Cantor Confusion
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From: mueckenh on 18 Oct 2006 15:58 jpalecek(a)web.de schrieb: > > The set of constructible numbers is countable. Any diagonal number is a > > constructed and hence constructible number. > > No. Definition (from MathWorld): Constructible number: A number which > can be represented by a finite number of additions, subtractions, > multiplications, divisions, and finite square root extractions of > integers. > > How do you represent the diagonal number (which is sort of a limit of a > series) > via FINITE number of +,-,*,/,sqrt( ) ? Which digit should not be constructible by a finite number of operations? > > > Every list of reals can be shown incomplete in exactly the same way as > > every list of contructible reals can be shown incomplete. > > No. A constructible number is a number which can be constructed. Definition obtained from Fraenkel, Abraham A., Levy, Azriel: "Abstract Set Theory" (1976), p. 54: "Why, then, the restriction to the digits 1 and 2 in our proof? Just to kill the prejudice, found in some treatments of the proof, as if the method were purely existential, i.e. as if the proof, while showing that there exist decimals belonging to C but not to C0, did not allow to construct such decimals." Definition (by me): A number which can be constructed like pi, sqrt(2) or the diagonal of a list is that what I call constructible. If you dislike that name, you may call these numbers oomflyties. Anyhow that set is countable. And that set cannt be listed. Therefore the diagonal proof shows that a set of countable numbers is uncountable. Regards, WM
From: mueckenh on 18 Oct 2006 16:00 Sebastian Holzmann schrieb: > mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: > > Further the result o the gedankenexperiment must not depend on > > switching numbers. Removing balls 1, 11, 21, ... does not change the > > quantities in fact, but according to set theory it does. Therefore set > > theory has been contradicted. > > No. In mathematics, there is no such things as "balls" or "vases". You > have tried to model some vaguely "real" problem in the world of > mathematics. We had this objection already. Translate balls and vase into elements and set. The result will not change. It yields a contradiction in any case, i.e., independent of its result. Regards, WM
From: mueckenh on 18 Oct 2006 16:06 Dik T. Winter schrieb: > > I used but a simple definition of the improper limit oo of the function > > x which is given by the fact that the function 1/x has the proper > > limit 0 (fo x --> oo in both cases). > > Let me ask. You *did* state that "The function f(t) = 9t is continuous, > because the function 1/9t is continuous", yes? Do you not see that that > reasoning is wrong? That if 1/f(t) is continuous that does not mean that > f(t) itself is continuous? > > > > Moreover, when we let t go to infinity, 1/9t is *not* continuous at > > > infinity (whatever that may mean). We can only define the limit, > > > not the function value. > > > > We can only define limits in *all* cases concerning the infinite. > > Nothing else is possible. > > Yes, so you can not talk about continuity. We can talk about continuity in the finite domain. We can calculate the limit in case of f(t) --> 0 and we can from that define a limit oo for the case 1/f(t). From these procedures we can obtain that the function 1/f(t) will never yield the limit 0. That is the continuity requirement I mentioned. > Any constructable list of constructable > numbers gives a constructable diagonal number, > so any *constructable* < list of constructable is necessarily incomplete. Any list gives a constructrible diagonal number. Any diagonal number is constructed: Fraenkel, Abraham A., Levy, Azriel: "Abstract Set Theory" (1976), p. 54: "Why, then, the restriction to the digits 1 and 2 in our proof? Just to kill the prejudice, found in some treatments of the proof, as if the method were purely existential, i.e. as if the proof, while showing that there exist decimals belonging to C but not to C0, did not allow to construct such decimals." If you cannot understand that, then try this: The set of diagonal numbers is countable. Therefore the diagonal proof proves the uncountability of a countable set. The diagonal proof proves the uncountability for a countable set. Regards, WM
From: mueckenh on 18 Oct 2006 16:13 Randy Poe schrieb: > > The cardinal numbers of the sets of balls residing in the vase > > Crucial phrase missing: The cardinality f(t) of the set of balls in > the vase at time t STRICTLY LESS THAN ZERO... Either we can conclude from f(t) on f(0) Or we canot at all talk about infinity. > > > are also > > natural numbers. f(t) = 9, 18, 27, ... which grow without end. How can > > such a function take on the value zero? > > Because the set of balls at t=0 is not one of these sets. t=0 is not > a time strictly less than zero. I agree. But then the set at t = 0 is undefined. There is no calculating and reasoning with infinite sets. Bijections between infinite sets are undefined. Regards, WM
From: mueckenh on 18 Oct 2006 16:18
imaginatorium(a)despammed.com schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > imaginatorium(a)despammed.com schrieb: > > > > > > But the function of balls/numbers removed from the vase is a > > > > continuously (stepwise) increasing one, containing all natural numbers > > > > at noon? > > > > > > Uh, yes, unless I mysteriously misunderstand you... If takenout() is a > > > function from time to the power set of the integers (i.e. it maps to a > > > set of integers) then each natural number m is included in the set that > > > takenout() maps to from time = -1/m. So by time zero, all natural > > > numbers are included. > > > > > > Was there a question with that? > > > > And this result would change, if the numbers of the balls were > > exchanged, for instance multiplied by 10 after having been inserted? > > Can you clarify what "exchanged" means? Obviously, if you do exactly > the same thing with exactly the same balls, it makes no difference what > is written on them, That is an astonishing statement, but I welcome it and agree. Why do you speak of exactly the same balls, however? They are all indistinguishable like atoms. There are only the numbers. > but perhaps that is not what you mean. Take off balls 10, 20, 30, ... instead of 1, 2, 3, ..., or change the numbers by multiplying them with 10. Regards, WM |