Galileo's Paradox
in [Math]
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From: Virgil on 27 Nov 2006 21:52 In article <456AF7C0.9080405(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/27/2006 3:30 AM, Virgil wrote: > > > > There is, in fact, a definition of infiniteness of sets (by Dedekind) > > which REQUIRES the bijection of the set with a proper subset of itself. > > Let's recall that neither Dedekind nor someone else was able to prove > Dedekind's basis assumption. Definitions do not need "basis assumptions", the are merely abbreviations of statements that without them take longer to express. > Maybe, his claim was not just ad odds with Euclid but also questionable. Maybe his definition, like "shibboleth", separates mathematicians from anti-mathematicians like EB.
From: Eckard Blumschein on 28 Nov 2006 04:41 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. > 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. 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. Dedekind's definition of an infinite system (he did not use the term set) belongs to the potential infitiniy. Fraenkel (2nd ed. 1923, p. 17) wrote: "... dass eine Menge gewissermassen gleichviel Elemente enthalten kann wie eine echte Teilmenge von ihr steht in einem gewissen Kontrast zu dem bekannten Satz: Das Ganze ist stets gr��er als ein Teil." Notice: Even Fraenkel wrote "gewissermassen" (=quasi) "gleichviel" (= equally much). Euclid is of course still correct if we abandon the unfounded quantification of the property countably infinite. Removal of the 1 ... n^2 would not alter the property of the numbers 1... n being countably infinite. {2,3,5,6,7,8,10,11,12,13,14,15,17,...,n} is countably, i.e. potentially, infinite as well.
From: Six on 28 Nov 2006 06:33 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. 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. > 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. >>>>>> 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? >>> >>>> It's just the awkwardness of plain text. The members of the sqaures >>>> set are meant to be circled, if you like, the missing intergers being >>>> supplied for comparison. >>> >>>But why should you bother listing the missing integers? If you >>>are comparing two sets, including elements that are not in one >>>of the sets in the comparision seems strange to me. If >>>I am comparing the sets {Bob, Dave, George} and {Eileen, Barb, Jill} >>>should I write out >>> >>> Adam, Bob, Chuck, Dave, Edgar, Fred, George, Hank, Igor, Jack >>> ^^^ ^^^^ ^^^^^^ >if you insist it matches some vague unspecified notion of "same size". >It may seem unintuitive, If {Adam, Bob, Chuck, Dave, Edgar, Fred, George, Hank, Igor, Jack} are all the people that exist, then it seems to me perfect sense to say that {Adam, Dave} misses out Bob, Chuck etc. >>> ^^^^ ^^^^^^ ^^^^ >>> >>>?? 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. 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. 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? 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
From: Six on 28 Nov 2006 06:59 On Tue, 28 Nov 2006 10:41:05 +0100, 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. Excellent. That puts much more clearly what I was trying to suggest is a coherent position. And we did not have to appeal to Spinoza or any special discrimination awarded to us by His Nibs. snip Six Letters
From: Bob Kolker on 28 Nov 2006 07:27
Eckard Blumschein wrote: > > 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 What imagination? n <-> n*n is a bijection between integers and squares of integers. Each n goes to one and only one square. Different integers to to different squares. Each square comes from one and only one integer (that is 0, 1, 2, ...). This is a clear one to one correspondence between natural integers and squares of natural integers. Plain and simple. Bob Kolker |