Obections to Cantor's Theory (Wikipedia article)
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From: Virgil on 27 Jul 2005 22:36 In article <MPG.1d519a73b20454b989fb7(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Virgil said: > > 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. > The difference being that mathematical 3-space doesn't model anything real? > Bull. > > > > > > > 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. > That idea is also rejected by me, especially when one line segment is a > subsegment of the other. It's nonsensical and unrealistic. And geometrically imperative for all standard axiomatizations of Euclidean geometry, among others. > > > > Which is why intuition about infinities should be mistrusted. > Uh, no, they shouldn't. Just look at how TO's intuitions have mislead him into self-contradiction! Too much faith in intuition is dangerous. > > > > > 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. > Derived from this ill infinite set theory nonsense. Dewrived from a number of different sets of axioms each very carefully thought out and much revised to avoid the kind of trivial self-contradictions that TO is repeatedly advocating. > > > > > 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. > Maybe yours does. Any intuition that claims infallability has already made one big mistake. > > > > If intuition were perfect, we would not need logic to correct it when it > > goes wrong. > Much intuition is based on logic, in conjunction with non-verbal forms of > thought. Much intuition is based on analogies, but every analogy can be stretched too far. > > > > So that across the gates of mathematics there might well be written: > > > > "Abandon intuition, all ye who enter here." > > > That's just stupid. No comment. That comment is a comment, and a stupid one. Those who enter mathematics seriously agree in advance to be convinced by certain types of arguments satisfying certain criteria of validity. TO neither accepts that agreement nor produces anything that satisfy those requirements for validity.
From: Daryl McCullough on 27 Jul 2005 22:23 Tony Orlow (aeo6) wrote: >If I prove that there is no n in N for which the set of all strings of length >less than or equal to n is infinite, then I have proven that no finite limit on >the length of strings can produce an infinite set. Right. The set of all finite strings has no maximum string length. If there was a maximum string length, then the set would be finite. But since there is no maximum string length, the set is infinite. There is no number n such that all finite strings have length less than or equal to n. -- Daryl McCullough Ithaca, NY
From: Virgil on 27 Jul 2005 22:42 In article <MPG.1d519be9313eee8b989fb8(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Robert Low said: > > Tony Orlow (aeo6) wrote: > > > > > 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. > > > > OK, so how many elements are there in the set of all finite > > natural numbers? > > > 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. If it doesn't exist, there is no finite set size to the set of all finitely generated naturals. Since TO is the only one declaring that there is any finite bound on the number of finitely generated naturals, it is TO's, and only TO's, problem to produce such a bound, not that of those who reject that claim. Does TO's argue that because we do not provide him what we claim does not exist, it must exist? About par for TO's level of competence.
From: Daryl McCullough on 27 Jul 2005 22:34 Tony Orlow writes: >> >If you place no restriction on the length of strings, then they can be >> >infinitely long. >> >> No, there is only one restriction on the length of strings, and that >> is that the length is finite. >Oh, well you had said there was NO restiction, and they could get arbitrarily >large. To say that they can get arbitrarily large is just to say that there is no largest string length in the set of finite strings. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 27 Jul 2005 22:42
Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > >Daryl McCullough said: >> >> S_1 = the set of strings of length 1 >> >> S_2 = the set of strings of length 2 >> >> S_3 = the set of strings of length 3 >> The sequence S_1, S_2, ... is an *infinite* sequence; >> it's a sequence that has no end; it has no last set. >> So no, the last set is not S_oo because there is no >> last set. >But there are an infinite number of them, so your subscript must become >infinite at some point, or are you repeating finite subscripts? No, the subscript never becomes infinite, and never repeats. >But there are an infinite number of them, right? If there were >a million, then the last would be S_million. Infinity is not a natural number! To say that there are 1 million sets is to say that the last set is set number 1 million. To say that there are infinitely many sets is to say that there *is* no last set. That's all that "infinitely many" means. There is no last set. It's not that the last set has subscript oo, it is that there *is* no last set. >And when you have an infinite number of them, you still have finite >subscripts? How is that possible? You get an infinite sequence by never ending, and never repeating. That's exactly what the operation x --> x+1 does for you. It never ends (you can always go to x+1). It never repeats (x+1 is never equal to x or anything smaller than x). -- Daryl McCullough Ithaca, NY |