "Right by virtue of being unable to see the future," but "Wrong, no matter how noble their intentions." No that's simply false. The computable numbers are a subset of M, and we can show that the antidiagonal E0 is not computable, and all we've shown is that a noncomputable number exists. We have NOT shown that M is uncountable. I truly don't know why you keep claiming without proof that this shows M is uncountable, when it clearly does not. Look at it this way. You have the countably infinite set of computable numbers. You adjoin to that set one noncomputable number. The resulting set is still countable. You adjoin another countably infinite many noncomputable numbers, and the augmented set is still countable. How do you know you can't exhaust the reals this way, proving them (or M) countable? Where is your proof that M is uncountable? You haven't got one. You start from a countable set and keep adding elements, but you don't have a proof that you won't end up exhausting M that way, proving M is countable. To prove M uncountable, you have to start from an arbitrary set of strings and show that there's one missing. That shows there's no enumeration of all the strings. If you start from a specialized countable set like the computables, you lose the uncountability proof. ps -- I think I sort of see what you might be doing. You start with a countable set and you keep adding elements, one at a time. (What happens after you've added countably many? You seem to be reinventing the ordinal numbers). Now if at some point we claim we have ALL the reals, that's falsified by the antidiag. So at best you have proved that you can't reach all the reals starting with a countable set and adding one real at a time. This is a far cry from the CDA. pps -- I just wanted to say that I find talking about the noncomputable numbers far more interesting, and far more within my own wheelhouse of competence, than trying to figure out what Russell meant by laws or knew about the axiom of choice. I'm no Russell scholar by any means.

I will respond point-by-point to your post later. Before I do that, I want to give a little background, and then pose and solve a seeming Puzzler about the computability of the antidiagonal of L. In short, you are equivocating two different meanings of "determined," and that is causing your reasoning to go astray. First, as we agree, a computable real number is one whose decimal (or binary, makes no difference) digits can be cranked out by an algorithm. By algorithm we include words and phrases like "law," "rule," "effective procedure," "effective calculation," "computation," and the like. These are all terms that were used in the 1920s as mathematicians sought their holy grail, a mechanical process by which they could determine the truth or falsity of any given mathematical statement. That holy grail turned out not to exist, as Gödel and Turing and others of that era showed. Here is a Wiki article giving an overview of the subject. The vague concepts of law, rule, effective procedure, etc., were clarified once and for all by Turing in his famous 1936 paper, "On Computable Numbers, with an Application to the Entscheidungsproblem." Here's the relevant Wiki page. The key properties any such law, effective procedure, algorithm, computation, etc. must have is: (1) The rule, or algorithm, is finite. (2) It's entirely mechanical. The answer can be cranked out by a machine or rote procedure; and (3) It always ends in finitely many steps, giving an answer. Two examples illustrate the concept. 1. The number [math]\pi[/math] is computable. That's because there are lots of formulas for it, such as the famous Leibniz series pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 ... It's a beginning programming exercise to implement that formula in Python or Java or C++ or any other popular programming language. So despite the fact that pi is irrational and transcendental, and that its digits are often regarded as "random," they are nothing of the sort. The digits are completely deterministic, and a beginning programming student could crank out as many digits of pi as desired, subject only to physical limitations on computing resources. [In the definition of computability we ignore such physical constraints]. 2) Let's make up a completely random binary sequence that has no rule, law, algorithm, effective procedure, etc., to generate its digits. Suppose we generate the binary digits of a real number by flipping a hypothetical fair coin infinitely many times, once for each digit position to the right of the binary point. If you insist on realism, perhaps we can use the oddness or evenness of the low-order bit of the femotosecond timestamp of the next neutrino to hit our detector. Any mechanism that's as random as we can imagine in this world. We will generate some noncomputable real number. Its digits are still "determined," in the sense of a mathematical function. If you give me a natural number n, I'll tell you whether there's a 1 or a 0 in that position. But it's not computable. There's no mechanical rule or procedure that tells us what bit is at digit position n. So you see that there is an ambiguity in the use of the word "determined." Any mathematical function takes an input and produces an output. The output is therefore "determined" by the input. But the mapping from inputs to outputs may or may not be computable. I believe that in your reasoning, you are often equivocating between these two meanings of "determined." The output of a mathematical function is always determined by the input. That's the definition of a functional relationship. But the mapping is only sometimes computable; and other times, it isn't. Now with all this in mind, I'm going to present a Puzzler. We know (because there are at most countably many possible computer programs, given a finite or countably infinite alphabet, and given that any program is a finite string of symbols) that there are countably many computable real numbers. Therefore there is an enumeration of ALL the computable numbers. You have been calling such an enumeration L. Given such an enumeration, we can form the antidiagonal in a perfectly deterministic, rule-based manner, by the rule "whatever symbol is in row n, column n, replace it with a different symbol." Now, since the antidiagonal is clearly not in L; that is, it is not on our list, which contains ALL the computable reals; therefore the antidiagonal can NOT be computable. Simple enough. But wait! We DO have a mechanical procedure to generate the antidiagonal. Step one, enumerate the computable reals; and step two, form the antidiagonal according to the "change the symbol" rule. So the antidiagonal is computable after all! Paradox! Mystery! What happened? What is the answer? Well, the answer is kind of subtle, and interesting. There is indeed an enumeration of the computable numbers. But there is no computable enumeration of the real numbers. In other words in step 1, when we say, "Form an enumeration of the computable real numbers," we can do so mathematically. Such an enumeration exists. Given a natural number n, we can say what is the n-th computable number. But that enumeration is itself NOT computable. There is no law, no rule, no algorithm, no effective procedure, no computation that can crank out the list! So the antidiagonal is not "law-determined." It's mathematically determined, in the sense that there's some symbol in each digit position. But there is no mechanical procedure to crank out the digits of the antidiagonal, because there is no computable enumeration of the computable numbers. Without going into detail, the reason there can be no computable enumeration of the computable numbers is related to the unsolvability of the Halting problem. No computer program exists that can input an arbitrary program and determine whether that program will halt after a finite number of steps. Turing proved this in his 1936 paper. If we could mechanically list all the computable numbers, that would amount to being able to determine which algorithms halt, and Turing showed that we can't do that. I believe this clears up some of the ambiguity in your post. You said the antidiagonal of L is "determined," but it is not computably determined. It's determined as the output of a mathematical function, but NOT as the output of a computation, algorithm, rule, law, effective procedure, etc. I will leave this post as it is, and reply specifically to your most recent post a little later in the day. But I hope that I have already cleared up the ambiguity in your reasoning. The antidiagonal of L is not "determined" by a law or rule or algorithm. It is not, after all, a computable number.

