Obections to Cantor's Theory (Wikipedia article)
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From: David Kastrup on 20 Jul 2005 15:33 Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > David Kastrup said: >> Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: >> >> > Barb Knox said: >> >> In article <MPG.1d4726e11766660c989f2f(a)newsstand.cit.cornell.edu>, >> >> Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: >> >> [snip] >> >> >> >> >Infinite whole numbers are required for an infinite set of whole numbers. >> >> >> >> Good grief -- shake the anti-Cantorian tree a little and out drops a >> >> Phillite. Here's a clue: ALL whole numbers are finite. Here's a >> >> (2nd-order) proof outline, using mathematical induction (which I >> >> assume/hope you accept): >> >> 0 is finite. >> >> If k is finite then k+1 is finite. >> >> Therefore all natural numbers are finite. >> >> >> >> >> > That's the standard inductive proof that is always used, and in >> > fact, the ONLY proof I have ever seen of this "fact". Is there any >> > other? I have three proofs that contradict this one. Do you have any >> > others that support it? >> > >> > Inductive proof proves properties true for the entire set of >> > naturals, right? >> >> For each of its members. >> >> > That entire set is infinite right? Therfore, the number of times you >> > are adding 1 and saying, "yep, still finite", is infinite, right? >> >> No. He is not adding 1 more than a single time, just to check that >> for n finite (which means that the set 0..n obeys the pigeon-hole >> principle) n+1 is still finite (the case of one additional pigeon-hole >> can be reduced to the case n if you check for the hole in position >> n+1 and in 0..n both before and after permutation). > How do you know 6598367 is finite? Because you got it by adding 1 to > 6598366, the 6598366th number. Certainly not. Don't tell me that you have spent the effort of counting to 6000000 manually just to be able to talk about that number. I know it is finite because it has a form (a finite number of digits) that can be shown to always refer to a finite number. I see that the number has 7 digits, and I know it is finite. That is, I know it is finite because it obeys certain laws, and those laws can be shown to hold for all natural numbers, by induction. I don't need to carry out any individual steps to make use of that knowledge. > So how to you get the aleph_0th number? What IS that number? it's > aleph_0. But that number does not obey a form I can recognize as belonging to a finite number. And indeed, taking a look at its properties, it becomes clear that it can't be a finite number. >> > So, you have some way of adding an infinite number of 1's and >> > getting a finite result? >> >> No, he is just checking that the conditions for the fifth Peano axiom >> hold. The whole point of the axioms is not to have to check an >> infinite number of steps in order to get a statement about the members >> of a particular infinite set, the naturals. > And yet, one cannot apply an increment an infinite number of time > without adding infinity. It is neither necessary nor feasible to increment "an infinite number of time" or "add infinity". -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: imaginatorium on 20 Jul 2005 15:39 Tony Orlow (aeo6) wrote: > imaginatorium(a)despammed.com said: > > Tony has no clue what mathematics is, nor how it is done, so he doesn't > > normally bother with definitions. The closest we got from him for a > > definition of "finite" was that a finite number is less than an > > infinite one. And you can guess the "definition" of infinite. > Well, that's about as close to a lie as one can get, eh? I asked for a > definition of infinite, and no one could give me a definition of that word. The > best I could get was that an infinite set can have a bijection with a proper > subset, which is hardly a definition of the word "infinite". Here's the real problem: you do not understand what "definition" means in mathematics. It does not mean something that gives the reader an etymologically warm feeling; it is not seeking to convey an intuitive grasp of some concept. Rather it seeks to provide some sort of mechanical test that can be used to divide objects into those that fall in the scope of the definition from those that do not. Of course, in a natural language dictionary, the definition of "infinite" will be vague (to a mathematician), because it is seeking to transfer to the reader the whole set of ideas for which the word may be used. But in a mathematical treatise, the definition above is extremely effective: given a set, it is a straightforward matter to determine whether there is or is not a 1-1 mapping from the set to a proper subset of itself. All of these terms, "set", "mapping", "proper subset", and so on, are clearly defined without relying on anything about being "infinite", so the definition 'works' - we determine whether a particular set is an infinite set or a finite set by investigating mappings. I have said this many times, but I do not think there is any hope you will ever