Cantor Confusion
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From: Virgil on 8 Oct 2006 17:18 In article <1160295577.704177.115480(a)i3g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > Ok. Let's call a day a "numbered day", if we are able > > to associate the day with a specific natural number. So day > > 5,341,134,322, is a numbered day, but the present day is not > > a numbered day. The question is now: "Can X write about > > each numbered day?" > > The question is easy to answer, but this X is a poor example, because > there are far better ones like Tristram shandy and the vase, yielding > sharper contradictions. > > 1) Every ball will have left the vase at noon. > 2) At noon there are more balls in the vase than at any time before. "Mueckenh"'s conclusion 2 does not follow from anything in ZF, ZFC or NBG without additional assumptions which those axiom systems do not justify.
From: Virgil on 8 Oct 2006 17:28 In article <1160296113.211935.299880(a)c28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > We know, not by > > > intuition, but by logic, that the vase at any time contains more balls > > > than have escaped. > > > > Absolutely false. > > Bold words, but unfortunately simply wrong. The contents of the vase > increases any time by 9 balls. "Mueckenh" is wrong on two counts. (1) The vase starts empty, and at that time does NOT contain more balls than have.escaped (2) At and after noon, every ball has escaped, so the vase is again empty, at least without assumptions that do not follow from the axioms of ZFC or MNG. Let A_n(t) be equal to 0 at all times, t, when the nth ball is out of the vase, 1 at all times, t, when the nth ball is in the vase, and undefined at all times, t, when the nth ball is in transition. Note that noon is not a time of transition for any ball, though it is a cluster point of such times. let B(t) = Sum_{n in N} A_n(t) represent the number of balls in the vase at any non-transition time t. B(t) is clearly defined and finite at every non-transition point, as being, essentially, a finite sum at every such non-transition point. Further, A_n(noon) = 0 for every n, so B(noon) = 0. Similarly when t > noon, every A_n(t) = 0, so B(t) = 0 > We have a contradiction n ZFC > 1) Every ball will have left the vase at noon. > 2) At noon there are more balls in the vase than at any time before. (2) does not follow from ZFC alone. "Mueckenh" needs to assume other things which need not be the case in ZFC. > > > To assert that at > > > noon (finished infinity, actualized transfinity) the vase is empty is > > > not counter-intuitive but it stupid. > > > > But nowhere near as stupid as claiming that balls that have been removed > > are still there. > > You claim that all have been removed although you do not know anything > about noon. > > > > 1) Before noon every ball comes out of the vase > > > 2) Before and at noon there are more balls in the vase than have come > > > out > > > then the axioms yield a contradiction. "Mueckenh" must be adding his own axioms, as it does not follow merely from ZFC or NBG. > > > > "Mueckenh" presumes some sort of "continuity" at noon, but the whole > > process is discontinuous at each time of change. > > If it is meaningful to talk about *all* elements of an infinite set, > then it is meaningful to c Let's do just that: Let A_n(t) be equal to 0 at all times, t, when the nth ball is out of the vase, 1 at all times, t, when the nth ball is in the vase, and undefined at all times, t, when the nth ball is in transition. Note that noon is not a time of transition for any ball, though it is a cluster point of such times. let B(t) = Sum_{n in N} A_n(t) represent the number of balls in the vase at any non-transition time t. B(t) is clearly defined and finite at every non-transition point, as being, essentially, a finite sum at every such non-transition point. Further, A_n(noon) = 0 for every n, so B(noon) = 0. Similarly when t > noon, every A_n(t) = 0, so B(t) = 0 onsider the result of all transactions. > > > > The function which counts balls in the vase as a function of time jumps > > discontinuously when any ball is inserted and also when any ball is > > removed. Since noon is a condensation point of such discontiuities, it > > is bootless to insist that it be a point of psuedocontinuity such as > > "Mueckenh" claims. > > Therefore it is also false to conclude from the bijection of natural > numbers and outcoming balls that no ball remains in the vase. You apply > just this pseudocontinuity. Actually, I insist on a discontinuity at noon, at least a left-discontinuity. > > > "Mueckenh" is certainly hopeless. > > With respect to become an adherent of your religion, yes. At least I do not try to sneak in extra assumptions over and above ZFC or NBG and claim that they are inherent.
