Obections to Cantor's Theory (Wikipedia article)
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From: malbrain on 27 Jul 2005 17:53 Tony Orlow (aeo6) wrote: > malbrain(a)yahoo.com said: > > Tony Orlow (aeo6) 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. > > > > >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. karl m > > > > > Indeetrminate means the opposite, so: without defined limits; uncertain or > arbitrary; not fixed; unestablished; indefinite. > That sounds about right, especially the first definition. Then you agree that the number of natural numbers is not both finite and indeterminate since they mean the opposite, and that this number is up-for-grabs? (in 1913)? karl m
From: Virgil on 27 Jul 2005 18:06 In article <MPG.1d51931a2be0c263989fb1(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Virgil said: > > In article <MPG.1d500de16aaa9f18989f85(a)newsstand.cit.cornell.edu>, > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > > > that is not at all my logic. Pay attention. > > > > > > Since we have never seen anything from TO that qualifies as logic, there > > has been nothing to pay attention to, at least of logic. > > > > TO claims that an iterative process, n -> n+1, that has no end must > > achieve actual infinity, but cannot say at which iteration such a > > transformation occurs. At the same time TO claims that the iterative > > process, 1/n -> 1/(n+1), never actually reaches zero. > > > Again you snip everything and then make false statements. Since without snipping, posts, particularly TO's, grow like topsy (but never to TO's naturally infintie natural length), snipping irrelevancies is good manners. > You really need to grow up. What axioms doe that follow from? > Inductive proof doesn't "achieve actual infinity" and I never said > that, And I did not said TO did say that. <QUOTE> > > TO claims that an iterative process, n -> n+1, that has no end must > > achieve actual infinity, but cannot say at which iteration such a > > transformation occurs. <UNQUOTE> > Why don't you try talking about what YOU think instead of trying to > malign others Whenever I do that TO maligns me. As is quite common, there is nothing the least mathematical or logical in TO's posting, it is all ad hominem.
From: malbrain on 27 Jul 2005 18:08 Tony Orlow (aeo6) wrote: > malbrain(a)yahoo.com said: > > I think you mean that the machine has finite runtime. That's what the > > OPERATING SYSTEM enforces. My java-virtual-machines strike the program > > dead with a BOLT OF LIGHTNING when it runs amok. karl m > > > > > OOOPS!!! Well, MY computer only has limited funtime, unfortunately. Heh! > > Ummm, doesn't all that lightning damage the motherboard? No. That's the beauty of mathematics. It's a VIRTUAL-BOLT. karl m
From: Virgil on 27 Jul 2005 18:14 In article <MPG.1d519425a540e748989fb2(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Daryl McCullough said: > > Tony Orlow (aeo6) says: > > > > >Except for the fact that somehow you got an infinite set in > > >a finite number of steps, producing 1 element at a time. > > >How does that work? > > > > No, if you produce one element at a time, then there is never > > a time in which you will have produced an infinite set. Once > > again, you're having quantifier problems. > Stop saying that. It's bullshit. > > > > Let enum(e,s,t) mean "enumeration e produces element s at > > or before step t". Then > > > > 1. S is finite > > <-> exists enumeration e, exists step t, forall s in S, > > enum(e,s,t) > > > > 2. S is countable > > <-> exists enumeration e, forall s in S, exists step t, > > enum(e,s,t) > > > > Note the difference between 1. and 2. The difference is in > > the order of quantifiers. That difference is important. > > The definition of finite says that there is some *maximum* > > number of steps t, independent of the element s, at which > > you know that you will have enumerated s. The definition > > of countable says that for each s, there is a corresponding > > t (depending on s) at which you will have enumerated s. > > > > -- > > Daryl McCullough > > Ithaca, NY > > > > > That has nothing to do with it. Do you produce the naturals through this > inductive process ala Peano, adding 1 each time to produce the next natural > number? Is it an infinite set? So, are you not adding 1 an ifninite number of > time sto generate an infinite number of naturals? I can't argue this anymore. > It's too obvious. It is certainly too obvious that TO is wrong! In order to produce TO's result, you must be able to add 1 to a finite quantity to get an infinite quanity. There must be some point at which one crosses the line between finite and infinite. Unless TO proposes some kind of continuous gradation from purely finite to purely infinite with every level from 100% finite at one end to 100% infinite at the other. And even then there is a 50% point.
From: malbrain on 27 Jul 2005 18:18
Tony Orlow (aeo6) wrote: > Barb Knox said: > > In article <MPG.1d503dd82269292f989f93(a)newsstand.cit.cornell.edu>, > > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > > [snip] > > >> > >> >> >Barb Knox wrote: > > >> > >> >> >> In article > > >> > >> >> >> <MPG.1d4ecd45545679a8989f6b(a)newsstand.cit.cornell.edu>, > > >> > >> >> >> Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > >> > >> >> >> [snip] > > >> > >> >> >> > > >> > >> >> >> >keep in mind thatinductive proof IS an infinite loop, > > >> > >> >> >> >so that incrementing in the loop createsinfinite values, > > >> > >> >> >> >and the quality of finiteness is not maintained > > >> > >> >> >> >over thoseinfinite iterations of the loop. > > >> > >> >> >> > > >> > >> >> >> Using your computational view, consider the following > > >> > >> >> >> infinite loop (using some unbounded-precision arithmetic > > >> > >> >> >> system similar to java.math.BigInteger): > > >> > >> >> >> > > >> > >> >> >> for (i = 0; ; i++) { > > >> > >> >> >> println(i); > > >> > >> >> >> } > > >> > >> >> >> > > >> > >> >> >> Now, although this is an INFINITE loop, every value printed will > > >> > >> >> >> be FINITE. Right? > > [snip] > > > > >In any case, sure, the program will spit out finite numbers, > > > > Right. > > > > >snce it is a finite machine running in finite time. > > > > Wrong. Just like your notion of "infinite loop" over the naturals is > > not encumbered by limited memory, neither is this program. Consider it > > to be running on a machine with unbounded memory, so it could literally > > run forever. > > > > >If the machine had infinite capacity > > > > Yes, that's the appropriate model. > > > > >and infinite funtime, it could conceivably produce infinite results. > > > > HOW?? No matter how long it runs, EVERY printed value is finite. > > WHEN EXACTLY do you think it would start producing "infinite" values > > (whatever those might look like). > > > > > At the point that the runtime became infinite, which is obviously not an > identifiable point. At what point DOES the runtime become infinite? A million > years? a billion? If you have infinite runtime, starting with a finite amount > of time, then you have infinite numbers, starting with finite ones. You can't > have it both ways. There is no need for infinite run times. You are assuming that each step requires a CONSTANT amount of time. Mathematics makes no such requirement. See Zeno's paradox. Each of the values that the "java" program produces is finite, yet there are an infinite number of them. karl m |