Obections to Cantor's Theory (Wikipedia article)
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From: Chris Menzel on 27 Jul 2005 00:32 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.
From: malbrain on 27 Jul 2005 00:55 Chris Menzel wrote: > 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. > (...) > > > 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. Obviously, he doesn't. Perhaps using the definition for all will help: The set made by taking each and every (finite) L sequence of elements of S is an infinite set. karl m
From: malbrain on 27 Jul 2005 01:07 Martin Shobe wrote: > On 26 Jul 2005 17:15:10 -0700, malbrain(a)yahoo.com wrote: > > >Chris Menzel wrote: > >> On Tue, 26 Jul 2005 16:39:58 -0400, Tony Orlow <aeo6(a)cornell.edu> said: > >> > ... > >> > then that function needs to be taken into account. This nonsense > >> > about an infinite set of finite whole numbers is pretty bad too, but > >> > probably without any real consequences. > >> > >> You seem to agree that the set of whole numbers is infinite. But there > >> was an inductive argument a few posts back that all the whole numbers > >> are finite, and hence that the set of finite whole numbers is infinite. > >> There was some real mathematics there. > > > >How does it follow that the count of finite whole numbers is infinite? > >How is this established by the Peano axioms? > > A set, A, is infinite if, and only if, there exists a one-to-one > function, f:A -> A, such that f(A) is a proper subset of A. > > Or equivalently, > > A set, A, is infinite if, and only if, there exists a function, f:A -> > A, such that > 1) for all x,y in A, f(y)=f(x) => x=y. > 2) there exists an x in A such that for all y in A, f(y) =/= x. > > Since there are no sets, and we are interested only in the domain of > PA, we have > > The domain of PA is infinite if, and only if, there exists a function, > f, such that > 1) for all x,y f(y)=f(x) => x=y. > 2) there exists an x such that for all y, f(y) =/= x. > > The successor function meets those criteria. Therefore, the domain of > PA is infinite. > > The only problem that I can see with this is that it's a theorem about > PA instead of a theorem of PA. It's ABOUT PA because you back-filled a definition for infinite to PA? > > You have Tony agreeing to > >the axiom of infinity apriori, when this is not indicated. > > The axiom of infinity is not needed to prove that a set is infinite. > The axiom of infinity is needed to prove that infinite sets exist. This KOAN is going to be a tough sell. karl m
From: Jiri Lebl on 27 Jul 2005 01:29 Han de Bruijn wrote: > OK. It seems that we finally have arrived somewhere. I have one final > question, though. Is it "legal" (according to mainstream mathematics) > to call a set "countable" if it can be brought in 1:1 correspondence > with the naturals? And uncountable otherwise? Perhaps it's trivial for > anyone, but I'm a bit at lost (due to those heated "Cantor's diagonal > argument" threads I think :-) That is the DEFINITION of the word "countable". A set is called countable if it can be brought to 1-1 correspondance with the naturals. This is the most important distinction IMO for sets in mathematics. There's lots of things one can do wih countable sets (such as sum over them etc...) that one can't do otherwise. You can do induction on countable sets. On the other hand one needs uncountable sets to do any kind of meaningful probability theory / measure theory (with continuous data). If the reals were countable for example, we wouldn't be able to integrate. Just because people didn't know this when they came up with integration theory doesn't mean that the completenes of the reals (which implies uncountability) wasn't at play. Jiri
From: MoeBlee on 27 Jul 2005 05:02
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 |