Obections to Cantor's Theory (Wikipedia article)
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From: Robert Low on 27 Jul 2005 17:15 Daryl McCullough wrote: > Tony Orlow <aeo6(a)cornell.edu> said: >>So, the sum of a finite number of finite terms is infinite. Sure. > No. The sum of a finite number of finite terms is finite, and the > sum of an infinite number of finite terms is infinite. I've lost the context here, but you know as well as I do that that isn't quite right: the sum of an infinite number of finite terms is infinite if the terms don't approach zero sufficiently fast. (Unless 'finite' meant 'non-zero natural number' here, in which case please disregard this letter...)
From: Tony Orlow on 27 Jul 2005 17:18 Daryl McCullough said: > Tony Orlow says... > > >despite the fact that an infinite set of whole numbers requires > >infinite whole numbers > > That's false, no matter how many times you say it. No finite > set can contain every (finite) natural. Why? Because every finite > set of naturals has a largest element, and there is no largest > finite natural. That's false, no matter how many times you say it. > > -- > Daryl McCullough > Ithaca, NY > > -- Smiles, Tony
From: David Kastrup on 27 Jul 2005 17:18 Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > Daryl McCullough said: >> Tony Orlow says... >> >> >despite the fact that an infinite set of whole numbers requires >> >infinite whole numbers >> >> That's false, no matter how many times you say it. No finite >> set can contain every (finite) natural. Why? Because every finite >> set of naturals has a largest element, and there is no largest >> finite natural. > That's false, no matter how many times you say it. Daryl told you _why_. That won't go away by whining. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: Robert Low on 27 Jul 2005 17:19 Tony Orlow (aeo6) wrote: > Daryl McCullough said: >>Tony Orlow (aeo6) <aeo6(a)cornell.edu> 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. I guess that wasn't true. For somebody who claims to have passed courses in discrete maths with As, you have a funny way of arguing. The definition of a string over an alphabet is that it is a finite sequence of the symbols in that alphabet. Then saying that there is no restriction on their length is saying that the length can be any finite number, no matter how large. Note that (except in your own private conceptual universe) saying that there is no upper bound to the length of strings is the same as saying they can become arbitrarily long, but different from saying that they can be of infinite length.
From: malbrain on 27 Jul 2005 17:21
David Kastrup wrote: > malbrain(a)yahoo.com writes: > > > David Kastrup wrote: > >> malbrain(a)yahoo.com writes: > >> > >> > David Kastrup wrote: > >> >> malbrain(a)yahoo.com writes: > >> >> > >> >> > malbr...(a)yahoo.com wrote: > >> >> >> Tony Orlow (aeo6) wrote: > >> >> >> > 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. > >> >> > > >> >> > Ooops. "indeterminate" is the exact opposite of finite. You've > >> >> > uncovered a contradiction about the count of elements in an > >> >> > infinite set that cannot be resolved from the definitions of > >> >> > finite and indeterminate. karl m > >> >> > >> >> Uh, no. "Indeterminate" just means unspecified, not infinite. > >> > > >> > Read the definition. Determinate=defined limit; > >> > indeterminate=undefined limit=infinite. > >> > >> Undefined limit is not the same as infinite. For example, in > >> programming languages indeterminate loop forms are those for which you > >> can't say in advance how often they will be run (i.e., while-loops, as > >> opposed to the determinate for-loops). > > > > Programming languages weren't invented in 1913. Please use > > analogies that are pre-Cantor. The opposite of "undefined-limit" is > > infinite. > > Algorithms are definitely pre-Cantor. Euklid's algorithm (a few > thousand years old) for finding the greatest common divisor of two > numbers takes an indeterminate number of iterations. Ok, here's a definition of "indeterminate" from the 1913 dictionary: Indeterminate problem (Math.), a problem which admits of an infinite number of solutions, or one in which there are fewer imposed conditions than there are unknown or required results. karl m |