I trust you are sincere. But you have gone astray here already. You say, ""Russell's restriction of the elements of the list" does not invalidate the construction of such lists." Correct. Agreed. We can certainly make a list of computable numbers. By using knowledge from a few decades after Cantor, we may note that the set of computable strings -- strings whose progress is determined by a law, which I take to be equivalent for purposes of this discussion to be an algorithm -- that set of computable strings is countably infinite. And therefore, we can place ALL the computable strings on a list. Now the antidiagonal isn't on the list. What is it? Since we listed all the computable numbers, the antidiagonal must be noncomputable! Now that's an achievement. Using a diagonal argument we've proven that there exists at least one noncomputable number. That's a great achievement in thought. If Russell had seen the implications of his own rhetorical inaccuracy, he'd have scooped Turing in the naming and discovery of noncomputable real numbers (or strings). But you see the problem. We CAN of course form lists of SOME and even ALL of the computable strings. But now the antidiagonal only furnishes proof of the existence of a noncomputable number. We completely lost our proof of the uncountability of the reals. So you may be right that Russells use of laws "does not invalidate the construction of such lists," whatever it is that you mean by that. But Russell's use of laws DOES kick the legs out from under any hoped for proof that you can't enumerate the reals. Having found a noncomputable number, you still have no way to show that there is no list or enumeration of all real numbers. So no matter how you slice it, Russell's remark destroys the point of his proof. He can no longer prove what he hoped to. Your argument here is: "Russell's proof is ok because he didn't KNOW at the time that it was wrong. Well that does not make Russell's proof become correct. It means that Russell made a mistake in the light of something that was only discovered later. That still counts as a mistake, in the scheme of things. Past that, what does it matter? That's the point I was making, and it's the only point I was making, and now that I've made it, I have nothing else to add. I can only keep reiterating the same argument. I don't see why you think you need the "laws" restriction in order to have access to all the characters. In a completely random (ie noncomputable) string there is nonetheless a fact of the matter as to what character is in slot 1, and what character is in slot 2, and in general, what character is in slot n, for all n. The characters that make up each string are "known" in the sense that they exist mathematically. There is just no algorithm that cranks them out. You mentioned a couple of times that you think Russell thought he needed the axiom of choice simply to postulate an infinitely long character string. That's not true, whether or not it's something he ever thought. So this seems to be a point of error in your thinking. You don't need a "law" to know what's in slot n. There's is something there. There is in fact a function: you put in a natural number n, and it spits out the character in the n-th slot. The only issue is that this function is not computable. But it does have mathematical existence; and the axiom of choice is not needed. I didn't try to respond to the rest of what you wrote about L. Since L (the set of all computable strings) is countable, finding a real number that's not in L only proves that L isn't all of the reals. But it DOESN'T prove that there is no list of reals. So Cantor's proof is no longer there. I call that a serious flaw. But as I've said, I can be charitable and stipulate that Russell made an honest mistake, and didn't realize how technically significant the idea of a law is; and that we should just ignore it. Then his proof works. I'm perfectly happy to leave it at that.

I agree with you that this distinction is the heart of the matter. Mathematicians have a beautifully involved and highly abstract theory of the infinite. Yet nobody has ever demonstrated a completed infinity in the physical world in which we live. Some cosmologists speculate that beyond the visible universe, the universe as a whole may be infinite. If so, then it would be a question of physics as to whether set theory applies to physical infinities. I believe this is a question for many decades in the future. It has often been the case that crazy abstract math later becomes necessary to describe the world we live in. Riemann and others pioneered the math of non-Euclidean geometry in the 1940s, and everyone considered it a great curiosity but of no use to the physicists. Then when Einstein was struggling to formulate his general theory of relativity, one of his math buddies said, "Hey, math has just the thing for you," and showed him Riemannian geometry, which is the perfect mathematical setting for relativity. Einstein reportedly said that when he got his own theory back from the mathematicians, he no longer understood it. So this is my thesis, or my belief. That some genius not yet born is going to find a use for infinitary set theory within physics; and that breakthrough, whatever it is, will enable the next huge leap in scientific understanding. I also think that answer to consciousness is NOT to be found in today's understanding of computability and AI; but rather within some future scientific revolution. Our current focus on computer AI is on the same level as the 18th century vision of the world as a great machine. Whatever is the technology of the age, we think the universe and/or our minds are that thing. We are not 18th century clockwork machines, and we are not computers as we currently understand them. I think the mathematics of the infinite is the key to the next revolution in physics; but that we're many decades or a century early.

Yes I agree. It's one of the most popular math topics online. However, "alternative" aka cranky discussions abound on this topic, probably (in my opinion) because it's one of the simplest math stories that is quite counterintuitive. So it gets a lot of play online, and it's also a crank magnet. I'm a grizzled internet veteran so I try not to get too involved in Cantor threads just as I avoid .999... = 1 threads. Just been in too many of these discussions over the years. In fact I'm all-to-eager to jump in to these kinds of discussions, which is why I try to avoid them. So I agree with you that a lot of people are interested, and I hope you understand why I've mostly given this thread a pass. I gave at the office, as it were. My picky self has a couple of quibblets here. First, the diagonal argument is a proof, not a "result," of the fact that there's an injection but not a surjection from the naturals to the reals. But when you say, "There are more real numbers than natural numbers," in my opinion this phrasing is one of the leading causes of confusion among people. Because "more than" is being overloaded with a specific technical meaning that does not necessarily correspond with the everyday meaning of the phrase. When we say there are "more reals than naturals," what we mean, by definition, is that * There is an injection from [math]\mathbb N \to \mathbb R[/math]; and * There is no surjection from [math]\mathbb N \to \mathbb R[/math] With those definitions clearly having been stated and explained, THEN we can casually say "more than," knowing that our listener will hear, or can be trained to hear, the technical definition in their head. But without that technical preparatory work, we are swindling the naive listener a bit, by telling them something that sounds impossible, based on the everyday meaning of "less than" or "smaller than." In fact it's manifestly nonsense that any infinite set is less than or smaller than some other infinite set, until we specify the context We might be talking about cardinality :|[0,1]| = |[0,2]|; or measure, or Euclidean length: m[0,1] = 1/2 m[0,2], or asymptotic density: the evens occur half the time in the naturals, or the subset relationship: the evens are a proper subset of the naturals, or some other way of characterizing the sizes of infinite sets that mathematicians have devised. So for these reasons, I object to the casual claim that one infinite set is "smaller than" another, unless you note that you are using the phrase in its technical sense of an injection but no surjection. Well you asked for my input and there you go A little mini-rant about telling the uninitiated that one infinite set is "smaller than" another, without pointing out that "smaller than" is a term of art in the math biz. It doesn't mean what it does in every day natural language. And that when spoken to a naive listener, it obfuscates more than it elucidates. So I'm against this particular lack of clarity. One should always make it clear that "less than" or "smaller" are true in a technical sense, once the proper definitions have been made; and likewise for "greater than" or "larger."

Interesting point. Uploadees reach a state of eternal bliss, in the moment. Could be true for all we know. Still, I can't help noticing that digital futurism always seem to converge with religion. If you pass a preacher on a streetcorner who says, "We will all ascend to heaven and reach eternal bliss," people will say Oh, that's religious superstition. Then that very night they attend a TED talk where a hipster scientist tells them that "We'll all ascend into the Great Computer, and reach eternal bliss," and everyone goes Ooooh, deep. If your eternal bliss idea is correct, can you distinguish between mind uploading and Christian theology? Jesus was uploaded for your sins. Interesting point. I googled "is mind uploading christian theology?" and up popped this: Heaven on Earth: The Mind Uploading Project as Secular Eschatology https://www.tandfonline.com/doi/abs/10.1080/14746700.2019.1632554?journalCode=rtas20 The article is paywalled, but the headline makes the point. Mind uploading is just a modern spin on a very ancient idea. In the end, I take your point. Nobody will care that reality keeps recycling in digital heaven, because we'll all be too blissed out to care.

If you posit an infinite universe it still wouldn't help unless your physical computers are themselves infinite. But if you want to imagine computers with infinite storage I'll grant you the point. But you see the leap of imagination required to even make the premise conceivable. In an observable universe known to be finite, you want an infinite universe in which we can build computers having infinite amounts of memory. At what point does this go from an exercise in rationality to pure a-scientific fantasy?

Your argument is that "cyber-nirvana" answers the objections that I've raised? I can't say this is much of a basis for conversation. It's like that old cartoon where two professors are standing in front of a blackboard full of symbols and diagrams, and one of them points to the phrase, "And here, a miracle occurs." There are finitely many atoms in the observable universe. Any device or machine, and I'm happy to include the human body in that category, can necessarily only attain a finite number of states. If your response is that a "cyber-nirvana" will ensue that somehow allows a machine containing N states to achieve N + 1 distinct states, that is not a substantive response to the point I've made. It violates the pigeonhole principle. It's like arguing that "space fairies" may someday provide the means for teleportation. It's not a serious argument. You're arguing that nobody can ever criticize a speculative idea, because after all, magic might occur. https://en.wikipedia.org/wiki/Pigeonhole_principle There are only finitely many possible states in any physical machine created in the observable universe. You don't see duplications because your lifespan is too short. If you run the computer long enough, you'll see your life experiences duplicate, over and over. If you allow interactions among the programs, that changes nothing. There are only finitely many possible states. Which part of this simple argument is unclear? If you have a standard American egg container with 12 slots, you can only put an egg into one of the twelve slots. Those are your only choices. If a machine has a finite number of states, say N, you cannot put it into N + 1 distinct states. This is a very basic point.

