Galileo's Paradox
in [Math]
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From: Tony Orlow on 29 Nov 2006 21:52 Virgil wrote: > In article <456DCC3F.5020700(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 11/29/2006 3:56 PM, Tony Orlow wrote: >> >>> Cardinality is generalized from the simple count of finite sets to the >>> infinite case. In the finite case, the cardinality of a set is exactly a >>> natural number, a quantity. In the infinite case, cardinality becomes >>> something more ephemeral, >> Epheremal means shortlived. We have a saying: Lies live short. >> >> but it still has its roots in the count of a set. >> >> Let's rather say in Cantor's illusion of allegedly being able to count >> the uncountable. >> >> >>>> What about when there is more than one type of measure that can be >>>> applied to a set, or none at all? What happens then? >> Then perhaps a red light will indicate logical error. > > Does volume completely determine mass? Different measures measure > different things and need have any correlation. Given a density of the material, indeed, volume determines mass. Given a set density, value range determines count.
From: zuhair on 29 Nov 2006 22:41 Six wrote: > GALILEO'S PARADOX > > 1 2 3 4 5 ................. > 1 4 9 16 25 ............... > > There is a paradox because the 1:1 Correspondence suggests the sets > are equal in size, by extension from the finite case, and yet clearly the > second set is contained in the first set. That an infinite set can be put > into 1:1 C with a proper subset is not by itself paradoxical. That is only > the beginning, the facts of the case. The paradox is that the squares seem > to be both smaller than N and the same size as N. > > 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. Yea, a quite negative approach. But it is not without intuitive backround. Intuitivelly speaking the idea that an infinite set has no fixed size comes to ones mind. That idea that infinity makes all infinite sets equal in size is also beautiful, and I think it was the idea before Cantor showed that there can be infinite sets of different sizes, the alephes and the powers are different in size, though infinite. If you want to change the definition of infinity to a one like saying, infinity is that quality which cause all sets that possess it to be equal in size, instead of the current definition of an infinite set, that is a set injectable to some proper subset of it, then you are free to do that,provided you bring a new definition of set size, other than cardinality. But this definition that looks to be their in your mind, is a negative one, I mean it canceal the chance of having meaningful comparisons of sizes of sets when they are infinite. If you bring a more positive claim, for example a method by which you can detect that there can exist difference is size of infinite sets that are currently considered to have equal size, then this idea would be somewhat chanllenging, but as I said you should bring a different rule of size comparison than cardinality. People here desire infinity to be determined by sets and desire the size of an infinite set to be also solelly determined by sets, i.e. knowledge of the members in a set is enough for you to know that they are infinite and let you know their set size, once apon a time I suggested the idea of generational size, which seems to be a measure of the generational size of sets as they are generated from themselfs or from other sets, a quality that is determined by the generational function from one set to the other. However even generationally speaking there are some types of generational size comparison that is solelly determined by sets only without the need to know the generational function of it from the other set. Example a set and its power set, whatever generational function that generated P(x) from x, then this function is strictly serjective from P(x) to x. and accordingly P(x) has always a bigger generational size than x. Anyhow this idea of generationl size was not apealing to the majority people in this forum, and it is certainly in the opposite direction to what you are suggesting here. However, two ideas are strong when intuitivelly speaking of infinite sets size, the first is that there nothing called infinite set. i.e to state that they are contradictive since bijection to a proper subset somewhat seems unapealing intuitivelly. the other idea is that if infinite sets exists then they should be equal in size. Both of these ideas though negative, yet can be true. I will discuss the idea of generational size again in a separate thread. Zuhair > > > My line of thought is that the 1:1C is a sacred cow. That there is > no extension from the finite 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................ > ^ ^ ^ > > (The intention here is to highlight the squares in the second row of > integers.) > > Then we would notice that the relative size of the squares set > becomes ever smaller as n increases, that increasingly large numbers of > integers are missed out. In fact, if we wanted to find a plausible > candidate for a set eual in size to N, then we would choose not the > squares, but the non-squares. > > The contrived nature of the 1:1C becomes more obvious when we > compare with N, sets that appear to be larger than N. The clearest example > is Z. > > We have: > > 1 2 3 4 5 6 7 ......... > 1 -1 2 -2 3 -3 4 ......... > > which is mildly clever, but again if we wanted to compare the two sets for > size we would write: > > 0 1 2 3 4.... > ...-4 -3 -2 -1 0 1 2 3 4.... > > with a perfect 1:2 Correspondence. > > Here one would like to say, since not only is there 1:1C between Z > and a proper subset, but an identity (1 2 3 4 ......), that however you > define infinity there has got to be more in Z than in N. Not, of course, if > you make all countable sets equal in size by definition. But for me, that > doesn't relieve the paradox at all. On the contrary it builds it into the > foundation of the mathematics. > I would like to suggest that the existence of 1:1C between the two > sets is a CONSEQUENCE of the fact that they are both infinite. The > infinities are what gives one room to manoeuvre, to manufacture a 1:1C. It > has no bearing on their relative size. > > Can one make sense of Z = 2N, of Q = N^2, etc.? (Incidentally the > number of squares would be sq.rt. of N, since after n^2 integers there are > n squares.) Maybe it's complete rubbish, but my argument is that the > alternative is the ineffable infinity. If it does make sense, there is no > place for a diagonal argument, or a power set argument, since it would > already be conceded that 10^N > N, that 2^N > N, or in general that k^N > > N, just as Z > N and Q > N. > > There remains of course Cantor's proof that R cannot be put into a > 1:1C with N, which is very interesting. But what does it mean? > Maybe something like this: > > So-called denumerable sets can be represented on a > finite-dimensional lattice, so that a self-avoiding walk can be shown to > systematically cover the entire line, are, volume or hyper-volume. For R > understood as a set of decimals (to choose that -- perfectly good -- > representation), by contrast, every decimal place can be construed as an > axis. > > In any case what I don't understand is how this affects the simple > paradox with which we began. > However, it may very well be that my insufficiently tutored brain > has flown its coop again, in which case I would be very grateful for any > illumination. > > > Six Letters 24/11/06
From: Virgil on 30 Nov 2006 00:15 In article <456e4621(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <456d9f90(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Mike Kelly wrote: > > > >>> If you think you have some wonderful notion of integrated count and > >>> measure that applies to all sets then even if you're correct (you're > >>> NOT) then cardinality is still a valid definition and still as useful > >>> as it is now. > >> Yes, and the square wheel will always be as useful as it ever has. > > > > Any squareness is all in TO's wheels. > > > >>> What about when there is more than one type of measure that can be > >>> applied to a set, or none at all? What happens then? > >> Where count can be calculated from either of two measures, then one has > >> a choice in that matter. Hopefully, one gets the same result either way. > >> Do you have an example you'd like to explore? > > > > "Outer measure" of sets in R^n, defined as the LUB of the content of a > > covering by open intervals, for one. > > Where standard measure is the same, there still may be an infinitesimal > difference, such as between (0,1) and [0,1], if that's what you mean. The outer measure of those two sets is exactly the same.
From: Virgil on 30 Nov 2006 00:20 In article <456e46b7(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Any uncountable > set with element indexes expressed as digital numbers will require > infinitely long indexes for most elements, such as is the case for the > reals in any nonzero interval. There is nothing in being a set, including being a set of reals, that requires its members to be indexed at all.
From: Virgil on 30 Nov 2006 00:25
In article <456e475e(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Given a > set density, value range determines count. Compare the "set densities" of the set of naturals, the set of rationals, the set of algebraics, the set of transcendentals, the set of constructibles, and the set of reals. |