Galileo's Paradox
in [Math]
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From: Tony Orlow on 29 Nov 2006 13:40 Eckard Blumschein wrote: > On 11/29/2006 3:56 PM, Tony Orlow wrote: > >> Cardinality is generalized from the simple count of finite sets to the >> infinite case. In the finite case, the cardinality of a set is exactly a >> natural number, a quantity. In the infinite case, cardinality becomes >> something more ephemeral, > > Epheremal means shortlived. We have a saying: Lies live short. > > but it still has its roots in the count of a set. > > Let's rather say in Cantor's illusion of allegedly being able to count > the uncountable. > > >>> What about when there is more than one type of measure that can be >>> applied to a set, or none at all? What happens then? > > Then perhaps a red light will indicate logical error. > > > > 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.
From: Virgil on 29 Nov 2006 13:59 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.
From: Virgil on 29 Nov 2006 14:03 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.
From: Virgil on 29 Nov 2006 14:06 In article <456d9e96(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > >> To tie measure with count in the infinite is the task here, regarding such > >> sets. > > > > Why? What is the utility of performing this "task"? What new > > mathematics does it allow for? > > > > It allows for a rich ordering of infinite sets which satisfies intuitive > notions of set density in a completely generalized way. Granted, IFR > only works for sets of elements with inherent measure, but where the > elements have such measure, why ignore it? Because To's games ignore the whole panoply of measure theory and much of the rest of modern analysis.
From: Bob Kolker on 29 Nov 2006 14:09
Lester Zick wrote:> > Actually not because "infinity" a well defined mathematical concept > whereas "cardinality" is only an ambiguously defined concept > mathematically restricted to undemonstrable set analytical techniques. Nonsense. Two sets have the same cardinality if and only if there exists a one to one onto mapping from one to the other. That is a plain definition. Same cardinality produces and equivalence relation defined on sets. The cardinal number of a set is the equivalence class of sets with the same cardinality as the the given set. Don't give up your day job Lester. Bob Kolker |