understand the mathematical concept of infinity, unless you first go through a stage of studying it under a different name, because you simply have too many preconceived (wrong) notions about it. > ... In fact I went to > the etymology, which literally means "without end". Finite means with a known > end or bound, and infinite means without end. Of course, I got all sorts of > flack for my definition, from those that couldn't even suggest one outside of > the set theory they were regurgitating. let's try to be straight here, and no > more insulting than necessary, so it doesn't come back to bite us, why don't > we? > > > > As best I can grasp it, the central principle of Orlovian pseudo-maths > > is that "infinite numbers", never clearly being defined, but reached by > > continuing from finite numbers through a "twilight zone" (whose > > existence is anecdotally stated), have essentially the same sort of > > properties as "ordinary numbers". The Orlow-refutation of the diagonal > > proof rests on the "fact" that a list of infinite sequences of digits > > is not a quarter-plane, as one might imagine, but a rectangle, of width > > P and height Q (P and Q being some Orlovian infinite numbers), so of > > course the diagonal hits the (infinite) side at some point. The > > mathematical concept of a sequence being endless means (to us) that > > there is no end, but that doesn't stop Tony using the end to prove > > something. You'll notice he gets a bit irritable when people point out > > that one of his "proofs" doesn't work because there *isn't* a largest > > integer (or whatever). > (sigh) Yes you are on a bullshit roll here, Brian. At least I offered a > definition for infinite, which none of you did. The "twilight zone" we > discussed is that impassable zone between finite and infinite, where your > impossible largest finite and you fictitious omega meet, which I repeatedly > agreed could not be transcended through finite addition or incrementation. "fictitious"? I think all of maths is fictitious. If it were not fictitious, we could measure it, or do experiments to find out (for example) the precise value of 3. "meet"?? There is no "meet". If you read page 1* of Conway's ONAG again, you'll see that he defines 0, 1, 2, and so on, using set notation. Then he defines w (omega, the "first ordinal"), by writing the whole set of {0, 1, 2, ...} on one side of the number notation. Later on, he creates (the surreals form a field, remember, even if you don't know what that means yet) w-1. But w-1 does not 'arrive' until *AFTER* w. There is no twilight zone; there is no sense [UIMM] in which one could meaningfully "start up the pofnats (0, 1, 2, ...) and somehow later find oneself arriving at w having come (in reverse order) from w-1, w-2, w-3, w-4, ... . * Value of 1 may be increased somewhat as required. I > asserted that many of the same properties hold on either side of this zone, in > mirror image, which I still contend is true. You assessment of my objection to > the diagonal proof is essentially correct, and I still stand by it. Yes, but you are "asserting" and "contending". There is no mathematical basis to these assertions and contentions - you are just reciting your intuitions. > Yes I get sick of the "largest finite" mantra, ... It's only a "mantra" because you keep making false claims, which rely on the existence of a "last pofnat". These are refuted by the mantra. If you kept relying on dividing by zero, you might get annoyed by being reminded that you can't. Tough. > ... especially in the context of > your equally impossible omega, your smallest infinite, which you only maintain > through the use of "non-standard" arithmetic where omega-1=omega. I offered > three proofs regarding the naturals, only one of which had anything to do with > a largest member. One of the others was "refuted" by saying that induction > doesn't prove things for an infinite set (bullshit), and that I was trying to > prove things about sets, not numbers, which is also bullshit, since I was > proving a property regarding a set DEFINED by a natural number, which is > ultimately a property of that number. The one using digital representations has > not been refuted at all, but largely ignored, since you CANNOT have an infinite > number of digital numbers, or strings on any finite alphabet, without allowing > infinitely long strings. That's (if I've remembered correctly) exactly the one I went through and pointed out the precise error. Consider the set of finite string-lengths over this finite alphabet: you claim that there are only a finite number of these lengths, right? But is this the result you are trying to prove, or is it something you rely on in your proof? (It's both, of course.) > That's how it went, for the record. What really > "irritates" me is deliberate bullshit, and lies regarding what I have said. I > do not need people summarizing my position, thank you. I can do that very well > myself. > > > > Tony has any number* of "proofs" that an infinite set of natural > > numbers must include "infinite naturals", but these are generally > > circular. The