Galileo's Paradox
in [Math]
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From: Six on 26 Nov 2006 15:23 On Fri, 24 Nov 2006 22:03:49 +0100, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >On 11/24/2006 1:20 PM, Richard Tobin wrote: >> >>>* If you have two sets of infinite size, is the union of these sets >>>than larger than infinity? What would larger than infinity mean? >> >> That this is not a stumbling block should be clear of you replace >> "infinite" with "big". The union of two big sets can be bigger than >> either of the original big sets. Like "big", "infinity" is not a >> number; it's a description of certain numbers. > >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.) Six Letters
From: Six on 26 Nov 2006 15:23 On Fri, 24 Nov 2006 18:26:37 +0000 (UTC), stephen(a)nomail.com wrote: >Six wrote: >> On Fri, 24 Nov 2006 16:04:12 +0000 (UTC), stephen(a)nomail.com wrote: > >>>Six wrote: >>> >>><snip> >>> >>>> I want to suggest there are only two sensible ways to resolve the >>>> paradox: >>> >>>> 1) So- called denumerable sets may be of different size. >>> >>>> 2) It makes no sense to compare infinite sets for size, neither to say one >>>> is bigger than the other, nor to say one is the same size as another. The >>>> infinite is just infinite. >>> >>>> >>>> My line of thought is that the 1:1C is a sacred cow. That there is >>>> no extension from the finite case. >>> >>>What do you mean by that? The one-to-one correspondence works >>>perfectly in the finite case. That is the entire idea behind >>>counting. Given any two finite sets, such as { q, x, z, r} and >>>{ #, %, * @ }, there exists a one-to-one correspondence between >>>them if and only if they have the same number of elements. >>>This is the idea that let humans count sheep using rocks long >>>before they had names for the numbers. > >> I love this quaint, homely picture of the origin of arithmetic. I >> am sure that evolutionary arithmetic will soon be taught in universities, >> if it is not already. Disregarding the anthropology, however, you have said >> absolutely nothing about whether !:!C is adequate for the infinite case. > >I was addressing your claim that there was "no extension from the >finite case". In the finite case, two sets have the same number >of elements if and only if there exists a one to one correspondence >between them. This very simple idea has been extended to the >infinite case. OK. The idea of a 1:1 corresondence is indeed a simple idea. The idea of infinity is not. >>>> If we want to compare the two sets for size we would write, not the >>>> above, but: >>> >>>> 1 2 3 4 5 6 7 8 9 ............... >>>> 1 2 3 4 5 6 7 8 9................ >>>> ^ ^ ^ >>> >>>Why would we write that? The second line seems to have nothing >>>to do with the second set. Why include elements that are not >>>in the set? > >> It's just the awkwardness of plain text. The members of the sqaures >> set are meant to be circled, if you like, the missing intergers being >> supplied for comparison. > >But why should you bother listing the missing integers? If you >are comparing two sets, including elements that are not in one >of the sets in the comparision seems strange to me. If >I am comparing the sets {Bob, Dave, George} and {Eileen, Barb, Jill} >should I write out > > Adam, Bob, Chuck, Dave, Edgar, Fred, George, Hank, Igor, Jack > ^^^ ^^^^ ^^^^^^ > > Amy, Barb, Cindy, Denise, Eileen, Faith, Gail, Helen, Ivy, Jill > ^^^^ ^^^^^^ ^^^^ > >?? How do I know what the missing elements are? > >The one-to-one correspondence idea is nice because it works for any >two sets. The idea you are looking at only works if one set >is a subset of the other. Yes, to set up the paradox we need to compare two sets for which there is a 1:1C and one is a subset of the other. It isn't a question of what works. It's a question of how the paradox is to be resolved. Thanks, Six Letters
From: Six on 26 Nov 2006 15:22 On 24 Nov 2006 15:08:20 GMT, richard(a)cogsci.ed.ac.uk (Richard Tobin) wrote: >In article <f41em2h1s4dv1qm70ntnv2jped6qekfd7s(a)4ax.com>, <Six Letters> wrote: > >>> (a) sets in 1-1 correspondence are the same size >>>and >>> (b) proper subsets are smaller than their supersets >> >>Exactly. It's throwing out (a) that I am trying to explore. > >This will result in bizarre consequences. For example, there will be >more decimal strings representing integers than binary strings, even >though they represent the same integers. On second thoughts I am not sure that I understand that. I rather suspect there are bizarre consequences, but I'd be really happy if you'd elaborate a little bit. Six Letters
From: Six on 26 Nov 2006 15:24 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. 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. 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. If the result is some wonderful mathematics, and the alternative is a dead end. 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. 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. 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: Six on 26 Nov 2006 15:24
On Fri, 24 Nov 2006 19:42:06 -0500, Bob Kolker <nowhere(a)nowhere.com> wrote: >Eckard Blumschein 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. > >Cardinal numbers are comparable. For all I know you may be a brilliant mathematician, but you just don't get it, do you? Six Letters |