Obections to Cantor's Theory (Wikipedia article)
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From: Virgil on 27 Jul 2005 16:08 In article <MPG.1d517b8a9cc7639a989fa4(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Daryl McCullough said: > > Tony Orlow (aeo6) wrote: > > > > > >imaginatorium(a)despammed.com said: > > >> ...That means that when I say "the pofnats", > > >> this expression could be replaced by the expression "the subset of the > > >> Tonats containing only those which are unambiguously finite"... > > >> Can I not select those of the Tonats that are finite? > > > > > >Sure you can talk about such a set. You just can't draw very much > > >in the way of conlusions about its size or upper bound. > > > > On the contrary, you can easily prove of this set (call it A) > > > > 1. There is no n in A such that size(A) = n. > That's funny, because I proved that, if N starts with 1, then at any point in > the generation of the set, the set size is also equal to the maximal element. > > 2. There is no n in A such that all n is the upper bound for A. > You mean ther eis no single n which is the upper bound for the set? True. The > only upper bound for the set of finite whole numbers is whatever smallest > infinity one can concoct. > > > > >I'll try, but there are so many times when it seems my words are > > >deliberately misrepresented, and grammer is misconstrued, and the > > >point obfuscated, and that it is a simple defenseive maneuver. > > >If people just say "define!", then it seems like not-picking nonsense. > > > > No, it's not. The whole point of defining one's terms and writing > > down the axioms for using those terms is that then *anyone* can > > prove theorems about the subject, and *everyone* will agree that > > those are indeed theorems. In contrast, if (as you prefer) you > > never give definitions for your terms, and you never write down > > axioms for using those terms, then *nobody* except you is able > > to prove anything about your concepts. > I understand that, but as far as defining "finite", "infinite", "string", and > "set", when I am using these terms, as far as I can tell, by their normal > definitions, it sounds like a deliberate attempt to wear me down. This is > already kind of draining, battling with this pack. I have started on a set of > axioms and definitions, and will keep your suggestions in mind, but don't > have > the system ready for delivery yet. Sorry. In the meantime, if it sounds like > I > am using a word in some non-standard way, then ask. That would help me > realize > what I may have to clarify. I am not used to putting things in such rigorous > verbal format. It's much easier for me to draw pictures. :) > > > > For example, you claim that if S is a set of bit-strings, and > > S is an infinite set, then S must contain at least one bit-string > > of infinite length. Nobody can prove such a claim except you. > > In contrast, with the usual definitions of "infinite set", > > "bit-string", and "length", plus some basic facts about naturals, > > *anybody* can prove the negation: That there exists a set S of > > bit strings such that (1) S is infinite, and (2) no string in S > > is infinite. > I gave you an axiom which is easily demonstrable inductively, and with which > no > one has taken issue (I hope): > > With a set of symbols of size S, one can create a set of unique strings of > length L, whose maximum size is N=S^L. And one can increase both S and L to larger and larger finite values without any finite limit, so that there is no finite limit to the possible finite values of S^L. Or does TO claim to be able to present us with some finite limit to the finite values of S^L? And the question has been asked (and not answered) if S^L is to be limited to some finite maximal size, why must S and L be limited to smaller sizes? If we can always increase S and L to the size of S^L, then A^L will be a new larger value and there is no finite limit to these increases.
