Obections to Cantor's Theory (Wikipedia article)
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From: Virgil on 26 Jul 2005 16:27 In article <MPG.1d503908f774ebb9989f91(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Not to mention the Banach-Tarski spheres. Doesn't that derivation > constitute a disproof by contradiction? Not if one understands the distinction between matheamatical 3-space and physical 3-space. > Isn't the result absolutely nonsensical? It is certainly counter-intuitive, but so is the statement that any two (purely geometric) line segments, regardless of length, contain exactly the same "number" or points, which as know to and accepted by the classic Greeks. Which is why intuition about infinities should be mistrusted. > And yet, it is accepted, somehow, as truth that one can > chop a ball into a finite number of pieces and reassemble them into > two solid balls, each the same size as the original, despite all > evidence and logic to the contrary. There is no "evidence" to the contrary, since no one claims that there is any physical way of doing this. Superficially, logic might seem to require that everything in mathematical geometry coincide with physical reality, but there are hundreds of ways in which it does not. The Banach-Tarski result is just another in a long line of such differences. > It's a clear sign of something > wrong in the system, when it produces results like that. The thing that is "wrong" is that our "intuition" often misleads us. If intuition were perfect, we would not need logic to correct it when it goes wrong. So that across the gates of mathematics there might well be written: "Abandon intuition, all ye who enter here."
From: Tony Orlow on 26 Jul 2005 16:35 Daryl McCullough said: > Tony Orlow <aeo6(a)cornell.edu> said: > > >Daryl McCullough said: > > >> 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. > > > >Yes, I understand the difference between those two statements, and > >in this case the two are equivalent. No, I misread those statements. The first, if you are referring to my arguments, should be that there is an infinite set of whole numbers, and the second is that there is an infinite whole number in the set. Those two statements imply each other because of the constant finite difference between whole numbers. This has nothing to do with quantifier dyslexia or other logical errors on my part. It has to do with dismissal of properties of the symbolic and numerical systems used in Cantorian arguments. > > Good. > > >If the length of strings is L and the symbol set has a > >finite size of S, then you have S^L strings, which is infinite > >IF AND ONLY IF L is infinite. Infinite Set <-> Infinite Element. > > You've just contradicted yourself. I thought you agreed that > there is a difference between the two sentences > > (1) There is no finite bound on the lengths of strings in S. > (2) There is a string in S that is infinite in length. > > In case (1), the set is infinite, but there is no infinite element. (sigh) There is a difference between those two statements. They are not equivalent. The two I just made, which you think contradict something else I said, ARE equivalent for whole numbers. I never said (no finite bound)<->(infinite). I said (infinite set size)<->(infinite whole number element) for natural numbers. > You seem to be thinking that there is some L characterizing the > lengths of strings. There isn't. There are strings of length 1, > there are strings of length 2. There are strings of length 3. > Each string has a length, but there is no length associated with > the set of finite strings as a *whole*. No, but if L can ONLY be finite, then S^L can ONLY be finite. If you do not allow L to be infinite, then you do not have infinite S^L. > > >> If you wrote these out as logical statements, you would see > >> that you are mixing up the order of quantifiers: > >> > >> (1) forall b, exists s in S, > >> (if b is a finite bound, then length(s) > b) > >> > >> (2) exists s in S, forall b > >> (if b is a finite bound, then length(s) > b) > >> > >> Statement (1) says that the *set* S has no finite bound. > >> Statement (2) says that S contains an *element* that has > >> no finite bound. Those are two different statements. > > > >I don't need a lesson in logic, thanks. > > Yes, Tony, you certainly do. Since you are at Cornell, there > are a number of courses you could take to fill the gaps in your > knowledge. To learn about logic, I suggest starting with > Math 281: Deductive Logic. To actually get up to speed on > the theory of natural numbers, you need to take > MATH 481 Mathematical Logic. If you weren't losing track of the argument, I might take the suggestion seriously, but you are attributing statements to me that I am not making. Perhaps I caused that by carelessly agreeing to the above statement that wasn't my position, but I guess I got tired and mistakenly assumed we were still discussing the same thing, and not digressing. I'll try to be more careful in the future. > > Or you could just take advantage of the excellent faculty. > Go to Dexter Kozen or Anil Nerode or Richard Shore and ask > them about your claim that an infinite set must contain an > infinite object. (But please don't tell them that I put you > up to it.) I believe it was Richard Shore I approached a while back. He dismissed the idea out of hand, and didn't want to discuss it then, since his father had just died, and he was in mourning. I'll bother him again when I get some of my pages finished. > > -- > Daryl McCullough > Ithaca, NY > > -- Smiles, Tony
