Cantor Confusion
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From: Alan Morgan on 15 Oct 2006 17:07 In article <1160944233.554227.5650(a)m7g2000cwm.googlegroups.com>, <mueckenh(a)rz.fh-augsburg.de> wrote: > >Virgil schrieb: > >> In article <1160814355.022151.205320(a)m73g2000cwd.googlegroups.com>, >> mueckenh(a)rz.fh-augsburg.de wrote: >> >> > > > Apply your knowledge to the balls of the vase. >> > > >> > > Which knowledge tells me that at noon each and every ball has been >> > > removed from the vase. >> > >> > You are joking? >> >> As I have inductively gone through the entire list of balls introduced >> into the vase and found that each of them has been removed before noon, >> why should stating that trivial fact be considered a joke? > >But you cannot go inductively through the cardinal numbers of the sets >of balls in the vase? They are 9, 18, 27, ..., and, above all, we can >show inductively, that this function can never decrease. You think that's bad? I have an even simpler situation! Add one ball at 1 minute to noon, another ball at half a minute to noon, another at 1/4 minute to noon, and so on. The number of balls in the vase before noon is always finite, but somehow, miraculously, at noon the number of balls in the vase becomes infinite. When, oh when, does that transition from finite to infinite happen? I submit that this is just as wierd a result as the original problem. Alan -- Defendit numerus
From: Virgil on 15 Oct 2006 17:12 In article <1160943747.258881.316180(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > David Marcus schrieb: > > > > > > > Han de Bruijn wrote: > > > > > David Marcus wrote: > > > > > > Is your claim only that set theory is not useful or is contrary to > > > > > > common sense? Or, are you claiming something more, e.g., that set > > > > > > theory > > > > > > is mathematically inconsistent? > > > > > > > > > > I said that set theory is not *very* useful. I have developed > > > > > (limited) > > > > > set theoretic applications myself, so I don't say it is useless. > > > > > > > > > Yes, a great deal of set theory is contrary to common sense. > > > > > Especially > > > > > the infinitary part of it (: say cardinals, ordinals, aleph_0). > > > > > > > > > > I'm not interested in the question whether set theory is > > > > > mathematically > > > > > inconsistent. What bothers me is whether it is _physically_ > > > > > inconsistent > > > > > and I think - worse: I know - that it is. > > > > > > > > What does "physically inconsistent" mean? Wouldn't your comments be > > > > better posted to sci.physics? Most people in sci.math are (or at least > > > > think they are) discussing mathematics. > > > > > > Even worse, most of them truly believe their ideas on mathematics and > > > the functions (of mathematics as well as of their brains) were > > > independent of physics > > > > > > > You are confusing levels. Brains are physically based. Concepts > > produced by those brains (beauty, mathematics, ...) need > > not be. > > They are based on brains which are based on what? Do you think that any > influence is zero unless it is direct? Do you think influence is everything? That is more like politics than physics.
From: Virgil on 15 Oct 2006 17:28 In article <1160944143.122919.243860(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > > > COLLECTED ANSWERS > > > > > > > Once you have chosen your set of 100 elements it will have > > > > a largest size. Questions as to the method of choice have > > > > no bearing on this. > > > > > > > Correct. Once you have chosen how to use all the bits of the universe, > > > then you will have a largest number. > > > > And crucial to this discussion, once you have chosen > > that you are going to use > > all the bits in the universe you know that you will have > > a largest number. > > No. I will not have a largest number unless I have chosen *how* to use > all the bits. In unary representation the largest number will clearly > be less than 10^100. But I am not indebted to focus on any fixed > representation. Therefore, there is no largest number. Then there is no set of numbers. > > As well as the largest integer that will be > > described there is also the largest integer that can be > > described. (you need to both give your representation > > *and describe your representation*. How about your largest possible integer plus one? > E.g. The string > > "1" in the base 1 billion, represents the integer one > > billion. However, the string "1" does not represent this > > integer, you need to add "in the base 1 billion". > > That is correct. It depends on the ingenuity how to define a large base > with few bits. There are no limits to ingenuity, therefore there is no > largest number. Therefore there is always a natural number larger that any one named. Which is our definition of infinitely many naturals actually existing. > > > We could > > do the usual formailization in terms of Turing > > machines, but this would do little except free us from > > such paradoxes as "let K be the largest integer that can be > > described plus 1".(note the usual interpretation is that K has > > not been described, you, however, claim that K cannot exist.)) > > > > And while the largest integer that will be described is not > > yet known (even in theory) > > I cannot see that. > > > the largest integer that > > can be described is known (at least in theory). So the > > largest integer that will be described cannot be chosen > > in an arbitrary manner. It must be less than or equal to > > the largest integer that can be described. > > > > As soon as you say "the set of integers consists of > > only those integers that will be described during the lifetime > > of the universe" you lose both actual and potential infinity. > > If you are correct, you are correct. But I doubt that you can > theoretically describe the limits of ingenuity. But "Mueckenh" can? > > You have produced a finite set without an upper bound. Remember the 100 > bits. That is not a set unless it is either finite with an upper bound or not finite. > > > > [You can of course, claim that N does not > > have an upper bound and N does not exist as a complete > > set. > > That is true. But claiming it doesn't make it true. > > > However, you wish to do more. You want to show > > that claiming "N does not have an upper bound and > > N exists as a complete set" leads to a contradiction.] > > > That is true too. But that proves the your "set" of all numbers expressible with 100 bits is also a contradic tion. > For all n e N we have {2,4,6,...,2n} contains larger natural numbers > than |{2,4,6,...,2n}| = n. > There is no larger natural number than aleph_0 = |{2,4,6,...