From: David Kastrup on 27 Jul 2005 16:08 Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > Chris Menzel said: >> On Tue, 26 Jul 2005 16:44:24 -0400, Tony Orlow <aeo6(a)cornell.edu> said: >> > I am saying that if L CANNOT be infinite, then S^L CANNOT be >> > infinite, >> >> No one disagrees with that, for fixed S and L. >> >> > and the fact that so many find this impossible to understand >> > demonstrates that Poincare was right, and Cantorian transfinite >> > cardinality is a disease in mathematics. >> >> Google for a recent post by Keith Ramsay for a correction of this >> historical myth. >> >> > For finite S, S^L can ONLY be infinite with infinite L. Why is this so >> > hard to understand? If S and L are both finite, then S^L is finite, >> > isn't it? >> >> Yes, of course. But that's for a fixed L, say 17. But for any given >> nonempty set S of natural numbers, the set of *all* finite sequences of >> elements of S -- i.e., S^1 U S^2 U S^3 ... -- is infinite. That's the >> part you don't seem to get. >> > Not unless you allow L to go to infinity. No. Just to arbitrarily large finite values. You don't understand the difference between infinite and arbitrarily large. "Infinite" would be a property of a single fixed number. "Arbitrarily large" is a property of an _unspecified_ member from a set. "Arbitrarily large" means that there is no fixed finite limit I can specify _before_ I pick a number. "Infinite" means that there is no fixed finite limit I can specify even _after_ I pick a number. As long as you don't understand this difference, you will be unable to understand how a set of finite members can have infinite size. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: David Kastrup on 27 Jul 2005 16:10 malbrain(a)yahoo.com writes: > David Kastrup wrote: >> malbrain(a)yahoo.com writes: >> >> > malbr...(a)yahoo.com wrote: >> >> Tony Orlow (aeo6) wrote: >> >> > Some finite, indeterminate number. You tell me the largest >> >> > finite number, and that's the set size. It doesn't exist? >> >> > Well, then, I can't help you. >> >> >> >> >From websters 1913 dictionary: >> >> >> >> De*ter"mi*nate (?), a. [L. determinatus, p. p. of determinare. See >> >> Determine.] >> >> >> >> 1. Having defined limits; not uncertain or arbitrary; fixed; >> >> established; definite. >> >> >> >> >> >> Thus "indeterminate" is the exact opposite of fine. You can't have it >> >> both ways. >> > >> > Ooops. "indeterminate" is the exact opposite of finite. You've >> > uncovered a contradiction about the count of elements in an >> > infinite set that cannot be resolved from the definitions of >> > finite and indeterminate. karl m >> >> Uh, no. "Indeterminate" just means unspecified, not infinite. > > Read the definition. Determinate=defined limit; > indeterminate=undefined limit=infinite. Undefined limit is not the same as infinite. For example, in programming languages indeterminate loop forms are those for which you can't say in advance how often they will be run (i.e., while-loops, as opposed to the determinate for-loops). -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: malbrain on 27 Jul 2005 16:12 David Kastrup wrote: > Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > > > Chris Menzel said: > >> On Tue, 26 Jul 2005 16:44:24 -0400, Tony Orlow <aeo6(a)cornell.edu> said: > >> > I am saying that if L CANNOT be infinite, then S^L CANNOT be > >> > infinite, > >> > >> No one disagrees with that, for fixed S and L. > >> > >> > and the fact that so many find this impossible to understand > >> > demonstrates that Poincare was right, and Cantorian transfinite > >> > cardinality is a disease in mathematics. > >> > >> Google for a recent post by Keith Ramsay for a correction of this > >> historical myth. > >> > >> > For finite S, S^L can ONLY be infinite with infinite L. Why is this so > >> > hard to understand? If S and L are both finite, then S^L is finite, > >> > isn't it? > >> > >> Yes, of course. But that's for a fixed L, say 17. But for any given > >> nonempty set S of natural numbers, the set of *all* finite sequences of > >> elements of S -- i.e., S^1 U S^2 U S^3 ... -- is infinite. That's the > >> part you don't seem to get. > >> > > Not unless you allow L to go to infinity. > > No. Just to arbitrarily large finite values. > > You don't understand the difference between infinite and arbitrarily > large. "Infinite" would be a property of a single fixed number. > "Arbitrarily large" is a property of an _unspecified_ member from a > set. Each and every "single-fixed-number" is finite, not infinite. Please be more careful. karl m
From: Tony Orlow on 27 Jul 2005 16:14
MoeBlee said: > Tony Orlow (aeo6) wrote: > > > MoeBlee said: > > > Tony Orlow: > > > > > > What is your logistic system, your primitive terms, and your axioms? > > > > > > MoeBlee > > > > > > > > I am trying to get together some pages concerning this, but don't have a lot of > > time. By "primitive terms" I assume you mean definitions, and I know what > > axioms are, but what exactly do you mean by "logistic system"? > > -- > > Smiles, > > > > Tony > > Hi Tony, > > At least in this particular post I hope not to be condescending, but I > should start by saying that I think you're making a fairly serious > mistake in your fights here. With your interests and energy, you could > accomplish so much more intellectually if you just took some time to > learn about this subject rather than fight over something you haven't > even studied. > > I don't think that all your philosophical intuitions about mathematics > are without merit. But your lack of familiarity with the rudiments of > this subject is causing you an inability to bring your ideas into > meaningful contest. Meanwhile, if you were actually informed about