From: Tony Orlow on 26 Jul 2005 16:36 Daryl McCullough said: > Tony Orlow <aeo6(a)cornell.edu> said: > > >I don't need a lesson in logic, thanks. > > Yes, Tony. That is exactly what you need. You seem to be trying > to learn some elementary facts about logic and mathematics by > arguing on the internet. That isn't a very efficient way for a > beginner to learn. If you're at Cornell, then go take a course > in logic. It's an excellent institution. > > -- > Daryl McCullough > Ithaca, NY > > I majored in Computer Science, and took plenty of logic and discrete math, and did quite well, thank you. Your inability to follow my argument is not an indication of my logical ineptitude. -- Smiles, Tony
From: Tony Orlow on 26 Jul 2005 16:39 Chris Menzel said: > On Thu, 21 Jul 2005 09:57:21 +0200, Han de Bruijn > <Han.deBruijn(a)DTO.TUDelft.NL> said: > > Virgil wrote: > > > >> Every bit of "Cantorianism" has been well enough defined for the > >> understanding of thousands upon thousands of people. That TO fails where > >> so many have succeeded says more about TO than about the adequacy of > >> "Cantorianism's" explanations. > > > > The fact that a faith has millions of adherants doesn't say anything > > about its validity. It says something about the society wherein it is > > accepted, though. > > Faith?? "Cantorianism" is embodied in a completely rigorous axiomatic > theory. The propositions you find unacceptable are demonstrably valid > in that theory. There is not a lick of faith involved. Instead of > tossing off idiotic comparisons to religious belief -- an inevitable > rhetorical haven for cranks and crackpots -- you might consider a > genuinely mathematical response: point out the axiom(s) of set theory > you consider unacceptable and defend your rejection of them with > arguments; or simply embark straightaway on the development of an > equally rigorous alternative. Responses like yours only show you > haven't the least clue what mathematics is. > > Chris Menzel > > The big problem in transfinite cardinality is the assumption that a bijection necessarily indicates equal sizes for infinite sets, as it does for finite sets. When the only way to form a bijection is with a mapping function, 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. -- Smiles, Tony
From: Virgil on 26 Jul 2005 16:41
In article <MPG.1d503a837ccdff15989f92(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Peter Webb said: > > > > > > e.g. we agree on the basis of our experience with the axiom's > > > veracity and viability. > > > > > > > > > You seem to think that somehow mathematics is a physical science, > > and the axioms are like physical laws, which can be true or false. > > You think that you can observe that zero does not have a suucessor > > just as you observe that every action has an equal and opposite > > reaction. > > > > Well, they are not like that. As David said, they are just the > > rules of the game. Mathematicians just pick the rules in order to > > make interesting games. The example I gave before - the axioms of > > group theory - make for a really interesting game. The axioms of > > set theory (ZFC) make a really, really interesting game. There is > > no question of "veracity". Is it true that there is an element of > > the group e such that for all g, g*e=g ? Yes, because that is an > > axiom of group theory. It is true *by definition*. It is part of > > the rules of the game called group theory. As David said, arguing > > about whether axioms are true makes about as much sense as arguing > > about how knights move in chess - if you think you can invent > > axioms which make better games, go for it. But you can't say - > > "that isn't how a knight moves" because we defined a knight in > > chess as moving that way. > > > > > > > > > Well, if this is how mathematicians feel about their study, and claim > it has no connection to reality, then Cantorian set theorists really > have no reason to claim that anything they say is more correct than > any competing claims. In fact, if they are not at all concerned with > understanding numbers as they pertain to reality, then I am not sure > what their motivation is besides playing games, and it is not > surprising that their solution to such a complex subject as infinity > is as nonsensical as it is. No "Cantorians" claim any more than that when playing by their rules you must play by their rules. > > If axioms are unquestionable, then I can prove anything true, just by > making it an axiom. But, where does that lead us? Certainly not > towards any deeper understanding of anything. The sets of axioms that mathematicians use did not spring into being from nothing. Before Hilbert, about the only 'axiom' systems were for geometry, and Hilbert showe them to be badly flawed. Axiomatics developed fairly slowly, often with many false starts and revisions, and axim sytems may still be revised from time to time or new ones may come into being in response to new questions being addressed. But axiomatics has proved to be much better than what went before. |