Galileo's Paradox
in [Math]
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From: Six on 24 Nov 2006 13:02 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. >> 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. Thanks, Six Letters
From: stephen on 24 Nov 2006 13:26 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. >>> 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. Stephen
From: Eckard Blumschein on 24 Nov 2006 15:54 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. 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 24 Nov 2006 16:03 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.
From: Eckard Blumschein on 24 Nov 2006 16:06
On 11/24/2006 2:03 PM, Bob Kolker wrote: > Six Letters wrote: > Same size (i.e. same cardinality) means there is a one to one > correspondence between the sets. That is the definition of "same size". Do you mean somebody here is not familiar with all of your wise utterances? |