the > subject, then you might be able to make some interesting arguments for > your tenets. > > If you took some time off to study and reflect, then, later, after > you've understood how mathematical logic and set theory really work > (not how, in complete darkness, you now only mis-imagine they work), > you could still critique what you found out about the subject. And > while you learn, you could try to see past your own personal > perspective to see how mathematics makes sense in ways you now deem > nonsensical. Keeping an open mind, with an intent on understanding set > theory on its own terms, you could still keep a journal of all the > things you disagree with, or, especially, find anathema. Then, later, > after you've understood the subject, you could return to your journal > to take up the sword anew against set theory. But I bet you wouldn't > take up the sword anew, because I bet you'd actually appreciate the > intellectual achievment, the logic, and the beauty of the subject. > > / > > To answer your questions, roughly put, with apologies for not being > perfectly correct in use of certain technical terms: > > The concept of a logistic system is one of the great inventions of > human reason. A logistic system is a formal language and a set of rules > of inference. The favorite logistic system for mathematics is first > order predicate logic. This gives a precise format for reasoning about > mathematics. If a theory is based in first order logic, then a proof > can rely upon no reasoning, no hidden assumption, and no appeal to > intuition that is not codified in the rules of first order logic. Now, > if one wants to use methods of reasoning other than first order logic, > then one would use another logistic system or codify one's methods in > one's own invented logistic system and declare that as the system one > is using. The beauty of the logistic method is that it is competely > objective. If we disagree about whether something has been proven, then > (at least in principle, and only extremely rarely not in practice too) > we could check the proof - line by line, symbol by symbol - to decide > whether the proof is correct. Senseless arguments about things like > what terms mean and if a purported proof is a cheat are avoided as we > just look at the formulas themselves to see if they hook up with one > another as dictated by the inference rules. > > (In practice, we usually don't stick to the pure symbolic formulations, > since they are usually cumbersome, but, if we're working correctly, we > watch out that any departure from pure symbolism is justified by our > being able to see that we could put our proofs into pure symbolism if > we wanted to. Also, working with a logistic system doesn't require that > one not use one's imagination and intuitions in thinking about > mathematics. All that the logistic method requires is that after the > imaginative thinking has been done, it needs eventually to come back > down to earth to be put in a formal lanaguage so that other people can > objectively check it out to see that it really does make sense, even if > only in its own theory.) > > But first order logic just gives us a codification of our reasoning; it > doesn't give us substantive premises. The substantive premises are > called 'non-logical axioms' because they're not statements, such as > 'not both A and not-A, that are logically true. And the axioms are > statements made purely with symbols of the formal language. But, one > might ask, aren't the axioms supposed to be true, and self-evidently > true? If you want to start your theory with only axioms that you > believe to be true or even self-evidently true, then you are welcome to > do so. However, your axioms will still be non-logical, since the rules > are that certain of the symbols you use are allowed to be taken by > other people to denote differently than you think of the symbols as > denoting. It's not that different from everyday language. If I say > 'Bill is a cowboy', then that is true depending on who I mean by 'Bill' > and what I mean by 'cowboy'. So you will choose axioms that can be > interepreted to denote states of affairs that you think are true, given > the denoations you have in mind. > > But what about this business of axioms not being true or false, but > just initial conditions in a game of proving theorems? Well, some > people work that way too. You don't have to accept their axioms as > true. But you cannot dispute that the theorems that have been proven > are indeed proven from the axioms. In other words, if one doesn't claim > axioms to be true, then one doesn't really claim the theorems derived > from the axioms to be true. Instead, one just says, truthfully, the > theorems are proven from the axioms, regardless the truth or falsehood > of the axioms. (And there's a range of other ways of looking at axioms > and mathematical truth.) You may have arguments to say why you don't > think someone's axioms or theorems describe what you belive to be > reality, but you can't rationally deny that an actual proof is a proof. > You need to be very clear about that point. Very very clear about it. > > You have a notion that the axioms should be derived inductively. That's > fine if you can find a way to do that. A problem, though, is that > induction, by its nature, is only tentative, while mathematical > theorems are usually supposed to be conclusive. In other words, if you > base your mathematics on induction, then you may later come across > observations that go against your earlier inductive inferences, and > this would make for a fickle mathematics. (If you say that it is not > possible that you'll come across observations that contradict earlier > inductive inferences, then