An uncountable countable set
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From: Han de Bruijn on 6 Oct 2006 04:10 Randy Poe wrote: > OK, if there's anyone out there reading this who knows > what Han means by "non-disciplinary mathematics", > could you please explain since Han is unable to? > > If *I* were to characterize undisciplined approaches to > mathematics, I would include something like "introduction > of terms which the author is unable to define but > nevertheless says 'the meaning is obvious'" Precisely! An example is Mike Kelly's function A(n,t) for t >= 0: > Let A(n,t) be 1 if the ball n is in the vase at time t, 0 if it is not > in the vase at time t. > > Let B(n) be the time that the nth ball is added to the vase and C(n) be > the time that it is removed. > > B(n) = -1/(2^(floor((n-1)/10))) > C(n) = -1/(2^(n-1)) > > Note that B(n) and C(n) are strictly less than 0. > > Now A(n,t) = { 1 if B(n) <= t < C(n) > 0 otherwise } > > Note that A(n,0) = 0. Sure. Simply "define" something that's undefined. And create the self fulfilling prophecy that suits you best. Han de Bruijn
From: mueckenh on 6 Oct 2006 04:18 MoeBlee schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > MoeBlee schrieb: > > > > > A CONTRADICTION? From what AXIOIMS? Please show a derivation of a > > > sentence P and ~P from the axioms. Oh, that's right, by "contradiction" > > > you don't mean a contradiction in the sense of a sentence and its > > > negation; you mean something that doesn't sit with your personal > > > intuition. > > > > > > > + 1 Gedankenexperiment: Put 10 balls in A and remove two, one of which > > is put in B and the other one is put in C. > > > > P: At noon all balls are in B. > > ~P: At noon all balls are in C. > > I take it that you mean that "At noon all balls are in C" implies ~P. Yes, because they are not in B, if C is not within B (which would be a special case to be mentioned) and if a normal form of logic is applied. > > Anyway, none of what you mentioned are formulas of set theory But the results are necessarily implied by the actual existence of omega. > nor have > you stated any axiomatic theory here. Just as I said about the other > poster, you find a conflict with your intuitions (here, regarding a > thought experiment), but no actual contradiction in an axiomatized > theory. I am sure, the results "all balls in B" and "all balls not in B" are not to be interpreted as an actual contradiction of set theory. It is just counter intuitive. Regards, WM
From: mueckenh on 6 Oct 2006 04:24 Virgil schrieb: > In article <1160044514.105544.245260(a)c28g2000cwb.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > Dik T. Winter schrieb: > > > > > > > > > > (There are exactly twice so > > > > > > much > > > > > > natural numbers than even natural numbers.) > > > > > > > > > > By what definitions? You never state definitions. > > > > > > > > By the only meaningful and consistent definition: A n eps |N : > > > > |{1,2,3,...,2n}| = 2*|{2,4,6,...,2n}|. > > > > Do you challenge its truth? > > > > > > No, I never did. But you draw conclusions about it about the set N. > > > Indeed, > > > for each finite n, it is true. > > > > And N is nothing but the collection of all finite n. > > That does not require that what is true for every member of a set be > true for the set itself. > > {2,4,6} is an odd sized set, despite all its members being of even size. And Mars looks red although all Marsians are green. Such analogies do not prove anything. In particular a set of finite natural numbers cannot be infinite, because the sum of differences of 1 between these numbers also makes up a finite natural number, as long as only finite numbers are present in the set. But this sum is nothing than the number of numbers (less 1). Regards, WM Regards, WM
From: mueckenh on 6 Oct 2006 04:27 Virgil schrieb: > In article <1160066751.825020.117740(a)h48g2000cwc.googlegroups.com>, > mueckenh(a)rz.fh-augsburg.de wrote: > > > MoeBlee schrieb: > > > > > A CONTRADICTION? From what AXIOIMS? Please show a derivation of a > > > sentence P and ~P from the axioms. Oh, that's right, by "contradiction" > > > you don't mean a contradiction in the sense of a sentence and its > > > negation; you mean something that doesn't sit with your personal > > > intuition. > > > > > > > + 1 Gedankenexperiment: Put 10 balls in A and remove two, one of which > > is put in B and the other one is put in C. > > > > P: At noon all balls are in B. > > ~P: At noon all balls are in C. > > > > > Q: some are in each. Then let the volume of C be a subset of the volume of A. Now A being no longer empty at noon? Regards, WM
From: mueckenh on 6 Oct 2006 04:36
MoeBlee schrieb: > > > > > > No? Who decides that? You see, I knew already that, according to > > > > > > Cantor, subtraction is possible. If I express this as addition of > > > > > > negative, what do you think did I meant? > > > > > > > > > > I did not know because there is no definition presented nor available. > > > > > Addition is defined between ordinal numbers. Ordinal numbers are (by > > > > > their very definition) larger than or equal to 0. But you want to > > > > > define addition between ordinal numbers and non-ordinal numbers. > > > > > > > > Wrong again. k and omega are ordinal numbers. > > > > > > But '-k' is *not* an ordinal number. > > > > "k" is an ordinal number and "-" is the advice to subtract it. > > > > > > In the meantime I have read it. You may note that in his notation > > > the solution for x + a = b is b - a (a < b). And the solution for > > > a + x = b (if that exists, and if there is more than one solution, > > > the smallest one) as b_(-a). > > > -- > > > > So you have learned that subtraction is possible, which was the main > > point that you originally doubted. But why should I stick to an > > old-fashioned and misleading notation? > > I'm coming into this part of the conversation late, so I hope I have > the context correct. > > We should be clear. Of course we can make various definitions, but you > can see that certain of them are conditional definitions. If the > conditions fail, then you cannot apply the definition as if the > condition holds. In this case, we don't have a definition of > subtraction that allows for substracting a finite ordinal k from w > (read 'w' as 'omega'). For a finite ordinal k, we don't have w-k as > some kind of ordinal such that there is an x such that k+x=w or even as > any kind of object (except by some method of assigning a default value > for otherwise non-referring terms, which is a whole other subject). Of course. The question was: Did Cantor use ordinal subtraction. And can one define it under certain circumstances. k + omega = {-k, -k+1, ..., 0, 1 , 2, 3, ...} is different from -k + omega = {k, k+1, k+2, ...} I used it as an abbreviation to explain that the set omega (or the set of digit indexes of a certain number: 0.111...) is not unuiquely defined. Regards, WM |