Galileo's Paradox
in [Math]
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From: David Marcus on 6 Dec 2006 14:06 Eckard Blumschein wrote: > On 12/5/2006 7:20 PM, Tony Orlow wrote: > > When we divide the line by this number, we get one point, > > This is the trick which provides anything. Are you reaaly so naive? The pot calling the kettle black. People who live in glass houses shouldn't throw stones. > > the 0D square. > > There's really no way you can convince me that the cube does not have > > infinitely more points than the square, or the square than the segment, > > or the segment than the point. > > Even Georg Cantor eventually accepted this while being hampered by the > same kind of intuitive thinking like you. He wrote: Je le vois, mais je > ne le crois pas. > > These are different levels of infinity. > > You did not even understand Cantor. How will you understand me? Actually, that's a good question. -- David Marcus
From: David Marcus on 6 Dec 2006 14:08 Tonico wrote: > Tonico ha escrito: > > Bob Kolker ha escrito: > > > David Marcus wrote: > > > > > > > > As Virgil pointed out, "no bijection to the naturals" is not a correct > > > > definition of "uncountable". > > > > > > Eh? > > > > > > See: http://en.wikipedia.org/wiki/Uncountable > > > > > > In mathematics, an uncountable or nondenumerable set is a set which is > > > not countable. Here, "countable" means countably infinite or finite, so > > > by definition, all uncountable sets are infinite. Explicitly, a set X is > > > uncountable if and only if there is an injection from the natural > > > numbers N to X, but no injection from X to N. > > > > > > From the wiki article on countability. > > > > > > Do you see in error in the wikipedia entry on uncountability > > > > > > > > > Bob Kolker > > ************************************************** > > Hi: > > I think you misunderstood something: we don't have a bijection from the > > set {x} to the naturals, and {x} is uncountable. > > So I think Virgil is right: no bijection to the naturals is NOT a > > correct definition of uncountable UNLESS we add the condition "...from > > an infinite set ...." > > Of course, perhaps you agree with this and I misunderstood something. > > Regards > > Tonio > *********************************** > Of course, I meant above "...and {x} is NOT uncountable" Right. -- David Marcus
From: David Marcus on 6 Dec 2006 14:10 Bob Kolker wrote: > David Marcus wrote: > > > > Analogy: mind is software, brain is hardware. > > underline the word -analogy-. Mind, as conceived of by the classical > philosophers, is a substance which differs from material substances. Res > Cogitens vs. Res extensa. This is essentially Cartesian Dualism and is > empirically unfounded. Another analogy. Brain is the instrument. Mind is > the music. In any case mind appears to be an epiphenomenon of the brain. > It is an effect of the physical activities of the brain. No brain, no > mind. Mind is not a stand-alone object. Agreed. Similarly, you can't have running software with hardware. > That is why ten thousand years of humans slicing and dicing each other's > bodies has never revealed a mind. -- David Marcus
From: Virgil on 6 Dec 2006 14:15 In article <4tnm3vF14s1e8U1(a)mid.individual.net>, Bob Kolker <nowhere(a)nowhere.com> wrote: > David Marcus wrote: > > > > > > As Virgil pointed out, "no bijection to the naturals" is not a correct > > definition of "uncountable". > > Eh? According to the standard of "no bijection to the naturals" every finite set would have to be labeled uncountable. > > See: http://en.wikipedia.org/wiki/Uncountable > > In mathematics, an uncountable or nondenumerable set is a set which is > not countable. Here, "countable" means countably infinite or finite, so > by definition, all uncountable sets are infinite. Explicitly, a set X is > uncountable if and only if there is an injection from the natural > numbers N to X, but no injection from X to N. > > From the wiki article on countability. > > Do you see in error in the wikipedia entry on uncountability > > > Bob Kolker No error, but also little relevance to the issue of whether "no bijection to the naturals" is a correct definition of "uncountable". According to that standard, the empty set is uncountable. A proper definition could be be phrased "no injection to the naturals", but NOT as "no bijection to the naturals".
From: David Marcus on 6 Dec 2006 14:15
Bob Kolker wrote: > Mike Kelly wrote: > > > > By "b)" I was referring to the statement > > > > "(b) proper subsets are smaller than their supersets " > > what do you mean by "smaller"? If you mean the cardinality, what you say > is just plain wrong. The set of even integers has the same cardinality > as the set of integers, for example. (b) is intended as a definition (or property) of "smaller" (whatever the latter means). -- David Marcus |