The OP, @phyti, has been making this same argument for years on various online forums. I have no interest in arguing the subject. I only suggested that rather than try to slog through a translation of 19th century German academic prose, he might find a more contemporary account more sensible. He responded that he finds contemporary accounts of the CDA equally objectionable. That closed the subject as far as I was concerned. I am not debating CDA. @JeffJo then suggested the Russell quote, which unfortunately contained a material error. Rather than take the sensible approach, to say that Russell just mis-spoke himself and we should ignore the error, @JeffJo has taken the view that Russell's restriction of the elements of the list to be symbol strings that can be generated by a law -- that is, computable strings -- does not fatally compromise the argument. It does, and I've pointed that out. I have no interest in unpacking the OP's issues with the CDA. My focus is intentionally narrow.

I can't correlate this paragraph to what I said that you were responding to. A physical digital computer has finitely many states. If you "upload a mind" to such a computer, it will eventually need to start repeating experiences, simply because there are only finitely many states. It's the old pigeon-hole principle. Once your digital avatar or machine consciousness exhausts all the available states, it must necessarily repeat. That's all I said. Perhaps I didn't understand your remark. I also didn't follow your hardware/software distinction. Hardware is a bunch of bits, and the software determines how they should be flipped moment to moment. The software runs on the hardware. Software is not magic, nor can it transcend the limitations of hardware. Oh I see. Storing your mind on a floppy disk to be installed on a future body. Ok. Nevermind about the finite states then. And now I understand your point about nanotech. In the future we'll be able to build general-purpose bodies that can then be programmed to the specifications found on the floppy that contains some particular individual's mind. Like a Field programmable gate array. https://en.wikipedia.org/wiki/Field-programmable_gate_array Anyway it seems that you were not talking about mind uploading, but rather only storage. If the computer runs long enough, it must necessarily exhaust the available states, and then the subjective experiences must duplicate. You can run the numbers however you like. [Subject to the assumption that uploading a mind to a digital computer is possible in the first place]. As far as your remark that some people might like such repetition, I find that hard to believe. You'd come to understand that nothing you do matters. You're a mindless puppet with no control over your strings. Every moment of your life would be determined, and you'd be aware of it. I can't imagine anyone being happy in such a circumstance. But I concede that there might be one or two people who are. You have no way to know that. Your life lasts only 80 or 90 years. Your body is made of finitely many atoms. There are only finitely many configurations you can be in. Say it takes a million years to exhaust all the possibilities. You never duplicate your experience because you can't live that long. But in the computer, you would. You'd live millions, billions, trillions of years. Maybe even in a very short amount of "outside time," if the computer runs fast enough. At that point, you would necessarily have to start duplicating experiences, because a physical digital computer has only finitely many states. If you lived long enough you'd find yourself in an episode of Groundhog Day, except WITHOUT the possibility of self-improvement and no hope of getting the girl. Every day of your life EXACTLY the same as it was the day before, moment by moment, without the slightest possibility of variation. You'd go mad and beg for unplugging.