I think most people would say that they're > not inductive inferences then.) (Note: This does not refer to > mathematical induction, which is something different.) (Eventually, you > might want to check out non-monotonic logic, which does allow for > adjustments to a theory as new information is obtained.) > > But isn't mathematics supposed to apply to the physical world? I can > only guess that the fundamental motivation for most people learning a > bit of math is to work out problems in the world. And mathematics, of > course, is crucial for the sciences and engineering as humans have been > motivated to learn and invent mathematics to conquer or just to cope > with the physical world. Meanwhile, many mathematicians study the > subject for its own sake, for an appreciation of the abstract > relations, regardless of application to the world. I don't see a > conflict. If mathematicians advocated that engineers apply mathematics > that doesn't work to hold up bridges, then there'd be a problem. Or, > even for less concrete endeavors, if mathematicians propose mathematics > that gives bad theories of physics, then there are important debates to > be had about that. But that's not what set theory does, at least as far > as I know. Set theory doesn't lead engineers to build faulty bridges, > and if set theory leads physicists to faulty theories, then, as has > been mentioned by other posters, the fault is with the physicists for > misapplication rather than with set theory, since set theory has great > value as a framework for mathematics but its practioners are not > exactly running around telling everyone to apply its theorems about > infinity to dirctly answer questions about the physical world. (And I > don't imagine it would be hard to document that set theory and > mathematical logic, especially as they lead to recursion theory, have > played a crucial role in computer science.) > > You asked about primitive terms. The primitive terms of a theory are > the terms that are NOT defined. Some terms must be undefined, otherwise > there'd be an infinite regress of definitions. The primitive terms get > their meaning not from definitions but from the axioms. The axioms > narrow down the possible states of affairs that the primitive terms can > refer to. Each added axiom narrows down the the kinds of states of > affairs that the axioms can apply to. For example, the axioms for a > complete ordered field (such as the the real numbers and their basic > operations and ordering) narrow down so that they don't apply to the > natural numbers, since, for example, natural numbers don't have > additive inverses. > > So there are axioms for natural numbers, axioms for real numbers, axoms > for certain kinds of algebraic structures, etc. What set theory does is > give one oveall theory in which all the individual theories can be > expressed. But to do this, it would seem that set theory needs to > provide infinite sets, as this need can be seen the minute one talks > about, say, a function on the reals. Now infinity is a strange thing to > think about in itself. So it shouldn't be surprising that axiomatizing > a theory that includes a concept of infinity might lead to odd looking > theorems about infinity. But the axioms of set theory are themselves > very straightforward and pretty much codify such common notions as > subset, power set, union, etc. But when we add just one axiom that says > there exists at least one inductive set (a set that has zero as an > element and is closed under the successor operation), then it seems we > can't avoid certain twists in the plot. In other words, if you can > devise a theory that has a way of talking about infinity, but avoids > any oddities about it, then, by all means, go for it. But I'll bet a > million bucks to one that what you'll learn is not that this can be > done, but rather an admiration for set theory for even giving us a (so > far as has been thus detected) consistent theory that addresses > infinite sets. > > I'm not expert about mathematics, but I do think I have some insight, > even if informal, into learning and understanding, and I think you've > been blowing it, especially as you've wasted the advice of some people > here who really are expert in mathematics. But you could refresh and > enrich yourself. Why not just grab a textbook on first order logic and > see what you find there? > > MoeBlee > > Thanks for the advice. I'll put up that dollar, if you have the million. Then I could afford to do this and other studies full time. As it is I have to work and take care of a partner, kids, and houses, and have little time, though I think this is important. By "logistical systems" I thought you were talking about predicate logics, but I don't know how to answer which system I would use. I guess I have to brush up on exactly what the difference is, as far as construction. The simpler the better, most likely. You may feel I am "blowing it" but, despite the fact that a better foundation in the methods and language would give me better leverage with those in the field, I have not had any mistake pointed out to me that I could identify as a mistake. My points about symbolic systems, about infinite series, trees and induction as a recursive process stand uncorrected, in my mind. Most of the objections are either due to misunderstanding, or confusion based on the axiom system I am rejecting. So, while I will continue my studies in this areas as time permits, I don't feel in the least bit like I am being foolish, except perhaps by trying to teach new tricks to old dogs. Anyway, thanks for the advice and concern. I don't suppose you have understood or agreed with any of my points? I must be on the wrong planet. -- Smiles, Tony |