Galileo's Paradox
in [Math]
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From: Six on 24 Nov 2006 09:46 On Fri, 24 Nov 2006 08:03:32 -0500, Bob Kolker <nowhere(a)nowhere.com> wrote: >Six Letters wrote: > >> >> I want to suggest there are only two sensible ways to resolve the >> paradox: >> >> 1) So- called denumerable sets may be of different size. > >Same size (i.e. same cardinality) means there is a one to one >correspondence between the sets. That is the definition of "same size". The question is: Is it a good definition? > >> >> 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. > >False. Where a correspondence can be established it makes perfectly good >sense. This is one horn of a dilemma I am presenting. >> My line of thought is that the 1:1C is a sacred cow. That there is >> no extension from the finite case. > >Dead wrong. You have been wrong for over a hundred years. Being dead wrong would not surprise me in the slightest. However, I am not one iota closer to understanding how the paradox is to be resolved. Thanks anyway, Six Letters.
From: Six on 24 Nov 2006 09:43 On 24 Nov 2006 12:20:11 GMT, richard(a)cogsci.ed.ac.uk (Richard Tobin) wrote: >In article <1164369693.301566.8460(a)l12g2000cwl.googlegroups.com>, >Peter van Liesdonk <peter(a)liesdonk.nl> wrote: > >>> 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. > >>This is exactly the case. All infinite sets have the same size: >>infinite. Actually, talking about size is a bit vague in this case. > >We don't usually go as far as that. Not all infinite sets are the >same size; there are different infinite sizes. But some infinite sets >that you might like to have different sizes don't. > >This is because - as I think the original poster realises - there just >isn't a consistent extension of size to infinite sets that satisfies >all our intuitions about size. If you want to have both > > (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. >you just can't do it, because (as the squares example demonstrates) >you can put infinite sets into 1-1 correspondence with certain of >their proper subsets. > >On the other hand, we don't have to throw everything out. It's not >always the case that an infinite set can be put into 1-1 >correspondence with a particular proper subset, as Cantor showed. It >turns out that we can have a number of different infinite sizes >without running into contradiction. > >>* 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. > >-- Richard Thanks, Six Letters
From: Richard Tobin on 24 Nov 2006 10:08 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. I doubt you can get very far with it. -- Richard -- "Consideration shall be given to the need for as many as 32 characters in some alphabets" - X3.4, 1963.
From: stephen on 24 Nov 2006 11:04 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. > 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? Stephen
From: Six on 24 Nov 2006 12:54
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. I haven't thought nearly enough about this, but a quick thought. Would 10^N have to be greater than 2^N in this sub-Cantorian arithmetic? Thanks, Six Letters |