Galileo's Paradox
in [Math]
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From: Virgil on 29 Nov 2006 18:13 In article <4t65lnF1193vaU2(a)mid.individual.net>, Bob Kolker <nowhere(a)nowhere.com> wrote: > Cardinality of set S is denoted | S | . Cardinality of its power set is > denoted 2 | S | . Shouldn't that be 2 ^ | S | ?
From: Six on 29 Nov 2006 18:11 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: >>>>>>>> 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? Is it 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 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. ><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 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. >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. 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. 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. 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. Thanks, Six Letters
From: Bob Kolker on 29 Nov 2006 18:30 Virgil wrote: > In article <4t65lnF1193vaU2(a)mid.individual.net>, > Bob Kolker <nowhere(a)nowhere.com> wrote: > > > >>Cardinality of set S is denoted | S | . Cardinality of its power set is >>denoted 2 | S | . > > > Shouldn't that be 2 ^ | S | ? Small typo. I copied the text from a Wiki article. Bob Kolker
From: MoeBlee on 29 Nov 2006 18:38 Six Letters wrote: > >>>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. > Is it a good definition? x is infinite <-> ~ x is finite is, in my opinion, a good definition (depending on what you mean by a 'good definition'). "Es(s is a proper subset of x & x and s are 1-1)" is equivalent to "~ x is finite" with the axiom of choice added to Z set theory. > There seems both to be as many squares as naturals (because of > correspondence) and less squares than naturals (because of containment). Sure, if you use 'as many' in two different ways. > 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. It's better that there are NOT multiple definitions of the same terminology. > It's not the layman that has the problem here, it's the > mathematician. You mean the problem of non-mathematicians not understanding certain mathematical defintions? Yes, I suppose this is a public relations problem. > 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. It's paradox only in a broad sense of the word 'paradox'. It is not a contradiction in any usually used mathematical theory. MoeBlee
From: Lester Zick on 29 Nov 2006 19:51
On Wed, 29 Nov 2006 14:11:57 -0500, Bob Kolker <nowhere(a)nowhere.com> wrote: >Lester Zick wrote: > > >> Just out of curiosity, Bob, why is cardinality in set theory not a >> measure? I mean if you ask "how much gas" and get the answer "two >> gallons" you've certainly measured the gas. Or if you ask "how much >> space" and get the answer "two inches" you've certainly measured the >> space. It seems to me that you can obviously superimpose cardinality >> on questions like "how much" without having to count or match things. > >Consider two measure sets, disjoint. The measure of the union is the sum >of the measures of each. So "two inches" is not the "measure" of the space in question? >Now consider two sets of the same cardinality, disjoint. The cardinality >of the union equals the cardinality of either. But not the cardinality of both? >In short cardinality does not add like measure. Well you say so, Bob, but I can't decipher the reason why. If you add the cardinality of "two inches" to the cardinality of "two inches" don't you get the cardinality of "four inches"? ~v~~ |