Cantor Confusion
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From: Dik T. Winter on 13 Nov 2006 10:16 In article <1163426814.926776.54020(a)i42g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: .... > May be so with some sets. A set of natural numbers includes its > cardinal number. I did not know that. Care to prove that the set {1, 3, 5, 7} includes the cardinal 4? -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: mueckenh on 13 Nov 2006 10:20 Dik T. Winter schrieb: > > > Well, my only advise is, read it. > > > > If he says so, then it wil not be a good idea to waste my time with it. > > Do you really think the node 1/3 is finitely far from the root in the tree? Dik, are you joking? There is no node yielding any number like 1/3. Every node represents one bit of the binary representation of many real numbers. The first two levels of the tree contain the following initial segments: 0.00... 0.01... 0.10... 0.11... which are to be continued in the third and following levels. I use the tree because it makes notation easy (and reduces the digits necessary to write down for infinitely many numbers because the first digit for all is either 0 or 1) and it assures that here is no number left out. (There is no diagonal argument possible.) > > > > > 2) Do you agree that this implies: There are bit positions infinitely > > > > far from the decimal point (or how this point may be called for binary > > > > numbers). > > > > > > No. > > > > What then do you mean by infinitely far from the root? > > Your node 1/3 is infinitely far from the root because there is no finite > (natural) number that can state the distance. There is no node 1/3. There is the number 1/3 consisting of infinitely many nodes 0.010101... None of them has infinite distance from the root. > BTW, in binary the point is called the binary point. Thanks. Regards, WM
From: mueckenh on 13 Nov 2006 10:29 William Hughes schrieb: > > Cardinal numbers like one, two, ... and ordinal numbers like first, > > second, ... are closely connected. So every iinitial segment of natural > > numbers is counted by a natural number, namely |{1,2,3,...,n}| = n. > > Therefore no initial segment of natural numbers can be counted by an > > unnatural number like omega. > > Only if we say that {1,2,3,...} is not an initial segment of natural > numbers. It is an initial segment of natural numbers because it contains only such numbers in well-ordering. > > No set of the form {1,2,3..,n} can be counted by an unnatural > number like omega. The set {1,2,3,...} is not a set of the form > {1,2,3,...,n} If only natural numbers are within, that it is of that form. Although we may not know the last one, we can be sure that it is a natural. > > > This leads to the problem |{1,2,3,...}| = > > omega and |{1,2,3,...,omega}| = omega + 1. So we have |{1,2,3,...,a}| = > > a or a + 1, corresponding to the kind of a. > > Yes. This is true. It is not however a problem. It shows that by setting omega we loose a number in between. It should disturb a mathematician. > > [Sometimes this is used to argue that starting at 0 is more elegent. That is nonsense. The first natural number is that one which is called the "first" and not the zeroth. The set {1} has 1 element, the set {1,2,} has 2 elements and so on. Regards, WM
From: mueckenh on 13 Nov 2006 10:44 MoeBlee schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > MoeBlee schrieb: > > > > > > > So let's be definite here. Do you have a strong belief that your > > > argument can be expressed in the language of set theory and uses only > > > first order logic applied to the axioms of set theory? > > > > I need no belief. If all consistent mathematics can be expressed in > > ZFC, then my argument can be expressed in ZFC too. > > First you said that your argument is about set theory. Then said your > argument has "nothing to do with" set theory. Now you're saying that if > "all consistent mathematics can be expressed in ZFC" then your > argument can be put in ZFC. > > If you would at least say clearly just what your argument about trees > is supposed to be about - to what theory it is supposed to apply - then > that would help. This argument simply shows that there are not more real numbers than natural numbers, where real and natural numbers are those defined in modern mathematics. BTW: You are correct in assuming the the finiteness of the univers does not play a role in this argument. Regards, WM
From: mueckenh on 13 Nov 2006 10:45
Franziska Neugebauer schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Franziska Neugebauer schrieb: > > > >> mueckenh(a)rz.fh-augsburg.de wrote: > >> > >> > Franziska Neugebauer schrieb: > >> [...] > >> >> Are there really three vertices in WM's "triangle"? > >> > > >> > If finished infinities [...] > >> > >> Verbiage. > > > > Yes. But, sorry to see, it is the fundament of modern mathematics. > > "Finished infinities" is your wording. Precisely describing the fundament of modern mathematics. Regards, WM |