Galileo's Paradox
in [Math]
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From: Virgil on 30 Nov 2006 16:09 In article <456EC9FA.5000603(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/29/2006 8:21 PM, Virgil wrote: > > > Does volume completely determine mass? Different measures measure > > different things and need have any correlation. > > Since measure theory is mathematics, it should be in position to > abstract from physics and have only one measure for let's say seven eleven. Nothing in measure theory requires that every quality of an object have the same standard of measure. For a 3D object, one can simultaneously and independently have measurements of surface area and volume. There are even such anomalies as "Gabriel's horn" of finite volume but infinite surface area: http://mathworld.wolfram.com/GabrielsHorn.html
From: Virgil on 30 Nov 2006 16:14 In article <456EF429.6080201(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/30/2006 6:25 AM, Virgil wrote: > > 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. > > Either discrete or continuous. Nothing in between. The natural ordering of the set of rationals is neither continuous not discrete, at least in the mathematical meanings of those words. It is dense, but not complete, and both density and completeness are required for the set to have a continuous ordering. So of the sets mentioned above, the reals and only the reals are continuous in amy mathematically acceptable sense.
From: Virgil on 30 Nov 2006 16:17 In article <456f334d$1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <456e4621(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > >> 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. > > Right, and yet, the second is missing two elements, and is therefore > infinitesimally smaller in measure. Except that in outer measure there are no infinitesimals, and the outer measure of the difference set, {0,1} is precisely and exactly zero.
From: Six on 30 Nov 2006 16:39 On 29 Nov 2006 19:41:11 -0800, "zuhair" <zaljohar(a)yahoo.com> wrote: > >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 > Lots of interesting stuff there which I need time to digest. I look forward to your post on generational size. Much appreciated, Six Letters
From: Six on 30 Nov 2006 16:36
On Thu, 30 Nov 2006 17:44:46 +0000 (UTC), stephen(a)nomail.com wrote: >Six wrote: >> On Wed, 29 Nov 2006 17:12:04 +0000 (UTC), stephen(a)nomail.com wrote: > >>>Six wrote: >>>> On Tue, 28 Nov 2006 15:14:22 +0000 (UTC), stephen(a)nomail.com wrote: >>> >>>>>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: >>>>>>> >>>>>>>> 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. >>> >>>> I'm talking about mathematical meaning. Specifically I'm talking >>>> about "How many?", more or less etc.. >>> >>>"How many" is not a technical term. Cardinality corresponds to our >>>notion of "how many" in the finite case, and that is likely what people >>>will think of when you ask "how many". I know that later on you complain >>>about the term "cardinality", but I will respond to that later. >>> >>> >>>>>> 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. >>> >>>> I know already. >>> >>>So what are you asking? > >> Is it a good definition? > >What do you mean by a 'good definition'? What makes something a good >definition, as opposed to a bad definition. > >Is 'a number is prime if it is only divisible by itself and 1' a good >definition. This is a serious question. You seem to find the definition >of 'infinite' somehow questionable. Is this unique to 'infinite', or >does it apply to mathematical definitions in general, such as 'even', >'prime', 'odd', etc.? > >The definition is "good" in the sense that there exist objects that >satisfy it. For example the set of natural numbers satisfies the >definition. It is a good definition because you can determine when >something meets the definition. It is also a good definition that >if you were to try to list the elements of an 'infinite' set one at >a time, you would never reach the end of them, and this corresponds >nicely with one of the common definitions of the word 'infinite'. > >So why do you think it might not be a good definition? > >>> That is the definition of 'infinite set'. >>>It means mathematically exactly what it says. >>> >>>>>>> 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? >>> >>>> See above >>> >>>I do not see an answer to the question above. > >> This is a prime example of not even reading what I wrote. Sorry, >> couldn't help it. > >I read what you wrote. I asked a question. You responded with 'See above', >which to me implies that the answer to the question was in the text above. There was a silly, harmless joke there. >> I responded to your imputation that I was smuggling in extraneous >> philosophical material well enough, I thought, that this rather facetious >> question of yours did not require an additional answer. > >It is not a facetious question. You seem to think there is something >wrong with the definition of 'infinite'. I am trying to determine if >your problem is soley with the word 'infinite', or with mathematical >definitions in general. If your objection is soley to the word 'infinite', >then I think you are making the mistake of worrying about philosophical >implications of the word that are irrelevant to mathematics. > ><snip> > >>>> Look at what you've written. It consists of repeating things I >>>> already know (definitions etc.) coupled with the suggestion that I'm mixing >>>> up different notions of size. Saying that