Galileo's Paradox
in [Math]
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From: Six on 28 Nov 2006 07:23 On Mon, 27 Nov 2006 06:46:24 -0500, Bob Kolker <nowhere(a)nowhere.com> wrote: >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 Making some allowances for native language, it is possible that Blumshein intended 'a bag of green sweets', and 'a bag of red sweets'. Then why don't you actually try to think about the problem. There's no need to embarrass yourself in front of everyone else. Just try it at home, and see if you can demonstrate to yourself that you have the capacity to think. You may not have done it for a long time, but I am sure it will come back. Six Letters
From: stephen on 28 Nov 2006 10:14 Six wrote: > On Mon, 27 Nov 2006 02:21:33 +0000 (UTC), stephen(a)nomail.com wrote: >>Six wrote: >>> 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. >> >>That depends on what 'idea of infinity' of you are talking about. >>The mathematical definition of 'infinite' is as simple as the >>idea of a 1:1 correspondence. > The mathematical definition of infinity may be simple, but is it > unproblematic? It seems to me that infinity is a sublte and difficult > concept. What concept of infinity? Note, I said 'infinite', not 'infinity'. You have been talking about Cantor and one-to-one correspondences, so you have been talking about set theory. The word 'infinity' is generally not used in set theory. It has no formal definition. 'infinite' is used to describe sets, and it has a very simple definition. > And that we are entitled to ask how well the simple mathematical > defintion captures what we mean by it, not necessarily in all its wilder > philosphical nuances, but what we mean by it mathematically, or if you > like, proto- mathematically. A set is infinite if there exists a bijection between the set and a proper subset of itself. That is what mathematicians mean when they say a set is infinite. There are other equivalent definitions. >> There is no point in dragging >>philosophical baggage into a mathematical discussion. > In my opinion the philsosopy is already there, and it impoverishes > mathematics to pretend otherwise. Do you have the same problem with prime numbers? Or even numbers? The words 'prime' and 'even' have meanings outside of mathematics. Do you feel obligated to drag those meanings into a discussion of prime or even numbers? <snip> >>>> >>>>?? 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 >> >>There is no need to resolve the paradox. There exists a >>one-to-correspondence between the natural numbers and the >>perfect squares. The perfect squares are also a proper >>subset of the natural numbers. This is not a contradiction. >> > I accept that. The contradiction comes about if the one notion > suggests equality of size and the other notion suggests inequality. Which > they do, so there is a prima facie paradox. The problem is that you are using a word 'size' that you have not defined. > I sense a cavalierness about common sense intuitions amongst > mathematicians (I don't mean you in particular, Stephen, it's just a > general comment.) Yes there is such a thing as conventional, accepted, > unexamined wisdom. Things are not always what they seem. But common sense > is, quite literally, where we all start. The articulation of it is > something else. "Common sense is the collection of prejudices acquired by age eighteen." -- Albert Einstein. Common sense is often wrong. Just think where physics would be if people relied on common sense. The problem here is not so much common sense, as the use of the word 'size' without first defining what you mean by 'size'. > Either a proper subset of a set can be the same size as the set > (for comparable sets or whatever technical qualification is needed), or it > must be smaller than the set, or it makes no sense to compare infinite sets > for size. (I suppose there could be some weirder alternative, such as the > size of a set might depend on how it is ordered, or something like that. > Haven't thought much about that.) Which is it, and why? Why use the word size at all? Two sets have the same cardinality if there exists a one-to-one correspondence between them. A set x is a proper subset of a set y if every element of x is an element of y, and there exist elements in y that are not in x. Those are two simple definitions that apply to any two sets. Of course people often use 'size' informally to mean 'cardinality'. In the finite case 'cardinality' corresponds exactly with the common sense notion of 'same number of elements'. Of course 'size' need not mean 'same number of elements' even in the finite case. Size is a very vague word, even when talking about physical objects. Does it mean height, weight, volume? If you use vague words, you are going to get vague results. > First option because Cantor says so might in a way be true, it > might be that that is where mathematicians are, but it I was going to join > them I would want to know why. > Thanks, Six Letters Noone is doing anything because 'Cantor says so'. Childish comments like that are a sure way to make this thread degenerate. Stephen
From: Eckard Blumschein on 28 Nov 2006 10:17 On 11/28/2006 3:46 AM, Virgil wrote: Meanwhile nothing of relevance in all of his replies. ..
From: Eckard Blumschein on 28 Nov 2006 10:18 On 11/28/2006 3:48 AM, Virgil wrote: > In article <456AF6F8.5020307(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 11/27/2006 3:21 AM, stephen(a)nomail.com wrote: >> >> > There is no need to resolve the paradox. There exists a >> > one-to-correspondence between the natural numbers and the >> > perfect squares. The perfect squares are also a proper >> > subset of the natural numbers. This is not a contradiction. >> >> What is better? Being simply correct as was Galilei or being more than >> wrong? (Ueberfalsch) > > Galileo was both right and wrong. He applied two standards to one > question and was confused when they gave different answers. Initially he was confused, yes. However, he found the correct answer: The relations smaller, equally large, and larger are invalid for infinite quantities.
From: stephen on 28 Nov 2006 10:17
Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/27/2006 8:47 PM, stephen(a)nomail.com wrote: >> Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >>> On 11/27/2006 3:21 AM, stephen(a)nomail.com wrote: >> >>>> There is no need to resolve the paradox. There exists a >>>> one-to-correspondence between the natural numbers and the >>>> perfect squares. The perfect squares are also a proper >>>> subset of the natural numbers. This is not a contradiction. >> >>> What is better? Being simply correct as was Galilei or being more than >>> wrong? (Ueberfalsch) >> >> Do you deny that there exists a one-to-one correspondence between >> the natural numbers and the perfect squares? > I just learned that mathematical existence means common properties. I do > not have any problem with the imagination of bijection between n and > n^2. I merely agree with Galilei that this bijection cannot serve as a > fundamental for ascribing a number to all n or all n^2. Both n and n^2 > just have two properties in common: They are countable because they are > genuine numbers, and they do not have an upper limit. So it is not 'more than wrong' to say that there exists a one-to-one correspondence between the natural numbers and the perfect squares. >> Or do you deny that the perfect squares are a proper subset of the >> naturals? > First of all, I do not consider the naturals and their squares identical > with the belonging sets. While the naturals are potentially infinite, > the term set is ambiguous in so far it claims to comprehend the naturals > looked at number by number, as well as _all_ naturals. So you do not think the perfect squares are a proper subset of the naturals? In which case you think there is a perfect square that is not a natural? > The latter ist something quite different and relates to actual infinity. > I consider any infinite set a fiction if one claims to include _all_ of > its elements. The natural numbers 1...n are potentially infinite, i.e. > they are no fiction, while the set of all natural numbers as a whole is > necessarily a fiction. That seems like a rather limited approach. Stephen |