Your latest post is very clear and I understand you much better now. I want to get to my other mentions on a couple of other threads before I write a detailed reply. If it takes me longer than expected just wanted to let you know that a response is all the way. I'll just say here that your post caused me to have a good think about all this; after which I am more certain than ever that Russell has undermined his own argument. What he has proven is that either: (a) If his initial list contains all computable numbers, then he's just shown that there exists a noncomputable number (namely the antigiagonal). But for all we know the reals are still countable, so he hasn't proved the reals are uncountable; or (b) If his initial list was just a countably infinite list containing some[/i] computable numbers, then all he proved is that the antidiagonal is some other real, maybe computable or maybe not. The core problem is that by restricting attention in the initial list to only computable reals, the argument falls apart. Only by starting with an arbitrary list of reals, as Cantor does, do we conclude that the antidiagonal is a real not on the list; thereby showing that no list of reals can contain all of them. Therefore the reals are uncountable. So Russell's casual use of the word law destroys his entire argument, and fails to prove the reals are uncountable. Now that I think of it, this little summary is actually a concise exposition of what I wanted to say. I was going to add in some background on computable real numbers, but I think I'll just leave it at this for now and see if you have any feedback to what I wrote here. So nevermind that I'll post something better later, I'll go with this for now. And like I say, the charitable thing to do is just ignore Russell's law remark and just go with the spirit of the proof. I'm fine with that. But if you are saying my objection is not relevant, then I have to reiterate that it is highly relevant. By restricting the rows to be computable reals, all the diagonal argument shows is that there's some other real not on the list, which may be noncomputable. But that proves nothing about the countability or uncountability of the reals. To show the uncountability of the reals it's necessary to allow the initial list to be an arbitrary list of real numbers. Restricting the list to only contain computable numbers (that is, numbers whose digits are chosen according to some particular law) wrecks the proof. It does prove something else, namely that there exists a noncomputable real number. But that's not what we're trying to prove.

Ok well I guess I'm done, I can't explain this any better. Regarding the quote of mine that you object to, I stand by it 100%. I "think" that? You disagree with Turing's 1936 paper? I would ask how you think "laws," interpreted as algorithms or effective procedures, can generate uncountably many distinct symbol strings. But they can't, so asking that question would not be productive. There are countably many Turing machines. They can generate only countably many distinct symbol strings over a finite or countably infinite alphabet. Russell didn't know that and he didn't think about it. His error entirely invalidates his version of the CDA unless, being charitable, we ignore it. You can have the last word. You are now denying a universally accepted 87 year old result. If it helps at all, I have not followed the details of your discussion with @phyti nor the details of Cantor's original argument. I have only discussed Russell's technical error in the passage you quoted to me. So your protestations that I'm missing the same point as @phyti and Cantor are misplaced. I don't even know what that conversation is about and have not discussed it.

Human experience is finite, but our lifespans are short. If you're uploaded to a computer, you'll inevitably reach the point where your experiences duplicate. Once that happens often enough, you'll beg to be unplugged. In no alternate universe, let alone this one, was Ray Kurzweil ever a founder of Google. He joined the company as an employee in 2012. Nano machines are not the subject of the thread.

Hi, I didn't want to just ghost the thread but I think I'd get frustrated if I tried to reply paragraph by paragraph, since we're talking past each other. So just to focus this down to one specific thing, where you say I "attributed content to CDA that is not there," I would say that I did no such thing. Can you explain what you mean by that? I called out an error, a howler in fact, committed by Russell in the passage you quoted. That's all I did. What content did I attribute to CDA that isn't there?

I have another objection or concern that comes to mind, would be interested in your feedback. Every physical computer has only finitely much memory. In the future it will be a lot, but it must be finite. That implies that the computer can only be in a finite number of states before repeating. So you are uploaded into digital heaven (more about that in a moment). You are served a seemingly endless supply of delightful digital heavenly experiences. Then one day you notice that you're in the same scene you've been in before. You realize that since the computer is only capable of being in finitely many states, you are condemned to loop forever. After the billionth time you've had sex with the celebrity of your choice, you realize it's not really fun anymore. It's boring. You realize you are not in heaven. You are in hell. There's a relevant Twilight Zone episode. Petty crook dies and goes to the beyond. His host (named Pip, played by the delightful Sebastian Cabot) tells him he can have anything he wants. Girls? They all fall for him. Gambling? He wins at everything. Crimes? Yes he can even commit crimes, and he gets away with them. One day he starts getting bored. He asks Pip if there could just be a chance he'd get caught committing a crime. Pip takes out his notebook, says, "Ok, you'd like to be caught." The guy says no, I just want there to be a CHANCE of getting caught. That's the excitement, to not know. He says, "If this is heaven, then I'd rather be in the other place." Pip lets out a devilish laugh and says, "Heaven? What ever made you think this is heaven? This IS the other place!" And so it would be for your existence inside a digital computer. After the millionth time through the same experiences, you would beg your machine maintainers to unplug you. You would beg for death. And then you remember -- you are not in digital heaven. In your life you committed a crime. You are CONDEMNED to your endless existence of the same predictable experiences, over and over for eternity. It's your punishment. You are in hell. What say you? By the way note that it wouldn't help to allow you to interact with the other uploadees. There's only so much memory and only so many states, and in the end, everyone's collective experiences must loop endlessly through the same finite set of experiences.

