Cantor Confusion
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From: David Marcus on 15 Oct 2006 18:51 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. > > > > > > - 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. > 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. Please let people know when you are not using standard terminology and when you do this, please define your terms. What does it mean to say a natural number "is not fixed"? > This shows that it is impossible to consider what you think is "the > complete set N". > > You cold see that by means of the vase if you had not been blinded > during your studies. -- David Marcus
From: David Marcus on 15 Oct 2006 19:02 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > 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. And it is easy to see: If we define Lim [n-->oo] > {1,2,3,...,n} = N, then we can see it easily: > > 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,...}|. > Contradiction, because there are only natural numbers in {2,4,6,...}. You appear to have written the following: Let N be the set of natural numbers. For all n in N, 2n > |{2,4,6,...,2n}| = n, n < |{2,4,6,...}| = alpheh_0, {2,4,6,...,2n} is a subset of N. I follow this. But, you have the word "contradiction" in your last sentence. Are you saying there is a contradiction in standard Mathematics? If so, what is it? I don't see it. -- David Marcus
From: William Hughes on 15 Oct 2006 20:44 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 My statement was not "you will have a largest number" but "you know that you will have a largerst number". The fact that you know that you will choose your largest number sometime in the future, and that you have not yet chosen the way you will choose this number does not change the fact that you know you will 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. > > > If you get your jollies from saying that > > there will be a largest integer, rather than there is a largest > > integer, far be it from me to stop you (but how you can > > have all the integers exist now, but only have the largest > > integer exist later is a problem you will have to solve). > > We have not all the integers exist now. > Or, to put it the other way, > all the integers which exist now are not those infinitely many integers > you are speculating about (without any rational reason). > > > > > Infinty does only come into the > > > play as long as this choice has not yet been fixed. > > > > > > Actually no. 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*. 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. Piffle. Ingenuity cannot change the fact that there is only a finite time to describe the method of representation. > > > 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. > You cannot know what the largest integer mentioned in the New York times in 2007 will be (even in theory). > > 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. Ingenuity may or may not be limited, but there is a limited amount of time to communicate the results of ingenuity. > > > > > > > > > It is true that in some cases the quantifier exchange is > > > >possible. However, the fact that the quantifier exchange > > > >is impossible in general means that you cannot use > > > >quantifier exchange in a proof without explicit > > > >justification of the step. Despite you protestations, > > > >this is exactly what you do. > > > > > > Despite of your belief I have proved that exchange is possible in > > > linear sets with only finite elements. If you state a counter example, > > > I can reject it by disproving it. You cannot, but only claim that for > > > an "infinite set" your position was correct - of course without being > > > able to show any infinity other than by the three "...". > > > > Let N be the set of finite integers. Either N has an upper bound > > or it does not. If you claim that N has an upper bound we will > > have to go our merry ways. Otherwise let K be subset of N that > > does not have an upper bound. There, I have produced an > > infinite set without using the three "...". > > You have produced a finite set without an upper bound. Remember the 100 > bits. What you described is not a set. It is a large collection of sets. None of these sets is arbitrary (arbirtray describes a way of choosing not what is chosen) Each of these sets has an upper bound. > > > > [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. > > > 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. And it is easy to see: If we define Lim [n-->oo] > {1,2,3,...,n} = N, then we can see it easily: > > For all n e N we have {2,4,6,...,2n} contains larger natural numbers > than |{2,4,6,...,2n}| = n. So we have something that is true for finite sets. > There is no larger natural number than aleph_0 = |{2,4,6,...}|. > Contradiction, because there are only natural numbers in {2,4,6,...}. > Piffle Wiithout any justification whatsoever you state something about infinite sets. Something that is true about finite sets does not have to be true about infinite sets. > In principle it is the same as the vase, Yes. There too you claim that something that is true about finite sets must be true about infinite sets. > but the latter is more > striking. and only hard-core set-theorists like Virgil maintain to see > no problem. > > > > > > > Your putative proof
From: William Hughes on 15 Oct 2006 20:51 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? The brain is contstrained by physical laws. The concepts produced by the brain are not contrained by physical laws. - William hughes > > Regards, WM
From: Dik T. Winter on 15 Oct 2006 21:56
In article <1160857922.727908.74990(a)i42g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: .... > > > We know that a set of numbers consisting altogether of 100 bits cannot > > > contain more than 100 numbers. Therefore the set is finite. The largest > > > number of such a set cannot be determined, as far as I know. > > > > That set is indeterminate. Just use Ascii notation. The string > > "Graham's number" fits in 100 bits. > > > > > Could you determine it? Or would you prefer to define that such ideas > > > do not belong to mathematics? Then I would not be interested in that > > > definition. > > > > It is an indeterminate set. > > The set of bits is determined: exactly 100. What you can build from 100 > bits belongs to the power set of this set. It is probably large but > certainly not infinite. The set of bits is determined. The set of numbers you can build from it is indeterminate. But whatever way you build your set of numbers, the size is certainly <= 2^100, and whatever way you build your set of numbers, there is a largest one that can be determined. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |