Obections to Cantor's Theory (Wikipedia article)
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From: Tony Orlow on 27 Jul 2005 17:32 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. -- Smiles, Tony
From: Daryl McCullough on 27 Jul 2005 17:14 Tony Orlow (aeo6) wrote: >If I am just winging it, I'm doing pretty good. In what sense? >Do you have a specific incosistency you'd like to mention? I don't see it. That's not surprising. The ability to see one's own mistakes requires a certain amount of competence. Here's an inconsistency: 1. You agree that every finite set has a size that is equal to some finite natural number. 2. You agree that for every natural number n, if a set has exactly n+1 elements, then that set has a largest element. 3. You claim that the set FN of all finite naturals is finite. 4. You deny that FN has a largest element. That's an out-and-out logical contradiction, Tony. -- Daryl McCullough Ithaca, NY
From: Tony Orlow on 27 Jul 2005 17:35 Daryl McCullough said: > Tony Orlow (aeo6) wrote: > > >Robert Low said: > > > >> 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. > > So you really think that there is some number n such that n is > finite, but if you add 1 you get an infinite number? (sigh) This is the last time I answer this question NOOOOOOOOOOO!!!!!!!!! > > Maybe it's 7? Maybe 7 is the largest finite number, and > 8 is actually infinite? Don't be stupid. > > >It doesn't exist? Well, then, I can't help you. See? I already answered the question. > > In fact, a set is finite if and only > if the number of elements is equal to a natural number. > There is no largest natural number, and there is no > largest finite set. The collection of all finite natural > numbers is an infinite set. The set of all finite numbers up to a given number has that number in it, which is also the set size. Any subset of N has a size that is in N. > > -- > Daryl McCullough > Ithaca, NY > > -- Smiles, Tony
From: Robert Low on 27 Jul 2005 17:50 Tony Orlow (aeo6) wrote: > Daryl McCullough said: > >>Tony Orlow (aeo6) wrote: >> >> >>>Robert Low said: >>> >>> >>>>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. >> >>So you really think that there is some number n such that n is >>finite, but if you add 1 you get an infinite number? > > (sigh) This is the last time I answer this question > NOOOOOOOOOOO!!!!!!!!! Or then again... Daryl wrote: So your claim that FN is finite implies that there is some number n that is the largest finite natural number. That means that n is finite n+1 is infinite By any sane definition of "infinite" I would think that would be impossible. And you replied: .... In the case of the finite naturals, it simply does not hold true. So is that NOOOOOOOOO! or YESSSSSSSSSSS! ? (I know it's more convenient if it's both, depending on what you're claiming at any given moment, but that's cheating.) > The set of all finite numbers up to a given number has that number in it, which > is also the set size. Any subset of N has a size that is in N. So which element of N is the size of the set of even numbers? Hang on a minute...there something sticking into my gum...I'll just pull it out. Well, I'll be damned, it's a hook! How did that get there?
From: David Kastrup on 27 Jul 2005 17:52
Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > Daryl McCullough said: >> Tony Orlow (aeo6) wrote: >> >> >Robert Low said: >> > >> >> 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. >> >> So you really think that there is some number n such that n is >> finite, but if you add 1 you get an infinite number? > (sigh) This is the last time I answer this question > NOOOOOOOOOOO!!!!!!!!! >> >> Maybe it's 7? Maybe 7 is the largest finite number, and >> 8 is actually infinite? > Don't be stupid. >> >> >It doesn't exist? Well, then, I can't help you. > See? I already answered the question. >> >> In fact, a set is finite if and only >> if the number of elements is equal to a natural number. >> There is no largest natural number, and there is no >> largest finite set. The collection of all finite natural >> numbers is an infinite set. > The set of all finite numbers up to a given number has that number > in it, which is also the set size. Correct. > Any subset of N has a size that is in N. Incorrect. The size of the set of positive even numbers, for example, is not in N. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum |