Galileo's Paradox
in [Math]
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From: Eckard Blumschein on 30 Nov 2006 05:27 On 11/29/2006 7:59 PM, Virgil wrote: > In article <456D7417.30000(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 11/28/2006 10:40 PM, Virgil wrote: >> > Eckard Blumschein wrote: >> >> >> The relations smaller, equally large, and larger are invalid for >> >> infinite quantities. >> > >> >> > All one needs do is divorce the "length" from the "number of points", >> > which is probably what Galileo did, as being different sorts of measures >> > (like weight versus volume), and the problem disappears. >> >> No. Infinite quantities include e.g. an infinite amount of points. >> Infinite means: The process of quantification has not been finished or >> cannot be finished at all. > > That may be your personal definition of infiniteness, but is not > everyones, and does not govern anyone but yourself. Cantor himself was the victim of his own stupid notion of infinity. He wrote: There are not more points in a cube than in a line. I see it but I cannot believe it. "Not more" was correct and logical.
From: Eckard Blumschein on 30 Nov 2006 05:41 On 11/29/2006 8:03 PM, Virgil wrote: > In article <456D7544.8090000(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 11/28/2006 10:31 PM, Virgil wrote: >> >> > There is no such thing as "genuine" for numbers in mathematics. >> >> Maybe it will exist in genuine mathematics. >> >> >> > So that EB has just refused to accept all of Analysis, including >> > calculus, which is based on just the sort of sets that EB denies exist. >> >> This is perhaps a lie. I feel well served by pre-Cantorian analysis and >> by modern mathematics which does not really rely on set theory. > > Since all of "pre-Cantorian analysis" is embeddable in "Cantorian > analysis" without loss, and with some gains (e.g., point-set topology > and measure theory), there is nothing to be gained by such retrograde > devolution. While colonies were changed when embedded in a Commonwealth, Cantorian distemper did effectively almost not at all affect mathematics. I just see some imperfections. I guess, point-set topology and measure theory do not require the claim of set theory to rule all mathematics. I wonder if they require aleph_2.
From: Eckard Blumschein on 30 Nov 2006 06:08 On 11/29/2006 8:13 PM, Bob Kolker wrote: > Tony Orlow wrote: >> >> Uncountable simply means requiring infinite strings to index the >> elements of the set. That doesn't mean the set is not linearly ordered, >> or that there exist any such strings which do not have a successor. > > Uncountable means infinite but not of the same cardinality as the > integers. For example the set of real numbers. It is an infinite set, > but it cannot be put into one to one correspondence with the set of > integers. Uncountable means: Counting is impossible. This property obviously belongs to fictitious elements of continuum. There is simply too much of them. So counting is not feasible. As long as one looks at a finite, just potentially infinite heap of single integers, one has to do with individuals. The set of all integers is something else. It is a fiction. It is to be thought constituted of an uncountable amount of non-elementary elements. Well this looks nonsensical. There is indeed a selfcontradiction within the notion of an infinite set. Non-elementary means not having a distinct numerical address. Element means "exactly defined by an impossible task". Eckard Blumschein
From: Eckard Blumschein on 30 Nov 2006 06:11 On 11/29/2006 8:15 PM, Bob Kolker wrote: > Eckard Blumschein wrote: > >> >> No. Infinite quantities include e.g. an infinite amount of points. >> Infinite means: The process of quantification has not been finished or >> cannot be finished at all. > > > A non-empty set is infinite if and only if it can be put in one to one > correspondence with a proper subset of itself. That is the standard > definition of infinite for sets. I consider Dedekind wrong, and he admitted to have no evidence in order to justify his basic idea.
From: Tonico on 30 Nov 2006 07:00
Eckard Blumschein wrote: > On 11/29/2006 7:59 PM, Virgil wrote: > > In article <456D7417.30000(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > >> On 11/28/2006 10:40 PM, Virgil wrote: > >> > Eckard Blumschein wrote: > >> > >> >> The relations smaller, equally large, and larger are invalid for > >> >> infinite quantities. > >> > > >> > >> > All one needs do is divorce the "length" from the "number of points", > >> > which is probably what Galileo did, as being different sorts of measures > >> > (like weight versus volume), and the problem disappears. > >> > >> No. Infinite quantities include e.g. an infinite amount of points. > >> Infinite means: The process of quantification has not been finished or > >> cannot be finished at all. > > > > That may be your personal definition of infiniteness, but is not > > everyones, and does not govern anyone but yourself. > > Cantor himself was the victim of his own stupid notion of infinity. > He wrote: There are not more points in a cube than in a line. > I see it but I cannot believe it. ************************************************************** So..?? You cannot see beyond your own nose: what's new, then? Everything came up clear to me when you wrote, black on white, that you are engineer. So clear now how it is possible that you know so little of maths! Of course, since most high school teachers are engineers, accountants, architects, etc., and in general people that doesn't have the faintest idea what maths REALLY can be (and not what it is, unfortunately, at HS level), no wonder they screwed you up a little too much in school. Unfortunately, somehow you gathered the nerve and chutzpah to try to deal with mathematical stuff that is way over your head...with mathematicians. And this when you're armed only with your....engineer "maths"!!!!!! Unbelievable, uh? So you cannot see it....hehe: what else is new? Tonio Ps Have you, and anyone else, noted how all the anticantorian cranks are NEVER mathematicians? But Internet welcomes all, and google's sci.math is an uncensored group, so anyone can offer his piece to all...and you know what? I think this is just fine. I'm convinced that also from the most stupid, dense and even annoying crank/troll we all can learn. |