one from "information theory" says that since there can > > only be a finite number of strings of finite length (even if the length > > has no limit), then to get an infinite set of numbers, you must include > > some that are infinitely long. The bit after "since" is a restatement > > of what he purports to prove, but he ignores people pointing this out. > > Ahem! That is another misrepresentation. The bit after "since" is a statement > about symbolic systems, and is a fact outside of the natural numbers. Given a > set of symbols of size S, one can construct a set of all strings of length L, > and the set of strings has size S^L. This is a fact, which when combined with > the fact that digital strings are strings on a finite alphabet (S is finite), > S^L can only be infinite if L is infinite. Therefore, an infinite set of > digital numbers MUST contain numbers with infinite numbers of digits. If there > are infinite numbers of significant digits to the left of the digital point, as > would be the case with infinitely long whole numbers, then by the definition of > digital systems, such strings represent infinite values. > > Refute that, specifically. It's obvious nonsense. You have quantifier dyslexia, which is generally incurable: for a particular (finite) value of L, there are S^L strings, and S^L has a finite value. But above you said: "[There can] only be a finite number of strings of finite length (even if the length has no limit)" If the length has no limit, then there is not a single value L, but any number of values L, and if these values are without limit, by the (etymological!) definition even, the number of values of "S^L" (well, there isn't a single value L at all, is there) is without limit. This is what we mean by infinite. If the strings can be any finite length, then no finite limit can be put on them. The absence of a finite limit (and a finite limit means a real, normal, finite pofnat, like 57, (or Mueck's latest 10^100^100 whatever it is) means that the value is infinite. That's how it works. Well, you see, typing is tiring, particularly as one has to keep inserting totally superfluous words like "finite" everywhere. Good night! Brian Chandler http://imaginatorium.org
From: Stephen Montgomery-Smith on 20 Jul 2005 15:40 Tony Orlow (aeo6) wrote: > Stephen Montgomery-Smith said: > I love the way mathematikers love to spew insults instead of replies. The > conclusion of the proof is that there isn't such a list, which is what I > disagree with, so your statement is either totally confused, or deliberate > obfuscation, which is more likely. > > Whatever you "got", you didn't catch it from me. I admit that I was poking fun at your argument. But I see so many other people trying to discuss things with you at a more logical level, and simply getting nowhere. You are definitely not "getting" what they have. Now let me, for a moment, put aside ideas like "correct" or "incorrect." There is something in my very depths of my being, that when I read one of the mathematikers proofs, that it just makes complete sense to me. I can try to justify why I think our approach is the correct approach using the collected thoughts of philosophers and logicians through the ages, but in the final analysis, I just *know* that we are correct. You do come across as sincere in your differing opinions. I can only suppose that in some strange manner that your brain is wired differently than ours are. What seems completely logical and sensible to us, seems to be nonsense to you, and conversely, what seems to be a proper argument to you, is so weird and strange to us that we seem unable to even know where to start it trying to disuade you from your point of view. Stephen
From: Robert Low on 20 Jul 2005 15:51 Daryl McCullough wrote: > But the Peano axioms say that *all* nonempty sets of naturals have > a smallest element. So if you say that there is no smallest infinite > natural, then that implies that there are *no* infinite naturals. I'm not entirely sure what you're claiming here, but order isn't even mentioned in the Peano axioms, so they certainly can't include a statement that the naturals are well-ordered. Do you mean that you can inductively define a notion of less than in terms of the successor function, and then show that the Peano axioms imply that the naturals are well-ordered?
From: Robert Low on 20 Jul 2005 15:55
Daryl McCullough wrote: > That's not true. If S is an infinite set of strings, then there > is a difference between (1) There is no finite bound on > the lengths of strings in S. (2) There is a string in S that is > infinite. Except that TO claims that (1) implies (2), though I can't even get far enough into his head to see why he thinks it, never mind finding his 'argument' convincing. Funnily enough, there is a similar sounding statement that is true in non-standard analysis: any set containing arbitrarily large finite integers must also contain an infinite integer. But in that game, the class of all finite integers isn't a set :-) I only mentioned this because I thought it might muddy the waters in an entertaining way... |