Galileo's Paradox
in [Math]
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From: Tonico on 28 Nov 2006 10:21 Eckard Blumschein ha escrito: > On 11/28/2006 3:46 AM, Virgil wrote: > > Meanwhile nothing of relevance in all of his replies. ************************************************************* Eckard's "Meanwhile nothing of relevance in all of his replies" = Eckard's too lazy/stupid to deal with mathematical stuff, so he'd rather dodge the question and shall continue to troll around. Tonio
From: Ross A. Finlayson on 28 Nov 2006 10:40 Six wrote: > ... > 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? > ... That's an interesting perspective. Consider a deck of cards, only visible through a card-sized slot under a token-operated flash strobe. When you look through the slot, insert a token, and see the cellophane-wrapped pack, you know their order and contents, or have some reasonable expectation thereof. After removing the cards and perhaps shuffling them, only a few, or say, one at a time can be seen. Are they thus "ordering-sensitive", in a sense? When it's a loose deck, without some a priori knowledge of the disposition of the elements, it takes 52 coins or thereabouts to know both the order and contents of the poker deck. The notion of a set, for example the real numbers, being in a sense ordering-sensitive is a reasonable one for quite a few considerations of how and why they are, as they are. Boy the trolls in this newsgroup sure have themselves been degrading lately, I find it rather despicable. Ross
From: Tony Orlow on 28 Nov 2006 15:30 Eckard Blumschein wrote: > 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. > Galileo's conclusions notwithstanding, there are certainly relationships among many countably and uncountably infinite sets which indicate unequal relative measures. I certainly consider 1 inch to be twice as infinitely many points in a row as 2 inches, nonstandard as that may sound. It's not a question in my mind whether there are more than any finite number of points, between finitely distant points in space, or that there is some relationship which reflects the density of the space in question. It also seems reasonable to use measures of set density, and more sophisticated methods of comparison, such as are employed in the converse situation, with infinite series. It seems natural to say that, if half the elements of A are in B, and all elements in B are in A, then B is half the size of A, as is the case where A=N and B=E. The proper subset as a smaller set should not be a notion violated by set theory, in my opinion. Tony
From: Virgil on 28 Nov 2006 16:31 In article <456C0431.4040809(a)et.uni-magdeburg.de>, 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: > > 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. There is no such thing as "genuine" for numbers in mathematics. > > > > 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 that EB has just refused to accept all of Analysis, including calculus, which is based on just the sort of sets that EB denies exist. Such a non-mathematician, or more properly anti-mathematician, can have nothing relevant to say about serious mathematics.
From: Virgil on 28 Nov 2006 16:40
In article <456C5361.40706(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > 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. For the lengths of line segments, longer, equally long, and shorter, are essential to Euclidean geometry. To deny that is to "throw out the baby with the bath water". And I doubt that Galileo did so. For the intersections of lines determining points, any two line segments can be shown to have a one to one correspondence of points. All one needs do is divorce the "length" from the "number of points", which is probably what Galileo did, as being different sorts of measures (like weight versus volume), and the problem disappears. |