From: Virgil on 8 Oct 2006 17:31 In article <1160296706.289632.205650(a)i3g2000cwc.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: > > > > > Hi, Dik, > > > > > > > > > > I would like to publish our result to the mathematicians of this group > > > > > in order to show what they really are believing if they believe in set > > > > > theory. > > > > > > > > > > There is an infinite sequence S of units, denoted by S = III... > > > > > > > > > > This sequence is covered up to any position n (included) by the finite > > > > > sequences > > > > > I > > > > > II > > > > > III > > > > > ... > > > > > > > > What do you mean by "cover"? > > > > > > A covers B if A has at least as many bars as B. A and B are unary > > > representations of numbers. > > > > > > Example: A = III covers I and II and III but not IIII. > > > > > > > But it is impossible to cover every position of S. > > > > > > > So: S is covered up to every position, but it is not possible to cover > > > > > every position. > > > > So, your conclusion is that no finite sequence of I's will cover S. > > Correct? > > > > Is this your entire theorem or is there more to the conclusion? > > My conclusion is: > Either > (S is covered up to every position <==> S is completely covered by at > least one element of the infinite set of finite unary numbers <==> S is > an unary natural) ==> Contradiction, because S can be shown to be not a > unary natural. > Or > S is not covered up to every position by unary naturals ==> The > positions of S are not defined ==> S does not exist. So that "Mueckenh" is asserting that ZFC and NBG are false, but has nothing but intuition on which to base his conclusion. > But as this example is very closely related to the vase just under > discussion here, we should switch to that one. Where "Mueckenh" is also assuming things about ZFC and NBG over and above what they include.
From: Virgil on 8 Oct 2006 17:32 In article <1160297070.054868.190530(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Tonico schrieb: > > > Han.deBruijn(a)DTO.TUDelft.NL wrote: > > > mueckenh(a)rz.fh-augsburg.de schreef: > > > > > > > Tonico schrieb: > > > > > > > > > It is nonsense, but it is the truth if infinity can be calculated. I > > > > said it merely to show that actual infinity is not only a huge nonsens > > > > but rather an infinite nonsense. > > > > > > And when you're debating with me (HdB). I'm on Mueckenheim's side > > > in these matters. > > > > ************************************************************* > > No wonder I confused between you two. You both use a lot of very weird, > > non-mathematical mumbo-jumbo, just like > > Tonico, you have shown here that you don't know mathematics history and > that you not even know present set theory concerning the most > fundamental definitions and technical terms. So shut up and go to > learn at least the most basic things. Then come back. > > Regards, WM This from someone who berated me for "impoliteness"?
From: Virgil on 8 Oct 2006 17:35
In article <1160302222.613036.300930(a)h48g2000cwc.googlegroups.com>, Han.deBruijn(a)DTO.TUDelft.NL wrote: > Dik T. Winter wrote: > > > I would say that all forms of mathematics are grounded on axioms (or dogmas > > as you prefer to say). But contrary to dogmas, axioms can be negated to > > get another form of mathematics. Dogmas are absolute truths, axioms are > > only absolute truths within some realm of discourse. > > If it is so simple, where then come these heated debates (about the > Balls in a Vase at noon) come from? And why then are some axiom systems > so much more dominant than others? > > > In the same way in most countries it is an axiom that you should drive on > > the right. But an Englishman would state, rightly, the right side is not > > the right side to ride. > > I wish it were so simple. > > Han de Bruijn If one chooses to work within ZFC or NBG, the vase is empty at noon. If one doesn't choose to work within them, one may get different results, but they tend to be just about as paradoxical. |