Cantor Confusion
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From: David Marcus on 15 Oct 2006 14:59 Han.deBruijn(a)DTO.TUDelft.NL wrote: > David Marcus schreef: > > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > > David Marcus schreef: > > > > Han de Bruijn wrote: > > > > > 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. > > > > > > ONE world or NO world. > > > > Sorry. No idea what you mean. Do you have anything to say about > > mathematics or would you prefer we all just ignore you? > > It would save me a lot of time if some people here start to ignore me. Does that mean you have nothing to say about mathematics? -- David Marcus
From: mueckenh on 15 Oct 2006 16:22 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? Regards, WM
From: mueckenh on 15 Oct 2006 16:29 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. > 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. > 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. > > > > > > 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. > > [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. There is no larger natural number than aleph_0 = |{2,4,6,...}|. Contradiction, because there are only natural numbers in {2,4,6,...}. 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. > > > > 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. > > > > > > This requires proof. Try to produce one without > > any unjustified quantifier exchange. > > > > I did never employ unjustified quntifier exchange, but if you dislike > > that example, then consider the binary tree which i just posted another > > time. > > 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. Regards, WM
From: mueckenh on 15 Oct 2006 16:30 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. 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? Regards, WM
From: mueckenh on 15 Oct 2006 16:35
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. 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. Regards, WM |