Galileo's Paradox
in [Math]
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From: Lester Zick on 6 Dec 2006 12:56 On Wed, 06 Dec 2006 12:08:34 -0500, Bob Kolker <nowhere(a)nowhere.com> wrote: >Eckard Blumschein wrote: > >> >> >> Fortunately, I was never trained in set-quasi-religion. > >Set theory is not a religion. You mean it's true? > It is one of many abstract mathematical >systems. In other words it's not true? > You might as well refer to group theory as a "quasi-religion". Exactly the point. >In a sense all mathematical systems are "quasi-religions" so calling one >of many mathematical systems a quasi-religion says nothing. Religion has many branches. Faith based mathematics is one. >Your use of pejoratives is unsabstantial and reveals to all who see that >you are a shallow silly person. Then he should feel right at home. ~v~~
From: Eckard Blumschein on 6 Dec 2006 12:59 On 12/6/2006 5:27 AM, David Marcus wrote: > Eckard Blumschein wrote: >> On 12/5/2006 1:12 AM, David Marcus wrote: >> > Eckard Blumschein wrote: >> >> On 11/30/2006 4:41 AM, zuhair wrote: >> >> > Six wrote: >> >> >> >> > and I think it was the >> >> > idea before Cantor showed that there can be infinite sets of different >> >> > sizes, >> >> >> >> He did not! Not! Not! He just misinterpreted uncountable as more than >> >> countable. Is incorrect more than correct? >> > >> > Are you saying that the cardinality of the reals is not greater than the >> > cardinality of the integers? >> >> While I do not consider the generalized size (= Maechtigkeit alias >> cardinatity) justified, I nonetheless fully agree with the distinction >> between countable and uncountable. >> >> The reals are as uncountable as is sauce because they are merely >> fictitious numbers, no distinct elements. >> On the other hand, the naturals, integers, and rationals are genuine, >> i.e. discrete numbers. >> >> Look up and recall Galilei's insight: The relations smaller, equally >> large, and larger are invalid among infinite quantities. >> >> So it would not make any difference when the order aleph_0 and aleph_1 >> were reversed or more reasonably the correct denotation was preferred: >> countable instead of aleph_0 and uncountable instead of aleph_1. > > Yes, yes, I know you like to take standard words and give them new > (vague) meanings. This is a praxis introduced by mathematicians to an extent that even rich languages get short of terms without distorted meaning. > But, I was asking you whether the cardinality of the > reals is greater than the cardinality of the integers if we use the > standard meanings for the words "cardinality", "reals", and "integers". I consider cardinality nonsense. Just say countably infinite instead of a_0 and uncountable instead of a_1 and forget the rest. > >> Of course, there is no room for aleph_2 or even higher alefs in my >> reasoning. Therefore I asked: Is there any use of aleph_2? So far I did >> not get any example for such use. Obviously, all alefs are fancies. Only >> in case of a_0 and a_1 there is a reasonable but misinterpreted background. > > Depends on how you define "any use". Certainly the hierarchy of alephs > is very useful in mathematics. Cantor's motivation for his work was his > study of Fourier transforms. I know. > As for aleph_2 itself, because of the > independence of the continuum hypothesis, it doesn't come up too often > in mathematics. = never? > However, there were some interesting articles recently > in the Notices of the AMS that discussed axioms to add to ZFC. There > seemed to be good reasons to add an axiom which would make the > cardinality of the reals equal to aleph_2. And which role has been envisioned for aleph_1?
From: Eckard Blumschein on 6 Dec 2006 13:11 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. > >> 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. > >> 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. >> 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. Using belonging methods back and forth has to return the original. > >> Feel free to suggest a better one. > > A better what? Perhaps a better checker.
From: Lester Zick on 6 Dec 2006 13:15 On Wed, 6 Dec 2006 00:14:34 -0500, 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. Well see, David, the problem here is that mathematikers make plenty of statements which they claim are precise and then simply go on to make deductions about them which are not precise at all. They make statements about a set "theory" which can't exactly be a theory at all because it can't be proven true. Is this an example of a precise statement? In fact I find mathematikers routinely refer to "mathematics" when all they've actually been talking about is set "theory". I see no reason to employ parochial private set "theory" definitions for mathematics in general. Axioms and definitions in modern math are held to a much lower standards of value than theorems. Axioms and definitions only have to be interesting (Dik) or utilitarian (Bob) whereas theorems are supposed to be true (at least with respect to the axioms and definitions from which they're drawn). So if axioms and definitions are only expected to be interesting or utilitiarian I find it curious the same standard should not also apply to theorems. Would the rac trisection of general angles not lead to interesting and perhaps even utilitarian results? Would it really matter that it contradicts various axioms and definitions whose only justification in turn is interest and utility? >The term "outer measure" has a precise meaning that is given in courses >and books on Real Analysis. ~v~~
From: Lester Zick on 6 Dec 2006 13:18
On Wed, 6 Dec 2006 00:35:38 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >Virgil wrote: >> In article <45706F34.1070809(a)et.uni-magdeburg.de>, >> Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >> > The word completeness is misleading. >> >> "Complete" for an ordered set has a precise mathematical definition. >> That mathematical meaning is the only relevant meaning in any >> mathematical discussion of ordered sets. Most words used in technical >> senses in mathematics mean something quite different from their common >> meanings, and those who conflate the common with the technical meanings >> demonstrate their mathematical incompetence in so doing. > >It is interesting that most cranks seem unable to deal with the fact >that common words have technical meanings. No problem at all where technical meanings are not true. > They also seem to have >trouble divorcing a word from its meanings. Whereas modern mathematikers have no difficulty whatsoever divorcing the meaning of a word from truth. ~v~~ |