Who runs the hardware? Assuming you can "upload your mind" into a digital computer -- a claim I consider preposterous for reasons I won't go into here, so for the moment I'll accept the hypothetical -- who runs the machines? Your mind-program runs on a digital computer made of chips and requiring electricity to function. Someone has to build the computers, run them, monitor them. Metals and plastics must be fabricated. Chips must be made in factories. Electricity must be generated. Massive hydroelectric dams must be built, or if you prefer, massive wind and solar farms. Windmills and solar panels require rare earth elements that must be mined, usually by children in third-world hellholes that urban futurists don't see but never mind that, let the kids mine the elements as they do today. See https://www.npr.org/sections/goatsandsoda/2023/02/01/1152893248/red-cobalt-congo-drc-mining-siddharth-kara just as one of many examples. But you're an elite first-worlder so you won't have to know about their lifetime of misery, just as right this moment you never stop to contemplate the children who mine the cobalt to power your electric car. So who are these people? Is there a huge class of slaves living underground whose lives consist of digging the earth to supply the raw materials for the computer chips, windmills and solar panels? They build out the electric grid, they sit in rooms monitoring computers to make sure your uploaded mind is having a pleasantly ethereal existence. As long as your claim is that "minds will be uploaded to computers" you have to consider who builds and runs the computers, builds the electric plants that run them, who digs the metals out of the earth. When a baby is born, who decides whether it's a privileged uploaded Eloi, or an underground-dwelling slave labor Morlock? What am I missing? It seems to me that even momentarily granting your premise of the possibility of "uploading your mind to a computer," the idea almost immediately falls into absurdity. Billions condemned to earthly slavery while you and your friends float in the ether. Will you in your uploaded heaven whip the third-world slaves to make more electricity to keep your digital hallucination running? Of course you will. You do that right now. Is your phone tainted by the misery of the 35,000 children in Congo's mines? https://www.theguardian.com/global-development/2018/oct/12/phone-misery-children-congo-cobalt-mines-drc Elitist technocrats. Don't get me started. And remember: Once in a while, the Morlocks eat the Eloi. Why shouldn't those child miners and slave laborers just unplug your sorry butt and save themselves the trouble? What's in it for them to keep your simulation running? They'd have no more compassion for you than I do for this browser tab when I close it. [Disclaimer: Overheated rhetorical invective aimed at your idea and at all those pseudo-intellectuals espousing these post-human delusions, not at you personally.]

