Galileo's Paradox
in [Math]
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From: Virgil on 5 Dec 2006 17:14 In article <457584A4.3000108(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/5/2006 12:16 AM, Virgil wrote: > > In article <4574755B.4070507(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> 2*oo is not larger than oo. Infinity is not a quantum but a quality. > > > > Which "infinity" is that? > > We do not need different infinities. Cantor did. And showed why with two separate proofs.
From: Virgil on 5 Dec 2006 17:21 In article <457586BB.9020406(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/5/2006 12:09 AM, Virgil wrote: > > In article <45745B16.40202(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > >> On 12/1/2006 9:59 PM, Virgil wrote: > >> > >> > Depends on one's standard of "size". > >> > > >> > Two solids of the same surface area can have differing volumes because > >> > different qualities of the sets of points that form them are being > >> > measured. > >> > >> Both surface and volume are considered like continua in physics as long > >> as the physical atoms do not matter. > >> Sets of points (i.e. mathematical atoms) are arbitrarily attributed. > >> There is no universal rule for how fine-grained the mesh has to be. > >> Therefore one cannot ascribe more or less points to these quantities. > >> > >> Look at the subject: Galileo's paradox: The relations smaller, equally > >> large, and larger are pointless in case of infinite quantities. > > > > Then length and area and volume comparisons of size must be fictional > > measurements. > > I do not think so. You deliberately mistook the sentence. No measured > length, area, or volume is infinite. But they are measures of fictional qualities in EB's rubric, so must be uncountable. > > > > >> > >> > >> > Sets can have the same cardinality but different 'subsettedness' because > >> > different qualities are being measured. > >> > >> I do not know a German equivalent to subsettedness. > > > > It is the quality of being a subset. It is one, but not the only, > > measure of set "size". > > Cantor claimed to have discovered different sizes of infinite sets. Who > introduced substtedness, and is it also quantified? The subset relation, as a partial ordering of sets, has been around as long as sets. > >> Standard mathematics may lack solid fundamentals. At least it is > >> understandable to me. However, I admit being not in position to likewise > >> easily understand what you mean with well-ordered subsettedness. > > > > The subset relation does not provide a well ordering of arbtrary sets. > > Nobody does provide a well-ordering of the irrationals. What has that to do with whether one can well order a set of sets by inclusion? EB seems compelled to go off on tangents.
From: Virgil on 5 Dec 2006 17:23 In article <4575b508(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Eckard Blumschein wrote:(deleted) > > Now, just a minute, Eckard. You're contradicting yourself If even TO is taking EB to task, then EB must indeed be in a bad skin.
From: Virgil on 5 Dec 2006 17:27 In article <1165333562.354313.109050(a)j44g2000cwa.googlegroups.com>, "Tonico" <Tonicopm(a)yahoo.com> wrote: > Eckard Blumschein ha escrito: > > > On 12/4/2006 11:52 PM, Virgil wrote: > > > > >> I am an electrical engineer > > > > > > Shocking! > > > > Why? We love mathematics. > *************************************************** > Oh, I bet elec. eng. love maths; the problem seems to be that maths > does not correspond AT ALL that love, at least in the case of several > engineers...**sigh**...tough. > Tonio Electrical stuff can be shocking to those who use it carelessly or abuse it, and judging by EB's careless abuse of mathematics, he is at risk.
From: Virgil on 5 Dec 2006 17:36
In article <45758F7C.8020107(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/4/2006 11:57 PM, Virgil wrote: > > In article <45744E30.8090207(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > >> On 12/1/2006 10:33 PM, Virgil wrote: > >> > In article <45706AD1.808(a)et.uni-magdeburg.de>, > >> > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> > >> > You claimed many "imperfectins" but did not justify those claims with > >> > anything mathematically valid. > >> > >> When I performed Fourier transform back and forth for a function > >> stepping at t=a, cf. > >> http://iesk.et.uni-magdeburg.de/~blumsche/M283.html > >> I correctly returned to the original function iff I ignored the > >> intermediate value at t=a and decided to extend integration from t<a > >> instead. > > > > That Fourier transforms do not do precisely as you wish they might, does > > not constitute an imperfection in the transforms so much as an > > imperfection of your understanding of what they can do. > > > > If you choose to hammer with a wrench, you may not always get the result > > a hammer would produce. > > You did not understand that I am using Fourier transform as an example. > 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. Experienced mathematicians should indeed know that > they must avoid this use. Some tables give the intermedite value for the > sake of putative mathematical correctness. 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. > > Feel free to suggest a better one. > > > > >> > >> >> >> I guess, point-set topology and measure > >> >> >> theory do not require the claim of set theory to rule all > >> >> >> mathematics. > >> >> > > >> >> > They cannot exist without a foundation of set theory. > >> >> > >> >> In this case they could not exist. Set theory does not have a solid > >> >> basis. So I doubt. > >> > > >> > There are a lot of textbooks on point-set, and other, topologies and on > >> > measure theory. I have yet to see one of them that is not based on set > >> > theory. If EB claims these books do not exist, he is even more foolish > >> > than usual. > >> > >> I do not claim this. I just guess that not a single one really needs the > >> transfinite numbers and nonsense cardinalities like aleph_2. > > > > They certainly need the cardinality of the (real) continuum. > > A continuum is continuous. Uncountable is a sufficient characterization. Then the Cantor set, which is totally discontinuous in that every two members are separated by a non-member, by being uncountable, is simultaneously continuous. > One has to be pretty miseducated in order to need the fancy aleph_1. That depends on whether one accepts or rejects the continuum hypothesis. |