Galileo's Paradox
in [Math]
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From: Bob Kolker on 29 Nov 2006 06:29 Tony Orlow wrote: > It is a direct consequence of the notion that a proper subset is always > smaller, in some sense, than the base set. And in another sense a proper subset of an infinite set has the same count as the set. Bob Kolker
From: Bob Kolker on 29 Nov 2006 06:30 Eckard Blumschein wrote:> > No. They are thought to contain infinitely many rational numbers, yes. > However, they do not contain infinitely many but uncountably much real > "numbers". Countable or uncountable is an alternative but no measure. > Incidentally, do not equate intervals and sets. Continuously filled > intervals are merely fictitious sets. And length, areas and volumes are delusions. Keep that in mind the next time you buy a rug or carpet. Bob Kolker >
From: Bob Kolker on 29 Nov 2006 06:36 Mike Kelly wrote: > 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? > > Forget integrating count and measure; it's not even possible to define > a measure that applies to all sets. The Banach-Tarski "paradox" is the example, par excellence, that what you say is the case. Just about any kind of additive measure leads to non-measurable sets. Bob Kolker
From: Eckard Blumschein on 29 Nov 2006 06:50 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.
From: Eckard Blumschein on 29 Nov 2006 06:55
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. |