I wonder if you might possibly be confusing me with the OP, @phyti I am not at all concerned about Russell's exposition, except for this howler: "where the a’s are each an m or a w in some determinate manner. (For example, the first p terms of Ep might be m’s, the rest all w’s. Or an other law might be suggested, which insures that the E’s of our series are all different.)" The fact that Russell regards the symbol strings as being generated by a "law" or "in some determinate matter" completely invalidates Cantor's proof, for reasons that I've already discussed at length. Now I agree that perhaps one could say, "Russell had a momentary brain fart, let's just pretend he didn't say that," and then all would be well. But you presented Russell's version as an improved exposition compared to Cantor, so I pointed out that the Russell quote contains a fatal flaw and is therefore worse, not better, than Cantor's version. I have no complaints at all about the CDA. I only suggested to @phyti that he might prefer a more modern exposition of the CDA. He replied to me that he finds modern versions equally objectionable. That answered my question and I left it at that. He did go too far. It's a fatal flaw in the argument to stipulate that the strings follow a law, since that absolutely rules out the uncountability of any collection of such strings. The axiom of choice has nothing to do with any of this. Summary: (1) I am not the OP. I have no objections to the CDA. (2) In general I prefer modern expositions to original ones, unless one is tracing the history of a given subject. (3) Russell committed a blooper that we can either pretend he didn't say, in order to get on with it; or else note that his blooper invalidates the very argument he's trying to explain. As for the rest, you seem to be trying to convince me that the CDA is true, which is entirely unnecessary; or that it's flawed, which it isn't. As far as the argument that Cantor assumes a list of "all" strings for purposes of a proof by contradiction, it's of course more clear to frame the argument as showing that ANY list of strings must necessarily be missing a string, showing that there is no list of all strings. That way no proof by contradiction is needed. So I agree with you that the reductio formulation of the CDA, so often reproduced online, is not the best way to put it. It's not wrong, but it seems to lead to confusion. Better to just use a direct proof to show that any arbitrary list can not contain all possible strings.

Yes that's true. The string of decimal digits representing a real number is countably infinite. Same for binary digits if we are using Cantor's original formulation of strings consisting of two symbols. Not sure I'm following your point in mentioning this, but perhaps I was not sufficiently clear. How many countably infinite strings (decimal or binary makes no difference) are there? Well, there are uncountably many. That's the content of Cantor's diagonal argument. But now Russell comes along in an attempt to "explain" the CDA, and claims that each such string is generated by a "law." And how many such strings can be generated by laws? If by law we mean algorithm, or effective procedure, or a rule written down in symbols, there are only countably many such strings. In this case, the CDA fails! In attempting to "clarify" the CDA, Russell entirely invalidates it. At best, he throws in a red herring that obfuscates and confuses the issue. The earlier point I was making was that original research, and expositions written near the time of that research, are often lacking in clarity and simplicity. It takes decades for a technical argument to get boiled down to its clear essence. You suggested Russell's exposition as an improvement in clarity. And I pointed out that (IMO) Russell's exposition is actually worse; because not only is it not more clear, it's actually wrong. It requires the symbol strings to be generated by laws; and there are only countably many such strings. We can never prove the uncountability of such strings, since there aren't uncountably many of them at all. There are only countably many such strings. Proof: What is a law? A law is a finite string of symbols ("Alternate 1's and 0s"; or "Write eight million zeros followed by all 1's," etc.). Given a finite alphabet (say standard English), there are only a finite number of laws of length 1; finitely many of length 2; finitely many of length 3; and so forth. Taking the union of countably many finite sets gives us at most countably many laws; and therefore only countably many symbol strings that can be generated by laws. A similar argument applies if by law we mean an algorithm, as formalized by Turing's concept of a Turing machine (TM). There are only finitely many TMs of length 1, finitely many of length 2, etc., and therefore at most countably many Turing machines. There are only finitely many algorithms, and therefore only finitely many real numbers that can bee generated by Turing machines. (For a TM you can substitute the more familiar notion of a computer program written in Python, Java, etc.) These are the computable real numbers. There are only countably many of them. Almost all real numbers, in the sense of all but countably many, can NOT be generated by an algorithm. They're entirely random. Cantor said no such thing as to require the symbol strings to be generated by laws. Perhaps he had insight a few decades into the future, realizing that there could only be countably many laws, algorithms, or procedures. So Russell (in the passage you quoted) has not only obfuscated the CDA; he has entirely invalidated it. If you restrict attention to symbol strings generated by laws, there are only countably many of them. The antidiagonal is then a noncomputable number, and it's unclear (to me, anyway) what Russell's point would then be. Is that more clear? Or did I misunderstand the intent of your remark?