}|. That would only be a contradiction if one claimed that aleph_0 were a natural number. > Contradiction, because there are only natural numbers in {2,4,6,...}. And no one , except possibly "Mueckenh", claims that aleph_0 is in {2,4,6,...} or even in {0,1,2,3,...} > > In principle it is the same as the vase, but the latter is more > striking. and only hard-core set-theorists like Virgil maintain to see > no problem. I see the problem that "Mueckenh" et al, are immune to the persuasions of logic and are addicted to imposing unwarranted assumptions. > > > > > > > Your putative proof of this "fact" depends on a step > > > in which quantifiers are switched without justification. > > > > > > Give a proof of that by a counter example using any concrete natural > > > numbers. > > > > > > > Let 0.111ppp be a number that has a 1 in every digit place > > corresponding > > to an element of the set N. Then every digit place is a finite > > natural number > > and therefore for every digit place n, there is an m(n) such that m(n) > > covers 0.111ppp up to digit place n. However, we cannot reverse > > the quantifiers. There does not exist a single M which covers > > 0.111ppp up to every finite natural number n. M would have to be > > an upper bound for the set N and we have assumed that such an > > upper bound does not exist. > > That shows that this assumption is wrong. That only sows that "Mueckenh"'s assumptions are wrong. > > > > This is a (slightly) obfuscated version of "{1,2,3,...,n} is bounded > > for all n in N. > > Therefore N is bounded". > > This conclusion is nearly true, in principle, though it should be > stated more carefully in order to avoid misunderstanding: N is > potentially infinite, i.e., always finite but without an upper bound. Then it cannot be an ordered set, as an ordered cannot be both finite and unbounded.
From: Virgil on 15 Oct 2006 17:35 In article <1160944233.554227.5650(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1160814355.022151.205320(a)m73g2000cwd.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > Apply your knowledge to the balls of the vase. > > > > > > > > Which knowledge tells me that at noon each and every ball has been > > > > removed from the vase. > > > > > > You are joking? > > > > As I have inductively gone through the entire list of balls introduced > > into the vase and found that each of them has been removed before noon, > > why should stating that trivial fact be considered a joke? > > But you cannot go inductively through the cardinal numbers of the sets > of balls in the vase? They are 9, 18, 27, ..., and, above all, we can > show inductively, that this function can never decrease. But there is nothing in the statement of the problem, nor in any part of any standard logic which requires the number of balls at noon to be a limit of numbers of balls at times strictly before noon. The numbers of balls as a function of time already has infinitely many discontinuities in every neighborhood of noon, so a discontinuity at noon is built into the problem and cannot be avoided. > > Ins't the impossibility of increasing of the function f(x) = 1/9x the > most important condition for the proof that the function f(x) = 1/9x > has the limit 0? What relevance have properties of a function continuous for all x < 0 have in analyzing properties of a function with infinitely many discontinuities for x < 0?
From: Virgil on 15 Oct 2006 17:42
In article <1160944503.060802.263540(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > My sympathies to his poor students. > > I will tell them your ideas about the vase and then ask them about > their opinion. But don't forget: They are not yet spoiled by what you > call logic. That is a cross they must bear without help. > > > > > > - plainly cannot > > > > comprehend the difference that swapping quantifiers makes. He cannot > > > > comprehend that there might be a difference between the significance of > > > > "every" in "Every girl in the village has a lover" and "John makes love > > > > to every girl in the village". > > > > > > Is the Imaginator too simple minded to understand, or is it just an > > > insult? The quantifier interchange is impossible in general, but it is > > > possile for special *linear* sets in case of *finite* elements. > > > > For example? > > > > Does "Mueckenh" claim that, say, > > "For every natural n there is a natural m such that m > n" > > and > > "There is a natural m, such that for every natural n, m > n" > > are logically equivalent? > > > > All the elements are finite and linearly ordered. > > The second statement is obviously wrong, because there cannot be a > natural larger than any natural. So is "Mueckenh" conceding that the two statements are not equivalent? > The quantifier exchange however is possible for sets of finite numbers > n the following form: > "For every natural n there is a natural m such that m >= n" > and > "There is a natural m, such that for every natural n, m >= n" > This natural m is not fixed. It is the largest member of the set > actually considered. But given that one can "consider" m to be a natural, is n = m + 1 then not a natural? > This shows that it is impossible to consider what you think is "the > complete set N". Not if one concedes that the two statements are not equivalent. > > You cold see that by means of the vase if you had not been blinded > during your studies. Since my "blindness" consists only in refusing the see the necessity of impossibilities, I am content with it. |