Galileo's Paradox
in [Math]
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From: Lester Zick on 30 Nov 2006 12:25 On Thu, 30 Nov 2006 11:13:38 -0500, Bob Kolker <nowhere(a)nowhere.com> wrote: >Eckard Blumschein wrote: >> >> The argument Cantors transfinite numbers are somthing positive something >> progressive is old and has proven wrong. Not even aleph_2 has found an >> application. > >So what? The criterion for goodness in pure mathematics is consistency, >not usability. Truth would be an even better criterion. > After that aesthetic issues dominate. Are the systems >interesting. Do they have a kind of beauty? etc. etc. Yes, yes, truth is beauty and beauty truth but I still haven't gotten anyone to answer whether set "theory" represents all of mathematics and if not why the term "cardinality" cannot have other equally valid mathematical definitions than used in set "theory"? ~v~~
From: Lester Zick on 30 Nov 2006 12:39 On Thu, 30 Nov 2006 09:52:49 +0100, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >On 11/29/2006 6:37 PM, Bob Kolker wrote: >> 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. Tony, you know we've been over this previously. All "infinite" means is lack of definition for a particular predicate such as numerical size. And when you add numerical finites to numerical infinites the result is still infinite. This problem mainly arises I suspect because mathematikers insist on portraying infinites as larger than naturals and somehow coming beyond the range of naturals such as George Gamow's famous 1, 2, 3, . . . 00. Then mathematkers try to establish certain numerical properties for infinities by comparative numerical analysis and mapping with numerically defined finites. However one cannot do comparative numerical analysis and numerical analysis with numerically undefined infinites anymore than one can do arithmetic. Infinites are neither large nor small; they're just numerically undefined. >> Not necessarily so, if it is an infinite set. >> >> Bob Kolker > >This time I agree with Bob. > > ~v~~
From: stephen on 30 Nov 2006 12:44 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. > 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. Stephen
From: Lester Zick on 30 Nov 2006 12:58 On Thu, 30 Nov 2006 12:08:03 +0100, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >On 11/29/2006 8:13 PM, Bob Kolker wrote: >> Tony Orlow wrote: >>> >>> Uncountable simply means requiring infinite strings to index the >>> elements of the set. That doesn't mean the set is not linearly ordered, >>> or that there exist any such strings which do not have a successor. >> >> Uncountable means infinite but not of the same cardinality as the >> integers. For example the set of real numbers. It is an infinite set, >> but it cannot be put into one to one correspondence with the set of >> integers. > >Uncountable means: Counting is impossible. This property obviously >belongs to fictitious elements of continuum. There is simply too much of >them. So counting is not feasible. As long as one looks at a finite, >just potentially infinite heap of single integers, one has to do with >individuals. The set of all integers is something else. It is a fiction. >It is to be thought constituted of an uncountable amount of >non-elementary elements. Well this looks nonsensical. There is indeed a >selfcontradiction within the notion of an infinite set. >Non-elementary means not having a distinct numerical address. Element >means "exactly defined by an impossible task". You make the same mistake of assuming "infinite" means "larger than" when it only means numerically undefined. Infinites are neither large nor small; they're only undefined. Consequently there are no numerical relations or operations possible between them and finites. The reason counting is not possible is not because infinites are huge or because they form a continuum but because there is no numeric metric defined for them and counting as well as every other arithmetic relation and operation requires some kind of numeric definitional metric. ~v~~
From: MoeBlee on 30 Nov 2006 14:24
Lester Zick wrote: > Yes, yes, truth is beauty and beauty truth but I still haven't gotten > anyone to answer whether set "theory" represents all of mathematics No, it doesn't. One can have mathematical theories other than set theory. What is usually said is that all of the usual theorems of classical mathematics can be expressed and proven in certain set theories (for example Z set theory with dependent choice). But that is not to claim that set theories, especially any given set theory, is the only mathematical theory, or only possible foundational theory, or that it exhausts all of mathematics. > and if not why the term "cardinality" cannot have other equally valid > mathematical definitions than used in set "theory"? I wouldn't use the word 'valid', but, for my own view, 'cardinality' is an English language nickname I use to talk about a defined symbol of certain theories. I don't demand that the word 'cardinality' may not be used in a different sense for different theories. In fact, even among different set theories, 'cardinality' is used in somewhat different ways. But if a conversation about mathematics is to be coherent, then if we use the word 'cardinality', we should be clear as to which definition we are working with at any given point in the conversation. MoeBlee |