Galileo's Paradox
in [Math]
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From: David Marcus on 6 Dec 2006 19:30 Eckard Blumschein wrote: > On 12/6/2006 5:35 AM, David Marcus wrote: > > Eckard Blumschein wrote: > > >> You did not understand that I am using Fourier transform as an example. > > > > Example of what? > > a typical mistake when using the immediate value. Be specific: what is the mistake? > >> I do not criticize FT but the integral tables, and I did not have a > >> problem myself but I recall several reported cases of unexplained error > >> by just the trifle of two. The integral tables suggest using the > >> intermediate value. > > > > You are criticizing integral tables? > > Some of them are rather eclectic and difficult to overlook. Others are a > bit slim. Sometimes the intermediate values are given, sometimes they > are omitted which i consider the better decision. I haven't a clue why you think integral tables have any relevance to set theory. > >> Experienced mathematicians should indeed know that > >> they must avoid this use. Some tables give the intermedite value for the > >> sake of putative mathematical correctness. > > > > Please give an example. > > I just have my old Bronstein-Semendjajew Teubner 1962 at hand. > On p. 351, number 13 does not give intermediate values, number 15 does. Please quote the entire example. Not everyone is next to a library. > >> Others omit it. > >> As long as one knows the result in advance, there is almost not risk. FT > >> and subsequent IFT may perform an ideal check of set theory. > > > > What do you mean "ideal check of set theory"? > > Set theory leads to intermediate values. What do you mean? Do try to be specific. -- David Marcus
From: David Marcus on 6 Dec 2006 19:38 Michael Press wrote: > In article > <jack-3EEBB8.13464706122006(a)newsclstr02.news.prodigy.com>, > Michael Press <jack(a)fake.net> wrote: > > In article <MPG.1fe01bc338e764a9899c8(a)news.rcn.com>, > > David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > > > Tony Orlow wrote: > > > > Virgil wrote: > > > > > In article <456f334d$1(a)news2.lightlink.com>, > > > > > Tony Orlow <tony(a)lightlink.com> wrote: > > > > >> Virgil wrote: > > > > >>> In article <456e4621(a)news2.lightlink.com>, > > > > >>> Tony Orlow <tony(a)lightlink.com> wrote: > > > > > > > > > >>>> Where standard measure is the same, there still may be an infinitesimal > > > > >>>> difference, such as between (0,1) and [0,1], if that's what you mean. > > > > >>> The outer measure of those two sets is exactly the same. > > > > >> Right, and yet, the second is missing two elements, and is therefore > > > > >> infinitesimally smaller in measure. > > > > > > > > > > Except that in outer measure there are no infinitesimals, and the outer > > > > > measure of the difference set, {0,1} is precisely and exactly zero. > > > > > > > > So, you're saying infinitesimals cannot be considered? You're saying one > > > > is not ALLOWED to consider the removal of a finite set from an ifninite > > > > set to make any difference in measure? I say you're wrong. > > > > > > I guess you still haven't figured out that in mathematics we make > > > precise statements and then make deductions. We don't just decide the > > > theorems based on what we wish. If a certain concept (e.g., cardinality, > > > measure, outer measure, Hausdorff measure, continuity, absolute > > > continuity, differentiability) doesn't do what we want, then we come up > > > with a new concept, state it precisely, and see if we can prove that it > > > does what we want. > > > > > > The term "outer measure" has a precise meaning that is given in courses > > > and books on Real Analysis. > > > > I mean to play devil's advocate. Mathematicians _do_ > > figure out what theorems they want, then construct > > minimal (hopefully) axiom sets to get those theorems. > > The intermediate value property was taken as the > > definition of continuous until folks started to examine > > functions like f(x) = sin 1/x, x <> 0; f(0) = 0. f has > > the ivp at 0, but does not satisfy the intuitive notion > > of continuous. Now ivp it a theorem, which is what we > > wanted, be it axiom or theorem. Agree 100%. That's what I was trying to say when I said "... come up with a new concept..." > s/axiom/definition/g > Please read `definition' where I wrote axiom. Thanks. -- David Marcus
From: imaginatorium on 6 Dec 2006 23:48 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. > > > that was made several posts earlier in the thread. SixLetters doesn't > > see why this leads to a contradiction. I tried to explain it. > > Whatever you "explanation", if you are referring to cardinality you are > just plain wrong. For goodness sake, Bob, if you want to play a part in fighting for truth, justice, and a crank-free future, learn to read a bit more carefully. Sixletters [sp?] may or may not be a crank, so let's just call him the OP. OP says (as quoted by MK, also on the side of TJAACFF): "(b) proper subsets are smaller than their supersets " All of us on the side of TJAACFF know that this statement is meaningless, unless you clarify what "smaller" means - and there are roughly two (2) ways of doing this: (1) "a is smaller than b iff a is a proper subset of b" (2) "a is smaller than b iff there is an injection of a to b, but not from b to a" In case (1), OP's statement is true, but (e.g.) a set of strings and the set of natural numbers are incomparable. In case (2) OP's statement is false. Since OP appears to think that some set of strings is the "same size" as the natural numbers, this implies case (2). Hence there is a contradiction in what OP said. MK is trying to explain this to OP. It is not really possible to explain what is wrong with an argument without reciting at least once some part of the fallacious argument, which is, of course, fallacious. It's bad enough trying to cope with the muddle of the genuinely confused cranks, without people like yourself repeatedly popping up, snipping bits from the demonstration of fallaciousness [is there a word 'fallacity'?] and chanting "WRONG"... Brian Chandler http://imaginatorium.org
From: Lester Zick on 6 Dec 2006 23:59 On Wed, 6 Dec 2006 14:10:20 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >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. I'm beginning to wonder if you can either one or both together. >> That is why ten thousand years of humans slicing and dicing each other's >> bodies has never revealed a mind. ~v~~
From: Lester Zick on 7 Dec 2006 00:01
On Wed, 06 Dec 2006 12:17:57 -0700, Virgil <virgil(a)comcast.net> wrote: >In article <4tnmr2F142pnkU2(a)mid.individual.net>, > Bob Kolker <nowhere(a)nowhere.com> wrote: > >> Virgil wrote:> >> > EB's arguments give us a plethora of examples of both unfoundedness and >> > the uselessness. >> >> That is why EB should be given the Zick Prize. >> >> Bob Kolker > >Isn't that pronounced "sick"? More like Aatu pronounced Virgil pathetic. ~v~~ |