Cantor Confusion
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From: Virgil on 14 Oct 2006 16:06 In article <1160813360.065002.214030(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > Dik T. Winter schrieb: > > > > > > > In article <1160647755.398538.36170(a)b28g2000cwb.googlegroups.com> > > > > mueckenh(a)rz.fh-augsburg.de writes: > > > > ... > > > > > It is not > > > > > contradictory to say that in a finite set of numbers there need not > > > > > be > > > > > a largest. > > > > > > > > It contradicts the definition of "finite set". But I know that you are > > > > not interested in definitions. > > > > > > 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. > > > > There is a big difference between saying we do not know what > > the value of the largest element of a set is and saying that > > a set does not have a largest element. > > The set discussed above does not have a largest element. Then it is not properly a set at all. At least in any standard set theory.
From: Han.deBruijn on 14 Oct 2006 16:14 stephen(a)nomail.com schreef: > Dik T. Winter <Dik.Winter(a)cwi.nl> wrote: > > In article <a1364$452f53c9$82a1e228$2828(a)news2.tudelft.nl> > Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > > ... > > > I don't know about Mueckenheim's opinion in these matters, but an old > > > Greek philosopher - Heraclitos - has said: "panta rei kai ouden menei" > > > (everything flows and nothing remains the same). And I agree with that. > > > Pray warn me when 2 has changed sufficiently to be the square of a rational. > > I would not like to miss that moment. > > The day the circle is squared cannot be to far behind. Come on, guys! You all know that, in the world of approximations, 2 _is_ the square of a rational and the circle _is_ squared. Han de Bruijn
From: Virgil on 14 Oct 2006 16:18 In article <1160813529.599591.166960(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > David Marcus schrieb: > > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > David Marcus schrieb: > > > > > > > I am sure you are able to translate brief notions like "to enter, > > > > > > > to > > > > > > > escape" etc. by yourself into terms of increasing or decreasing > > > > > > > values > > > > > > > of variables of sets, if this seems necessary to you. Here, > > > > > > > without > > > > > > > being in possession of suitable symbols, it would become a bit > > > > > > > tedious. > > > > > > > > > > > > Yes, I can translate it myself. However, that would only tell me > > > > > > how I > > > > > > interpret the problem. > > > > > > > > > > Hasn't it become clear by the discussion? > > > > > > > > > > I use two variables for sequences of sets. Further I use a function. > > > > > I > > > > > use the natural numbers t to denote the index number. The balls are > > > > > simply the natural numbers. I speak of "balls" in order to not > > > > > intermingle these numbers with the index-numbers. > > > > > > > > > > The set of balls having entered the vase may be denoted by X(t). > > > > > So we have the mathematical definition: > > > > > X(1) = {1,2,3,...10}, X(2) = {11,12,13,...,20}, ... with UX = N > > > > > There is a bijection between t and X(t). > > > > > > > > t is a number and X(t) is a set. If t = 1, then your sentence says, > > > > "There is a bijection between 1 and X(1)". But, X(1) = {1,2,3,...10}. > > > > So, I don't follow. What do you mean, please? > > > > > > That what is written. There is a bijection between the set of all > > > numbers t and the set of all sets X(t). 1 is mapped on X(1), 2 is > > > mapped on X(2), and so on. Is there anythng wrong? > > > > > > > > > The set of balls having left the vase is described by Y(t). So we > > > > > have > > > > > the mathematical definition: > > > > > Y(1) = 1, Y(2) = {1,2}, ... with UY = N > > > > > There is a bijection between t and Y(t). > > > > > > Here we have the same as above with Y instead of X. > > > > > > > > > > And the cardinal number of the set of balls remaining in the vase is > > > > > Z(t). So we have the mathematical definition: > > > > > Z(t) = 9t with Z(t) > 0 for every t > 0. > > > > > There is a bijection between t and Z(t). > > > > Sorry, but I'm not following. I asked you for a translation into > > Mathematics of the ball and vase problem. The problem in English ends > > with a question mark. I don't see the question mark in your translation > > above. Would you please state just the problem in both English and > > Mathematics? > > Sorry, I don't know how to state the question " Hasn't it become clear > by the discussion?" in what you think is mathematics. The only necessary constraint on insertions of balls into the vase and removals of balls from the vase is that each ball that is to be removed must be inserted before it can be removed, and, subject only to that constraint, the set of balls remaining in the vase at the end of all removals is independent of both the times of insertion and of the times of removal. To argue otherwise is to misrepresent the problem.
From: mueckenh on 14 Oct 2006 16:18 William Hughes schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > Tony Orlow schrieb: > > > > > > > > > > Why shouldn't it? If every digit position of 0.111... is a finite > > > > position then exactly this is implied. Your reluctance to accept it > > > > shows only that you do not understand how an infinite set can consist > > > > of finite numbers. In fact, nobody can understand it, because it is > > > > impossible. > > > > > > > > Regards, WM > > > > > > > > > > But Wolfgang, surely that consideration does not impact, say, the set of > > > reals in (0,1], which are all finite, yet whose number is infinite. It > > > is not a requirement that a set of all finite values be finite. That > > > conclusion follows from the combination of that fact with the fact there > > > is a constant positive unit difference between consecutive elements. > > > > Of course, Tony, you are right! > > But only a finite number of real numbers will every be described in > the lifetime of the universe. Surely by your reasoning there must be > a finite number of real numbers? Yes, but the reason you mentioned, correctly, is another one than Tony had in mind, correctly. Regards, WM
From: Virgil on 14 Oct 2006 16:20
In article <1160813834.753411.43510(a)k70g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > (To those who've had a course in mathematical logic: I'm aware that the > > preceding ignores the question of what first-order language the Peano > > axioms are stated in. You can assume I'm just using whatever language > > I'm using for all my Mathematics.) > > ((And to those who consider taking a second course in mathematical > logic: What are the foundations of logic? Where are they obtained from? > Is there a theory which yields the axioms of logic as theorems?)) > > Regards, WM Even for logic one has to start with an axiom system which is assumed rather than proved from even more basic assumptions. So that those who reject axiom systems entirely can never even et started. |