Galileo's Paradox
in [Math]
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From: Eckard Blumschein on 27 Nov 2006 05:54 On 11/25/2006 1:42 AM, Bob Kolker wrote: > Eckard Blumschein wrote: >> I already wrote elsewhere that Galilei eventually found out: >> The relations smaller, equally large, and larger do not hold for >> infinite quantities. > > Cardinal numbers are comparable. My father told me the story of a little poor boy who in the twenties of last century bought mixed sweets. The salesman gave him a red one and a green one and said: Do mix them yourself. Having some evidence that the green sweet means countable and red one uncountable, I can not even know that red is lager than green or vice versa. Eckard Blumschein
From: Bob Kolker on 27 Nov 2006 06:46 Eckard Blumschein wrote: > > Having some evidence that the green sweet means countable and red one > uncountable, I can not even know that red is lager than green or vice > versa. A totally incoherent analogy. You are in fine form today. Bob Kolker
From: Eckard Blumschein on 27 Nov 2006 08:58 On 11/26/2006 9:23 PM, Six wrote: > On Fri, 24 Nov 2006 22:03:49 +0100, Eckard Blumschein >>According to Spinoza, infinity is something that cannot be enlarged >>(and also not exhausted). It is a quality, not a quantity. >> >>There is only one such ideal concept, not different levels of infinity. >> > I understand that there is such a concept of infinity. But it is > not one I am appealing to. Something similar though is one way of > responding to the paradox. See comments below on your other post. > (Incidentally, I read, and admired Spinoza many years ago. I would now have > to re-read to see the relevance.) There is only one ideal concept of actual infinity. Spinoza was possibly the first one who clearly understood that infinity cannot be enlarged. If it cannot be enlarged, it can also not be exhausted. This concept is very useful in engineering. For instance, electrical engineers consider ideal sources of voltage independent of current.
From: Eckard Blumschein on 27 Nov 2006 09:21 On 11/26/2006 9:24 PM, Six wrote: > On Fri, 24 Nov 2006 21:54:48 +0100, Eckard Blumschein > <blumschein(a)et.uni-magdeburg.de> wrote: > >>On 11/24/2006 12:35 PM, Six wrote: >>> GALILEO'S PARADOX >> >> >>I already wrote elsewhere that Galilei eventually found out: >>The relations smaller, equally large, and larger do not hold for >>infinite quantities. > > Indeed this is one, radical, way of resolving the paradox. And it > is not clear to me (maybe in my ignorance) that there is any purely > MATHEMATICAL way of rejecting this. > I should point out that I really do not expect any part of > mathematics to fall over as a result of anything I post here. I am a > beginner (relatively), and the best I am hoping for is some enlightenment. You may hopefully also enlighten the putativly knowing and understanding all by means of set theory. > I am, if you like, as Descartes presents himself to be, looking to expand > his knowledge by a chain of certain, deductive links (futile, I know, but > it's the way I'm inclined to set about things). > > As Tolbin astutely points out, it may come down to a choice of > prejudices. We cannot keep both of the intuitions (fine for finite sets) > that a !:1 C means identity of size and that a set should be greater than > its proper subset. I thought I had presented some reasonable arguments to > the effect that it is the first intuition that should be dropped. Cantor was dominated by his first intuition when he came out with his claim to be able to quantify infinity. > It seemed > to me that accepted practice and the resultant prejudice is all the other > alternative has to recommend it. Perhaps, though, accepted practice should > not, in the end, be despised. Why not if it is unfounded? > If the result is some wonderful mathematics, > and the alternative is a dead end. Neither nor, on the contrary: Set theory did not create progress but caused endless botchering with CH, ZFC, ZF, MBG, etc., and it hinders purification of mathematics. There is not even a single application for aleph_2. > Perhaps that is all the answer there is. > It does not seem very satisfying, at least at first, and I doubt that all > of the mathemiticans themselves would understand it. But at least it would > tell you something about the nature of mathematics. Initially I met mathematical problems. Meanwhile most of them burned down to human stubborn predjudice and common biasedness. > I have every respect for the mathematicians here, and I'm sure (not > that I'd be able to tell) that some real, creative, practical mathematical > work is done here. I can understand that these questions about fundamentals > from a mathematical ignoramus are an annoying distraction (or source of > amusement) for many. Even so, one would expect a reasonable reply to a > reasonable argument. Hopefully this will be respected. > > > Thanks, Six Letters > >>Dedekind and Cantor intended to make possible the impossible and just >>enchant irrationals into numbers of the same kind as rational numbers. >>If this would be possible, then there were indeed more real than >>rational numbers. >> >>So the history of mathematics was stained by more than a century of >>unnecessary confusion. >> >>E. B.
From: Eckard Blumschein on 27 Nov 2006 09:27
On 11/27/2006 12:35 AM, Jesse F. Hughes wrote: > Six Letters writes: It isn't a >> question of what works. It's a question of how the paradox is to be >> resolved. > > But the resolution needs to also discuss set sizes for non-comparable > sets. Otherwise, it's a pretty dull notion of set size. We already > have the subset operation. Not in any case. |