Galileo's Paradox
in [Math]
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From: Tony Orlow on 29 Nov 2006 09:58 Bob Kolker wrote: > Tony Orlow wrote: > >> It is a direct consequence of the notion that a proper subset is >> always smaller, in some sense, than the base set. > > And in another sense a proper subset of an infinite set has the same > count as the set. > > Bob Kolker > It has the same cardinality perhaps, but where one set contains all the elements of another, plus more, it can rightfully be considered a larger set. Tony
From: Tony Orlow on 29 Nov 2006 10:01 Bob Kolker wrote: > Mike Kelly wrote: > >> 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? >> >> Forget integrating count and measure; it's not even possible to define >> a measure that applies to all sets. > > The Banach-Tarski "paradox" is the example, par excellence, that what > you say is the case. > > Just about any kind of additive measure leads to non-measurable sets. > > Bob Kolker Banach-Tarski is a proof by contradiction that set theory is out of whack. Point set topology loses measure, since points have no measure. The axiom of choice is abused. Additive measure of infinite sets is possible with infinitesimals.
From: stephen on 29 Nov 2006 12:12 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: >>>>>>> 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. > 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? 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. <snip> >> >>> 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. > True. I took it that people knew what I meant. And I think they do. No. I do not know what it means when applied to a set. Does it mean "cardinality"? If so then we would not be having this discussion. If it does not mean "cardinality", what does it mean? Can you give me a mathematical definition of "size"? <snip> >>Noone is doing anything because 'Cantor says so'. Childish comments >>like that are a sure way to make this thread degenerate. > Certainly I write things in the heat of the moment which I later > regret. But this wasn't meant as a cheap jibe. I've already conceded that > following Cantor might in some deep way be right, if it comes down to > following productive branches and forsaking dead ends. > 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 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. > 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
From: Bob Kolker on 29 Nov 2006 12:37 Tony Orlow wrote: > > It has the same cardinality perhaps, but where one set contains all the > elements of another, plus more, it can rightfully be considered a larger > set. Not necessarily so, if it is an infinite set. Bob Kolker
From: Eckard Blumschein on 29 Nov 2006 13:00
On 11/29/2006 3:58 PM, Tony Orlow wrote: > where one set contains all the > elements of another, plus more, it can rightfully be considered a larger > set. All of oo? |