people are confusing two >>>> different notions of X is a classic manoeuvre of 20th century philosophy in >>>> the moribund analytic movement, and in every case, I'd venture to say, it >>>> sells the argument short. As if anybody that disagreed with your point of >>>> view was a complete idiot. >>> >>>You seem to be taking this all far too personally. > >> You miss my point, I think. I was not suggesting that you were >> calling me an idiot. I was trying to typify your style of argument. > >I am just trying to get you to answer some questions. You have >still not provided a definition of 'size' or 'how many'. > >>>You have not provided >>>a definition of 'size'. You are using a vaguely defined word, which >>>is always going to get you into trouble in mathematics. > >> My God, it's a wonder mathematics ever got started! > >>> >>>> There is an intuition that there are less squares (even numbers, >>>> primes, whatever) than naturals. We are talking here precisely of >>>> intuitions about infinite sets. It is not good enough to say: You're >>>> getting mixed up with finite sets, or: You can't rely on common sense >>>> intuitions in maths. >>> >>>> So if there are less squares than naturals, then since they have >>>> the same cardinality, how can cardinality have anything to do with size >>>> (how many)? Why not just say there's a bijection and forget about >>>> cardinality. >>> >>>Why not just say 'having no factors other than itself and one' instead of >>>'prime'? Whe not just say 'divisible by 2' instead of even? Cardinality >>>has a very precise definition. Yes, we could replace the word 'cardinality' >>>with its definition. It would not change anything. >>> >>>Again, your problem is insisting that cardinality match some vague notion of 'how many' >>>that you have not defined. Until you come up with a precise definition of 'how many', >>>any questions about 'how many' elements are in a set simply cannot be answered. >>> >>>> You suggested I conduct my argument without using the term >>>> 'infinity'. I am quite happy to do that. I suggest you conduct the rest of >>>> your argument without using the term 'cardinality'. >>> >>>Why? Cardinality has a definition in set theory. 'infinity' does not have >>>a definition. Do you really think that the two words are on an equal footing? >>> >>>Stephen > >> If I'm questioning the fitness of a definition, it hardly makes >> sense to keep bashing me over the head with it. > >What definition are you questioning? 'infinite'? 'cardinality'? >What about those definitions are you questioning? > >> There seems both to be as many squares as naturals (because of >> correspondence) and less squares than naturals (because of containment). >> I don't see how anything could be clearer than that. I was tempted >> to prefix this with 'in exactly the same sense of "how many" '. But there >> aren't multiple meanings of 'how many', not at least until mathematicians >> get to work on it. > >Can you provide me with that single meaning of 'how many'? Until you >actually define what you mean by the phrase 'how many', then it is >impossible to answer any question about 'how many'. > >> It's not the layman that has the problem here, it's the >> mathematician. It is quite in order for me to question the mathematical >> response to this paradox. It is quite in order for you to defend it. Please >> begin. > >You seem to be arguing against a position I have not taken. >I cannot defend anything involving 'how many' until you define what 'how >many' means, especially with regard to infinite sets. You have repeatedly >refused to do this. My point is that the paradox is a result of >thinking that 'size' and 'how many' have a common sense definition that >applies to infinite sets. As far as I know, they do not. If you think >otherwise, say what the definition is. > > >> I suspect though that there may be no proof of the matter either >> way, as has been hinted at in other parts of this thread. It may come down >> to this: that someone who wants to take a different, but still very >> reasonable (maybe more reasonable) appoach to this paradox, would need to >> demonstrate that some interesting and viable mathematics can result from >> it. This would certainly involve having precise defintions and so forth; it >> is just that they would be more or less different ones. I certainly do not >> have the wherewithall to even begin such a task. But it's interesting to >> speculate. And it's good to keep an open mind. > >You are more than welcome to come up with a definition of 'how many' >that you like. If other people like it, they may use it. It is not >going to change the fact that the naturals and the squares have the >same cardinality, nor the fact that the squares are a proper subset >of the naturals. > That is exactly what it is going to change. Your insistence that I define my terms in advance is back to front. There is a primitive intuition that naturals exceed squares and that they are equinumerous. There is no simple confusion here. Though undoubtedly there is something to be learned. It seems to me that your understanding of the unclarity about how many is a phantom product, a reflection of the set theory you accept so uncritically. You want to pretend the ambiguity about how many pre-exists, whereas in fact it is a creation of the mathematics. Conventional set theory has no unique. proprietary rights over this paradox. and it seemed to me that, whatever else it does and no doubt does very well, it deals with this paradox rather poorly. Thanks, Six Letters |